elementary algebra ab

Elementary algebra abFrom Wikipedia, the free encyclopedia

Contents

1 Additive identity 11.1 Elementary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.1 The additive identity is unique in a group . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.2 The additive identity annihilates ring elements . . . . . . . . . . . . . . . . . . . . . . . . 21.4.3 The additive and multiplicative identities are dierent in a non-trivial ring . . . . . . . . . . 2

1.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Additive inverse 42.1 Common examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Relation to subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Algebraic expression 73.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 In roots of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.2 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4 Algebraic vs. other mathematical expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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ii CONTENTS

3.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Algebraic fraction 104.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Irrational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Algebraic operation 125.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Arithmetic vs algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Properties of arithmetic and algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Associative property 146.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Generalized associative law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.5 Non-associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5.1 Nonassociativity of oating point calculation . . . . . . . . . . . . . . . . . . . . . . . . . 196.5.2 Notation for non-associative operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Brahmaguptas identity 227.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Application to Pells equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 BrahmaguptaFibonacci identity 248.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Relation to complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4 Interpretation via norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.5 Application to Pells equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

8.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Elementary algebra 279.1 Algebraic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9.1.1 Alternative notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9.2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2.2 Evaluating expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2.4 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.3 Solving algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3.1 Linear equations with one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.3.2 Linear equations with two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.3.3 Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.3.4 Exponential and logarithmic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3.5 Radical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.3.6 System of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3.7 Other types of systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Proofs involving the addition of natural numbers 4610.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2 Proof of associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.3 Proof of identity element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.4 Proof of commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 1

Additive identity

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which,when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 fromelementary mathematics, but additive identities occur in other mathematical structures where addition is dened, suchas in groups and rings.

1.1 Elementary examples The additive identity familiar from elementary mathematics is zero, denoted 0. For example,

5 + 0 = 5 = 0 + 5 In the natural numbers N and all of its supersets (the integers Z, the rational numbers Q, the real numbers R,or the complex numbers C), the additive identity is 0. Thus for any one of these numbers n,

n + 0 = n = 0 + n

1.2 Formal denitionLet N be a set which is closed under the operation of addition, denoted +. An additive identity for N is any elemente such that for any element n in N,

e + n = n = n + e

Example: The formula is n + 0 = n = 0 + n.

1.3 Further examples In a group the additive identity is the identity element of the group, is often denoted 0, and is unique (see belowfor proof).

A ring or eld is a group under the operation of addition and thus these also have a unique additive identity 0.This is dened to be dierent from the multiplicative identity 1 if the ring (or eld) has more than one element.If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below).

In the ring Mmn(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by nmatrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices overthe integers M2(Z) the additive identity is

0 =

0 00 0

1

2 CHAPTER 1. ADDITIVE IDENTITY

In the quaternions, 0 is the additive identity. In the ring of functions from R to R, the function mapping every number to 0 is the additive identity. In the additive group of vectors in Rn, the origin or zero vector is the additive identity.

1.4 Proofs

1.4.1 The additive identity is unique in a groupLet (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G,

0 + g = g = g + 0 and 0' + g = g = g + 0'

It follows from the above that

(0') = (0') + 0 = 0' + (0) = (0)

1.4.2 The additive identity annihilates ring elementsIn a system with a multiplication operation that distributes over addition, the additive identity is a multiplicativeabsorbing element, meaning that for any s in S, s0 = 0. This can be seen because:

s 0 = s (0 + 0) = s 0 + s 0) s 0 = s 0 s 0) s 0 = 0

1.4.3 The additive and multiplicative identities are dierent in a non-trivial ringLet R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, or 0 = 1. Let r beany element of R. Then

r = r 1 = r 0 = 0

proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is thereforeshown.

1.5 See also 0 (number) Additive inverse Identity element Multiplicative identity

1.6 References David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3d ed.): 2003, ISBN 0-471-43334-9.

1.7. EXTERNAL LINKS 3

1.7 External links uniqueness of additive identity in a ring at PlanetMath.org. Margherita Barile, Additive Identity, MathWorld.

Chapter 2

Additive inverse

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This numberis also known as the opposite (number),[1] sign change, and negation.[2] For a real number, it reverses its sign: theopposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additiveinverse of itself.The additive inverse of a is denoted by unary minus: a (see the discussion below). For example, the additive inverseof 7 is 7, because 7 + (7) = 0, and the additive inverse of 0.3 is 0.3, because 0.3 + 0.3 = 0 .The additive inverse is dened as its inverse element under the binary operation of addition (see the discussion below),which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, doubleadditive inverse has no eect: (x) = x.

2.1 Common examplesFor a number and, generally, in any ring, the additive inverse can be calculated using multiplication by 1; that is, n= 1 n . Examples of rings of numbers are integers, rational numbers, real numbers, and complex number.

2.1.1 Relation to subtractionAdditive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:

a b = a + (b).

Conversely, additive inverse can be thought of as subtraction from zero:

a = 0 a.

Hence, unary minus sign notation can be seen as a shorthand for subtraction with 0 symbol omitted, although in acorrect typography there should be no space after unary "".

2.1.2 Other propertiesIn addition to the identities listed above, negation has the following algebraic properties:

(a + b) = (a) + (b)a (b) = a + b(a) b = a (b) = (a b)(a) (b) = a b

notably, (a)2 = a2

4

2.2. FORMAL DEFINITION 5

0

+i

i

1 +1

These complex numbers, two of eight values of 81, are mutually opposite

2.2 Formal denitionThe notation + is usually reserved for commutative binary operations, i.e. such that x + y = y + x, for all x, y . If suchan operation admits an identity element o (such that x + o ( = o + x ) = x for all x), then this element is unique ( o =o + o = o ). For a given x , if there exists x such that x + x ( = x + x ) = o , then x is called an additive inverse of x.If + is associative (( x + y ) + z = x + ( y + z ) for all x, y, z), then an additive inverse is unique

x = x + o = x + (x + x) = (x + x ) + x = o + x = x

For example, since addition of real numbers is associative, each real number has a unique additive inverse.

2.3 Other examplesAll the following examples are in fact abelian groups:

complex numbers: (a + bi) = (a) + (b)i. On the complex plane, this operation rotates a complex number180 degrees around the origin (see the image above).

6 CHAPTER 2. ADDITIVE INVERSE

addition of real- and complex-valued functions: here, the additive inverse of a function f is the function fdened by (f )(x) = f (x) , for all x, such that f + (f ) = o , the zero function ( o(x) = 0 for all x ).

more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then theidentity element of this group):

sequences, matrices and nets are also special kinds of functions. In a vector space the additive inverse v is often called the opposite vector of v; it has the samemagnitude as theoriginal and opposite direction. Additive inversion corresponds to scalar multiplication by 1. For Euclideanspace, it is point reection in the origin. Vectors in exactly opposite directions (multiplied to negative numbers)are sometimes referred to as antiparallel.

vector space-valued functions (not necessarily linear), In modular arithmetic, themodular additive inverse of x is also dened: it is the number a such that a + x 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is thesolution to 3 + x 0 (mod 11).

2.4 Non-examplesNatural numbers, cardinal numbers, and ordinal numbers, do not have additive inverses within their respective sets.Thus, for example, we can say that natural numbers do have additive inverses, but because these additive inverses arenot themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.

2.5 See also Absolute value (related through the identity | x | = | x | ) Multiplicative inverse Additive identity Involution (mathematics) Reection symmetry

2.6 Footnotes[1] Tussy, Alan; Gustafson, R. (2012), Elementary Algebra (5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.

[2] The term negation bears a reference to negative numbers, which can be misleading, because the additive inverse of anegative number is positive.

2.7 References Margherita Barile, Additive Inverse, MathWorld.

Chapter 3

Algebraic expression

Rational expression redirects here. For the notion in formal languages, see regular expression.

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraicoperations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational num-ber).[1] For example, 3x2 2xy+ c is an algebraic expression. Since taking the square root is the same as raising tothe power 12 ,

r1 x21 + x2

is also an algebraic expression. By contrast, transcendental numbers like and e are not algebraic.A rational expression is an expression that may be rewritten to a rational fraction by using the properties of thearithmetic operations (commutative properties and associative properties of addition and multiplication, distributiveproperty and rules for the operations on the fractions). In other words, a rational expression is an expression whichmaybe constructed from the variables and the constants by using only the four operations of arithmetic. Thus, 3x22xy+cy31is a rational expression, whereas

q1x21+x2 is not.

A rational equation is an equation in which two rational fractions (or rational expressions) of the form P (x)Q(x) areset equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undened, so that a solution causing formal division by zero is rejected.

3.1 Terminology

Algebra has its own terminology to describe parts of an expression:

1 Exponent (power), 2 coecient, 3 term, 4 operator, 5 constant, x; y - variables

7

8 CHAPTER 3. ALGEBRAIC EXPRESSION

3.2 In roots of polynomialsThe roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can alwaysbe written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such asolution of an equation is called an algebraic solution. But the Abel-Runi theorem states that algebraic solutions donot exist for all such equations (just for some of them) if n 5.

3.3 Conventions

3.3.1 VariablesBy convention, letters at the beginning of the alphabet (e.g. a; b; c ) are typically used to represent constants, andthose toward the end of the alphabet (e.g. x; y and z ) are used to represent variables.[2] They are usually written initalics.[3]

3.3.2 ExponentsBy convention, terms with the highest power (exponent), are written on the left, for example, x2 is written to the leftof x . When a coecient is one, it is usually omitted (e.g. 1x2 is written x2 ).[4] Likewise when the exponent (power)is one, (e.g. 3x1 is written 3x ),[5] and, when the exponent is zero, the result is always 1 (e.g. 3x0 is written 3 , sincex0 is always 1 ).[6]

3.4 Algebraic vs. other mathematical expressionsThe table below summarizes how algebraic expressions compare with several other types of mathematical expressions.A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient ofpolynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as x + 4.

3.5 See also Algebraic equation

Linear_equation#Algebraic_equations Algebraic function Analytical expression Arithmetic expression Closed-form expression Expression (mathematics) Polynomial Term (logic)

3.6 Notes[1] Morris, Christopher G. (1992). Academic Press dictionary of science and technology. p. 74.

[2] William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, TheRosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71

3.7. REFERENCES 9

[3] James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425,9780387985428, 221 pages, [James E. Gentle page 183]

[4] DavidAlanHerzog, Teach Yourself Visually Algebra, Publisher JohnWiley&Sons, 2008, ISBN0470185597, 9780470185599,304 pages, page 72

[5] JohnC. Peterson, TechnicalMathematicsWith Calculus, Publisher Cengage Learning, 2003, ISBN0766861899, 9780766861893,1613 pages, page 31

[6] JeromeE.Kaufmann, Karen L. Schwitters,Algebra for College Students, Publisher Cengage Learning, 2010, ISBN0538733543,9780538733540, 803 pages, page 222

3.7 References James, Robert Clarke; James, Glenn (1992). Mathematics dictionary. p. 8.

3.8 External links Weisstein, Eric W., Algebraic Expression, MathWorld.

Chapter 4

Algebraic fraction

In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Twoexamples of algebraic fractions are 3xx2+2x3 and

px+2

x23 . Algebraic fractions are subject to the same laws as arithmeticfractions.A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials. Thus 3xx2+2x3is a rational fraction, but not

px+2

x23 ; because the numerator contains a square root function.

4.1 TerminologyIn the algebraic fraction ab , the dividend a is called the numerator and the divisor b is called the denominator. Thenumerator and denominator are called the terms of the algebraic fraction.A complex fraction is a fraction whose numerator or denominator, or both, contains a fraction. A simple fractioncontains no fraction either in its numerator or its denominator. A fraction is in lowest terms if the only factor commonto the numerator and the denominator is 1.An expression which is not in fractional form is an integral expression. An integral expression can always be writtenin fractional form by giving it the denominator 1. A mixed expression is the algebraic sum of one or more integralexpressions and one or more fractional terms.

4.2 Rational fractionsSee also: Rational function

If the expressions a and b are polynomials, the algebraic fraction is called a rational algebraic fraction[1] or simplyrational fraction.[2][3] Rational fractions are also known as rational expressions. A rational fraction f(x)g(x) is calledproper if deg f(x) < deg g(x) , and improper otherwise. For example, the rational fraction 2xx21 is proper, and therational fractions x3+x2+1x25x+6 and x

2x+15x2+3 are improper. Any improper rational fraction can be expressed as the sum of

a polynomial (possibly constant) and a proper rational fraction. In the rst example of an improper fraction one has

x3 + x2 + 1

x2 5x+ 6 = (x+ 6) +24x 35

x2 5x+ 6 ;

where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rationalfraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions iscalled resolving it into partial fractions. For example,

2x

x2 1 =1

x 1 +1

x+ 1:

10

4.3. IRRATIONAL FRACTIONS 11

Here, the two terms on the right are called partial fractions.

4.3 Irrational fractionsAn irrational fraction is one that contains the variable under a fractional exponent.[4] An example of an irrationalfraction is

x12 13a

x13 x 12

:

The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrationalfraction in which the radicals are monomials may be rationalized by nding the least common multiple of the indicesof the roots, and substituting the variable for another variable with the least common multiple as exponent. In theexample given, the least common multiple is 6, hence we can substitute x = z6 to obtain

z3 13az2 z3 :

4.4 Notes[1] Bansi Lal (2006). Topics in Integral Calculus. p. 53.

[2] rnest Borisovich Vinberg (2003). A course in algebra. p. 131.

[3] Parmanand Gupta. Comprehensive Mathematics XII. p. 739.

[4] Washington McCartney (1844). The principles of the dierential and integral calculus; and their application to geometry.p. 203.

4.5 ReferencesBrink, Raymond W. (1951). IV. Fractions. College Algebra.

Chapter 5

Algebraic operation

Algebraic operations in the solution to the quadratic equation. The radical sign, denoting a square root, is equivalent toexponentiation to the power of . The sign represents the equation written with either a + and with a - sign.

In mathematics, an algebraic operation is any one of the operations addition, subtraction, multiplication, division,raising to an integer power, and taking roots (fractional power). Algebraic operations are performed on an algebraicvariable, term or expression,[1] and work in the same way as arithmetic operations.[2]

5.1 Notation

Multiplication symbols are usually omitted, and implied when there is no operator between two variables or terms,or when a coecient is used. For example, 3 x2 is written as 3x2, and 2 x y is written as 2xy.[3] Sometimesmultiplication symbols are replaced with either a dot, or center-dot, so that x y is written as either x . y or x y.Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol,[4]and it must be explicitly used, for example, 3x is written as 3 * x.Rather than using the obelus symbol, , division is usual represented with a vinculum, a horizontal line, e.g. 3/x + 1.In plain text and programming languages a slash (also called a solidus) is used, e.g. 3 / (x + 1).Exponents are usually formatted using superscripts, e.g. x2. In plain text, and in the TeX mark-up language, the caretsymbol, ^, represents exponents, so x2 is written as x ^ 2.[5][6] In programming languages such as Ada,[7] Fortran,[8]Perl,[9] Python[10] and Ruby,[11] a double asterisk is used, so x2 is written as x ** 2.The plus-minus sign, , is used as a shorthand notation for two expressions written as one, representing one expressionwith a plus sign, the other with a minus sign. For example y = x 1 represents the two equations y = x + 1 and y = x 1. Sometimes it is used for denoting positive-or-negative term such as x.

12

5.2. ARITHMETIC VS ALGEBRAIC OPERATIONS 13

5.2 Arithmetic vs algebraic operationsAlgebraic operations work in the same way as arithmetic operations, as can be seen in the table below.Note: the use of the letters a and b is arbitrary, and the examples would be equally valid if we had used x and y .

5.3 Properties of arithmetic and algebraic operations

5.4 References[1] William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "Algebraic Operations"

[2] Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison,Phinney, Blakeman, & Co., 1866, page 7

[3] Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, Algebraic notation, in Mathematics Matters Secondary 1 Express Text-book, Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, page 68

[4] William P. Berlingho, Fernando Q. Gouva,Math through the Ages: A Gentle History for Teachers and Others, PublisherMAA, 2004, ISBN 0883857367, 9780883857366, page 75

[5] Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153,9789380298153, page 212

[6] George Grtzer, First Steps in LaTeX, Publisher Springer, 1999, ISBN 0817641327, 9780817641320, page 17

[7] S. Tucker Taft, Robert A. Du, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual,Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352,page 13

[8] C. Xavier, Fortran 77AndNumericalMethods, PublisherNewAge International, 1994, ISBN812240670X, 9788122406702,page 20

[9] Randal Schwartz, brian foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140,9781449313142, page 24

[10] MatthewA. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, ISBN1598631586,9781598631586, page 46

[11] Kevin C. Baird,Ruby by Example: Concepts and Code, PublisherNo Starch Press, 2007, ISBN1593271484, 9781593271480,page 72

[12] Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: CengageLearning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, page 7

5.5 See also Elementary algebra Order of operations

Chapter 6

Associative property

This article is about associativity in mathematics. For associativity in the central processing unit memory cache, seeCPU cache. For associativity in programming languages, see operator associativity.Associative and non-associative redirect here. For associative and non-associative learning, see Learning#Types.

In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associa-tivity is a valid rule of replacement for expressions in logical proofs.Within an expression containing two or more occurrences in a row of the same associative operator, the order inwhich the operations are performed does not matter as long as the sequence of the operands is not changed. That is,rearranging the parentheses in such an expression will not change its value. Consider the following equations:

(2 + 3) + 4 = 2 + (3 + 4) = 9

2 (3 4) = (2 3) 4 = 24:Even though the parentheses were rearranged, the values of the expressions were not altered. Since this holds truewhen performing addition and multiplication on any real numbers, it can be said that addition and multiplication ofreal numbers are associative operations.Associativity is not to be confused with commutativity, which addresses whether a b = b a.Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups andcategories) explicitly require their binary operations to be associative.However, many important and interesting operations are non-associative; some examples include subtraction, exponentiationand the vector cross product. In contrast to the theoretical counterpart, the addition of oating point numbers in com-puter science is not associative, and is an important source of rounding error.

6.1 DenitionFormally, a binary operation on a set S is called associative if it satises the associative law:

(x y) z = x (y z) for all x, y, z in S.

Here, is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbollike for the multiplication.

(xy)z = x(yz) = xyz for all x, y, z in S.

The associative law can also be expressed in functional notation thus: f(f(x, y), z) = f(x, f(y, z)).

14

6.2. GENERALIZED ASSOCIATIVE LAW 15

A binary operation on the set S is associative when this diagram commutes. That is, when the two paths from SSS to S composeto the same function from SSS to S.

6.2 Generalized associative lawIf a binary operation is associative, repeated application of the operation produces the same result regardless how validpairs of parenthesis are inserted in the expression.[2] This is called the generalized associative law. For instance, aproduct of four elements may be written in ve possible ways:

1. ((ab)c)d

2. (ab)(cd)

3. (a(bc))d

4. a((bc)d)

5. a(b(cd))

If the product operation is associative, the generalized associative law says that all these formulas will yield the sameresult, making the parenthesis unnecessary. Thus the product can be written unambiguously as

abcd.

As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but theyremain unnecessary for disambiguation.

6.3 ExamplesSome examples of associative operations include the following.

The concatenation of the three strings hello, " ", world can be computed by concatenating the rst twostrings (giving hello ") and appending the third string (world), or by joining the second and third string(giving " world) and concatenating the rst string (hello) with the result. The two methods produce thesame result; string concatenation is associative (but not commutative).

In arithmetic, addition and multiplication of real numbers are associative; i.e.,

16 CHAPTER 6. ASSOCIATIVE PROPERTY

(((ab)c)d)e

((ab)c)(de)

((ab)(cd))e

((a(bc))d)e

(ab)(c(de))

(a(bc))(de)

(ab)((cd)e)

(a(b(cd)))e

a(b(c(de)))

a((bc)(de))

a(b((cd)e))

a(((bc)d)e)

a((b(cd))e)

(a((bc)d))e

In the absence of the associative property, ve factors a, b, c, d, e result in a Tamari lattice of order four, possibly dierent products.

(x+ y) + z = x+ (y + z) = x+ y + z(x y)z = x(y z) = x y z

for all x; y; z 2 R:

Because of associativity, the grouping parentheses can be omitted without ambiguity.

6.3. EXAMPLES 17

(x + z+ y)

x + z)+ (y=

The addition of real numbers is associative.

Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is alsoassociative, but multiplication of octonions is non-associative.

The greatest common divisor and least common multiple functions act associatively.

gcd(gcd(x; y); z) = gcd(x; gcd(y; z)) = gcd(x; y; z)lcm(lcm(x; y); z) = lcm(x; lcm(y; z)) = lcm(x; y; z)

for all x; y; z 2 Z:

Taking the intersection or the union of sets:

(A \B) \ C = A \ (B \ C) = A \B \ C(A [B) [ C = A [ (B [ C) = A [B [ C

for all sets A;B;C:

IfM is some set and S denotes the set of all functions fromM toM, then the operation of functional compositionon S is associative:

(f g) h = f (g h) = f g h for all f; g; h 2 S:

Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

(f g) h = f (g h) = f g h

as before. In short, composition of maps is always associative.

Consider a set with three elements, A, B, and C. The following operation:

is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative.

Because matrices represent linear transformation functions, with matrix multiplication representing functionalcomposition, one can immediately conclude that matrix multiplication is associative.


6.4 Propositional logic

6.4.1 Rule of replacementIn standard truth-functional propositional logic, association,[3][4] or associativity[5] are two valid rules of replacement.The rules allow one to move parentheses in logical expressions in logical proofs. The rules are:

(P _ (Q _R)), ((P _Q) _R)

and

(P ^ (Q ^R)), ((P ^Q) ^R);

where ", " is a metalogical symbol representing can be replaced in a proof with.

6.4.2 Truth functional connectivesAssociativity is a property of some logical connectives of truth-functional propositional logic. The following logicalequivalences demonstrate that associativity is a property of particular connectives. The following are truth-functionaltautologies.Associativity of disjunction:

(P _ (Q _R))$ ((P _Q) _R)

((P _Q) _R)$ (P _ (Q _R))Associativity of conjunction:

((P ^Q) ^R)$ (P ^ (Q ^R))

(P ^ (Q ^R))$ ((P ^Q) ^R)Associativity of equivalence:

((P $ Q)$ R)$ (P $ (Q$ R))

(P $ (Q$ R))$ ((P $ Q)$ R)

6.5 Non-associativityA binary operation on a set S that does not satisfy the associative law is called non-associative. Symbolically,

(x y) z 6= x (y z) for some x; y; z 2 S:

For such an operation the order of evaluation does matter. For example:

Subtraction

(5 3) 2 6= 5 (3 2)

6.5. NON-ASSOCIATIVITY 19

Division

(4/2)/2 6= 4/(2/2)

Exponentiation

2(12) 6= (21)2

Also note that innite sums are not generally associative, for example:

(1 1) + (1 1) + (1 1) + (1 1) + (1 1) + (1 1) + : : : = 0

whereas

1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + : : : = 1

The study of non-associative structures arises from reasons somewhat dierent from the mainstream of classicalalgebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associativelaw is replaced by the Jacobi identity. Lie algebras abstract the essential nature of innitesimal transformations, andhave become ubiquitous in mathematics.There are other specic types of non-associative structures that have been studied in depth; these tend to come fromsome specic applications or areas such as combinatorial mathematics. Other examples are Quasigroup, Quasield,Non-associative ring, Non-associative algebra and Commutative non-associative magmas.

6.5.1 Nonassociativity of oating point calculation

In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, theaddition and multiplication of oating point numbers is not associative, as rounding errors are introduced whendissimilar-sized values are joined together.[6]

To illustrate this, consider a oating point representation with a 4-bit mantissa:(1.000220 + 1.000220) + 1.000224 = 1.000221 + 1.000224 = 1.0012241.000220 + (1.000220 + 1.000224) = 1.000220 + 1.000224 = 1.000224

Even though most computers compute with a 24 or 53 bits of mantissa,[7] this is an important source of roundingerror, and approaches such as the Kahan Summation Algorithm are ways to minimise the errors. It can be especiallyproblematic in parallel computing.[8] [9]

6.5.2 Notation for non-associative operations

Main article: Operator associativity

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears morethan once in an expression. However, mathematicians agree on a particular order of evaluation for several commonnon-associative operations. This is simply a notational convention to avoid parentheses.A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

x y z = (x y) zw x y z = ((w x) y) zetc.

9=; for all w; x; y; z 2 Swhile a right-associative operation is conventionally evaluated from right to left:


x y z = x (y z)w x y z = w (x (y z))etc.

9=; for all w; x; y; z 2 SBoth left-associative and right-associative operations occur. Left-associative operations include the following:

Subtraction and division of real numbers:

x y z = (x y) z for all x; y; z 2 R;x/y/z = (x/y)/z for all x; y; z 2 R with y 6= 0; z 6= 0:

Function application:

(f x y) = ((f x) y)

This notation can be motivated by the currying isomorphism.

Right-associative operations include the following:

Exponentiation of real numbers:

xyz

= x(yz):

The reason exponentiation is right-associative is that a repeated left-associative exponentiation operationwould be less useful. Multiple appearances could (and would) be rewritten with multiplication:

(xy)z = x(yz):

Function denition

Z! Z! Z = Z! (Z! Z)x 7! y 7! x y = x 7! (y 7! x y)

Using right-associative notation for these operations can be motivated by the Curry-Howard correspon-dence and by the currying isomorphism.

Non-associative operations for which no conventional evaluation order is dened include the following.

Taking the Cross product of three vectors:

~a (~b ~c) 6= (~a~b) ~c for some ~a;~b;~c 2 R3

Taking the pairwise average of real numbers:

(x+ y)/2 + z

26= x+ (y + z)/2

2for all x; y; z 2 R with x 6= z:

Taking the relative complement of sets (AnB)nC is not the same as An(BnC) . (Compare material nonim-plication in logic.)

6.6. SEE ALSO 21

6.6 See also Lights associativity test A semigroup is a set with a closed associative binary operation. Commutativity and distributivity are two other frequently discussed properties of binary operations. Power associativity, alternativity and N-ary associativity are weak forms of associativity.

6.7 References[1] Thomas W. Hungerford (1974). Algebra (1st ed.). Springer. p. 24. ISBN 0387905189. Denition 1.1 (i) a(bc) = (ab)c

for all a, b, c in G.

[2] Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 0-471-51001-7. Ifa1; a2; : : : ; an (n 2) are elements of a set with an associative operation, then the product a1a2 : : : an is unambiguous;this is, the same element will be obtained regardless of how parentheses are inserted in the product

[3] Moore and Parker

[4] Copi and Cohen

[5] Hurley

[6] Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

[7] IEEEComputer Society (August 29, 2008). IEEE Standard for Floating-Point Arithmetic. IEEE. doi:10.1109/IEEESTD.2008.4610935.ISBN 978-0-7381-5753-5. IEEE Std 754-2008.

[8] Villa, Oreste; Chavarra-mir, Daniel; Gurumoorthi, Vidhya; Mrquez, Andrs; Krishnamoorthy, Sriram, Eects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems (PDF), retrieved 2014-04-08

[9] Goldberg, David, What Every Computer Scientist ShouldKnowAbout Floating Point Arithmetic (PDF),ACMComputingSurveys 23 (1): 548, doi:10.1145/103162.103163, retrieved 2014-04-08

Chapter 7

Brahmaguptas identity

In algebra, Brahmaguptas identity says that the product of two numbers of the form a2 + nb2 is itself a numberof that form. In other words, the set of such numbers is closed under multiplication. Specically:

a2 + nb2

c2 + nd2

= (ac nbd)2 + n (ad+ bc)2 (1)= (ac+ nbd)

2+ n (ad bc)2 ; (2)

Both (1) and (2) can be veried by expanding each side of the equation. Also, (2) can be obtained from (1), or (1)from (2), by changing b to b.This identity holds in both the ring of integers and the ring of rational numbers, andmore generally in any commutativering.

7.1 HistoryThe identity is a generalization of the so-called Fibonacci identity (where n=1) which is actually found in Diophantus'Arithmetica (III, 19). That identity was rediscovered by Brahmagupta (598668), an Indian mathematician andastronomer, who generalized it and used it in his study ofwhat is now called Pells equation. HisBrahmasphutasiddhantawas translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in1126.[1] The identity later appeared in Fibonacci's Book of Squares in 1225.

7.2 Application to Pells equationIn its original context, Brahmagupta applied his discovery to the solution of what was later called Pells equation,namely x2 Ny2 = 1. Using the identity in the form

(x21 Ny21)(x22 Ny22) = (x1x2 +Ny1y2)2 N(x1y2 + x2y1)2;

he was able to compose triples (x1, y1, k1) and (x2, y2, k2) that were solutions of x2 Ny2 = k, to generate the newtriple

(x1x2 +Ny1y2 ; x1y2 + x2y1 ; k1k2):

Not only did this give a way to generate innitely many solutions to x2 Ny2 = 1 starting with one solution, but also,by dividing such a composition by k1k2, integer or nearly integer solutions could often be obtained. The generalmethod for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was alsobased on this identity.[2]

22

7.3. SEE ALSO 23

7.3 See also Brahmagupta matrix BrahmaguptaFibonacci identity Indian mathematics List of Indian mathematicians

7.4 References[1] George G. Joseph (2000). The Crest of the Peacock, p. 306. Princeton University Press. ISBN 0-691-00659-8.

[2] John Stillwell (2002), Mathematics and its history (2 ed.), Springer, pp. 7276, ISBN 978-0-387-95336-6

7.5 External links Brahmaguptas identity at PlanetMath Brahmagupta Identity on MathWorld A Collection of Algebraic Identities

Chapter 8

BrahmaguptaFibonacci identity

In algebra, the BrahmaguptaFibonacci identity or simply Fibonaccis identity (and in fact due to Diophantus ofAlexandria) says that the product of two sums each of two squares is itself a sum of two squares. In other words, theset of all sums of two squares is closed under multiplication. Specically:

a2 + b2

c2 + d2

= (ac bd)2 + (ad+ bc)2 (1)= (ac+ bd)

2+ (ad bc)2 : (2)

For example,

(12 + 42)(22 + 72) = 262 + 152 = 302 + 12:

The identity is a special case (n = 2) of Lagranges identity, and is rst found in Diophantus. Brahmagupta provedand used a more general identity (the Brahmagupta identity), equivalent to

a2 + nb2

c2 + nd2

= (ac nbd)2 + n (ad+ bc)2 (3)= (ac+ nbd)

2+ n (ad bc)2 ; (4)

showing that the set of all numbers of the form x2 + y2 is closed under multiplication.Both (1) and (2) can be veried by expanding each side of the equation. Also, (2) can be obtained from (1), or (1)from (2), by changing b to b.This identity holds in both the ring of integers and the ring of rational numbers, andmore generally in any commutativering.In the integer case this identity nds applications in number theory for example when used in conjunction with oneof Fermats theorems it proves that the product of a square and any number of primes of the form 4n + 1 is also asum of two squares.

8.1 History

The identity is actually rst found in Diophantus'Arithmetica (III, 19), of the third century A.D. It was rediscovered byBrahmagupta (598668), an Indian mathematician and astronomer, who generalized it (to the Brahmagupta identity)and used it in his study of what is now called Pells equation. His Brahmasphutasiddhanta was translated fromSanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126.[1] The identitylater appeared in Fibonacci's Book of Squares in 1225.

24

8.2. RELATED IDENTITIES 25

8.2 Related identitiesAnalogous identities are Eulers four-square related to quaternions, andDegens eight-square derived from the octonionswhich has connections to Bott periodicity. There is also Psters sixteen-square identity, though it is no longer bilinear.

8.3 Relation to complex numbersIf a, b, c, and d are real numbers, this identity is equivalent to the multiplication property for absolute values ofcomplex numbers namely that:

ja+ bijjc+ dij = j(a+ bi)(c+ di)jsince

ja+ bijjc+ dij = j(ac bd) + i(ad+ bc)j;by squaring both sides

ja+ bij2jc+ dij2 = j(ac bd) + i(ad+ bc)j2;and by the denition of absolute value,

(a2 + b2)(c2 + d2) = (ac bd)2 + (ad+ bc)2:

8.4 Interpretation via normsIn the case that the variables a, b, c, and d are rational numbers, the identity may be interpreted as the statement thatthe norm in the eld Q(i) is multiplicative. That is, we have

N(a+ bi) = a2 + b2 and N(c+ di) = c2 + d2;

and also

N((a+ bi)(c+ di)) = N((ac bd) + i(ad+ bc)) = (ac bd)2 + (ad+ bc)2:Therefore the identity is saying that

N((a+ bi)(c+ di)) = N(a+ bi) N(c+ di):

8.5 Application to Pells equationIn its original context, Brahmagupta applied his discovery (the Brahmagupta identity) to the solution of Pells equation,namely x2 Ny2 = 1. Using the identity in the more general form

(x21 Ny21)(x22 Ny22) = (x1x2 +Ny1y2)2 N(x1y2 + x2y1)2;

26 CHAPTER 8. BRAHMAGUPTAFIBONACCI IDENTITY

he was able to compose triples (x1, y1, k1) and (x2, y2, k2) that were solutions of x2 Ny2 = k, to generate the newtriple

(x1x2 +Ny1y2 ; x1y2 + x2y1 ; k1k2):

Not only did this give a way to generate innitely many solutions to x2 Ny2 = 1 starting with one solution, but also,by dividing such a composition by k1k2, integer or nearly integer solutions could often be obtained. The generalmethod for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was alsobased on this identity.[2]

8.6 See also Brahmagupta matrix Indian mathematics List of Indian mathematicians Eulers four-square identity

8.7 References[1] George G. Joseph (2000). The Crest of the Peacock, p. 306. Princeton University Press. ISBN 0-691-00659-8.

[2] John Stillwell (2002), Mathematics and its history (2 ed.), Springer, pp. 7276, ISBN 978-0-387-95336-6

8.8 External links Brahmaguptas identity at PlanetMath Brahmagupta Identity on MathWorld A Collection of Algebraic Identities

Chapter 9

Elementary algebra

The quadratic formula, which is the solution to the quadratic equation ax2 + bx + c = 0 . Here the symbols a; b; c; x all arevariables that represent numbers.

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics.It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmeticdeals with specied numbers,[1] algebra introduces quantities without xed values, known as variables.[2] This use ofvariables entails a use of algebraic notation and an understanding of the general rules of the operators introduced inarithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realmof real and complex numbers.The use of variables to denote quantities allows general relationships between quantities to be formally and conciselyexpressed, and thus enables solving a broader scope of problems. Most quantitative results in science andmathematicsare expressed as algebraic equations.

9.1 Algebraic notation

Main article: Mathematical notation

Algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own termi-nology. For example, the expression 3x2 2xy + c has the following components:

27

28 CHAPTER 9. ELEMENTARY ALGEBRA

3 2 1 0 1 2 3

3

2

1

3

2

1y = x2x2

x

y

Two-dimensional plot (magenta curve) of the algebraic equation y = x2 x 2

1 : Exponent (power), 2 : Coecient, 3 : term, 4 : operator, 5 : constant, x; y : variables

A coecient is a numerical value which multiplies a variable (the operator is omitted). A term is an addend or asummand, a group of coecients, variables, constants and exponents that may be separated from the other terms bythe plus and minus operators.[3] Letters represent variables and constants. By convention, letters at the beginning ofthe alphabet (e.g. a; b; c ) are typically used to represent constants, and those toward the end of the alphabet (e.g.x; y and z ) are used to represent variables.[4] They are usually written in italics.[5]

Algebraic operations work in the same way as arithmetic operations,[6] such as addition, subtraction, multiplication,division and exponentiation.[7] and are applied to algebraic variables and terms. Multiplication symbols are usually

9.2. CONCEPTS 29

omitted, and implied when there is no space between two variables or terms, or when a coecient is used. Forexample, 3 x2 is written as 3x2 , and 2 x y may be written 2xy .[8]Usually terms with the highest power (exponent), are written on the left, for example, x2 is written to the left of x. When a coecient is one, it is usually omitted (e.g. 1x2 is written x2 ).[9] Likewise when the exponent (power)is one, (e.g. 3x1 is written 3x ).[10] When the exponent is zero, the result is always 1 (e.g. x0 is always rewritten to1 ).[11] However 00 , being undened, should not appear in an expression, and care should be taken in simplifyingexpressions in which variables may appear in exponents.

9.1.1 Alternative notation

Other types of notation are used in algebraic expressions when the required formatting is not available, or can not beimplied, such as where only letters and symbols are available. For example, exponents are usually formatted usingsuperscripts, e.g. x2 . In plain text, and in the TeX mark-up language, the caret symbol "^" represents exponents,so x2 is written as x^2.[12][13] In programming languages such as Ada,[14] Fortran,[15] Perl,[16] Python [17] andRuby,[18] a double asterisk is used, so x2 is written as x**2. Many programming languages and calculators use asingle asterisk to represent the multiplication symbol,[19] and it must be explicitly used, for example, 3x is written3*x.

9.2 Concepts

9.2.1 Variables

Main article: Variable (mathematics)

Elementary algebra builds on and extends arithmetic[20] by introducing letters called variables to represent general(non-specied) numbers. This is useful for several reasons.

1. Variables may represent numbers whose values are not yet known. For example, if the temperature today,T, is 20 degrees higher than the temperature yesterday, Y, then the problem can be described algebraically asT = Y + 20 .[21]

2. Variables allow one to describe general problems,[22] without specifying the values of the quantities thatare involved. For example, it can be stated specically that 5 minutes is equivalent to 60 5 = 300 seconds.A more general (algebraic) description may state that the number of seconds, s = 60 m , where m is thenumber of minutes.

3. Variables allow one to describe mathematical relationships between quantities that may vary.[23] Forexample, the relationship between the circumference, c, and diameter, d, of a circle is described by = c/d .

4. Variables allow one to describe some mathematical properties. For example, a basic property of additionis commutativity which states that the order of numbers being added together does not matter. Commutativityis stated algebraically as (a+ b) = (b+ a) .[24]

9.2.2 Evaluating expressionsMain article: Expression (mathematics)

Algebraic expressionsmay be evaluated and simplied, based on the basic properties of arithmetic operations (addition,subtraction, multiplication, division and exponentiation). For example,

Added terms are simplied using coecients. For example, x+ x+ x can be simplied as 3x (where 3 is thecoecient).

Multiplied terms are simplied using exponents. For example, x x x is represented as x3


Example of variables showing the relationship between a circles diameter and its circumference. For any circle, its circumference c, divided by its diameter d , is equal to the constant pi, (approximately 3.14).

Like terms are added together,[25] for example, 2x2+3abx2+ab is written as x2+4ab , because the termscontaining x2 are added together, and, the terms containing ab are added together.

Brackets can be multiplied out, using distributivity. For example, x(2x + 3) can be written as (x 2x) +(x 3) which can be written as 2x2 + 3x

Expressions can be factored. For example, 6x5 + 3x2 , by dividing both terms by 3x2 can be written as3x2(2x3 + 1)

9.2.3 EquationsMain article: Equation

An equation states that two expressions are equal using the symbol for equality, = (the equals sign).[26] One of themost well-known equations describes Pythagoras law relating the length of the sides of a right angle triangle:[27]

c2 = a2 + b2

9.2. CONCEPTS 31

Animation illustrating Pythagoras rule for a right-angle triangle, which shows the algebraic relationship between the triangleshypotenuse, and the other two sides.

This equation states that c2 , representing the square of the length of the side that is the hypotenuse (the side oppositethe right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented bya and b .An equation is the claim that two expressions have the same value and are equal. Some equations are true for allvalues of the involved variables (such as a+ b = b+ a ); such equations are called identities. Conditional equationsare true for only some values of the involved variables, e.g. x2 1 = 8 is true only for x = 3 and x = 3 .The values of the variables which make the equation true are the solutions of the equation and can be found throughequation solving.Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, orless, than the other. The symbols used for this are: a > b where > represents 'greater than', and a < b where have the property of transitivity:[29]

If a < b and b < c then a < c ; If a < b and c < d then a+ c < b+ d ;[30]

If a < b and c > 0 then ac < bc ; If a < b and c < 0 then bc < ac .

By reversing the inequation, < and > can be swapped,[31] for example:

a < b is equivalent to b > a

9.2.4 SubstitutionMain article: Substitution (algebra)See also: Substitution (logic)

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for a in the expressiona*5 makes a new expression 3*5 with meaning 15. Substituting the terms of a statement makes a new statement.When the original statement is true independent of the values of the terms, the statement created by substitutions isalso true. Hence denitions can be made in symbolic terms and interpreted through substitution: if a2 := a a ,where := means is dened to equal, substituting 3 for a informs the reader of this statement that 32 means 3*3=9.Often its not known whether the statement is true independent of the values of the terms, and substitution allows oneto derive restrictions on the possible values, or show what conditions the statement holds under. For example, takingthe statement x+1=0, if x is substituted with 1, this imples 1+1=2=0, which is false, which implies that if x+1=0 thenx can't be 1.If x and y are integers, rationals, or real numbers, then xy=0 implies x=0 or y=0. Suppose abc=0. Then, substitutinga for x and bc for y, we learn a=0 or bc=0. Then we can substitute again, letting x=b and y=c, to show that if bc=0then b=0 or c=0. Therefore, if abc=0, then a=0 or (b=0 or c=0), so abc=0 implies a=0 or b=0 or c=0.Consider if the original fact were stated as "ab=0 implies a=0 or b=0. Then when we say suppose abc=0, we havea conict of terms when we substitute. Yet the above logic is still valid to show that if abc=0 then a=0 or b=0 orc=0 if instead of letting a=a and b=bc we substitute a for a and b for bc (and with bc=0, substituting b for a and cfor b). This shows that substituting for the terms in a statement isn't always the same as letting the terms from thestatement equal the substituted terms. In this situation its clear that if we substitute an expression a into the a termof the original equation, the a substituted does not refer to the a in the statement "ab=0 implies a=0 or b=0.

9.3 Solving algebraic equationsSee also: Equation solvingThe following sections lay out examples of some of the types of algebraic equations that may be encountered.

9.3. SOLVING ALGEBRAIC EQUATIONS 33

A typical algebra problem.

9.3.1 Linear equations with one variableMain article: Linear equation

Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equationsto solve are linear equations that have only one variable. They contain only constant numbers and a single variablewithout an exponent. As an example, consider:

Problem in words: If you double my sons age and add 4, the resulting answer is 12. How old is my son?

Equivalent equation: 2x+ 4 = 12 where x represent my sons age

To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by thesame number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other sideof the equation is the value of the variable.[32] This problem and its solution are as follows:In words: my sons age is 4.The general form of a linear equation with one variable, can be written as: ax+ b = cFollowing the same procedure (i.e. subtract b from both sides, and then divide by a ), the general solution is given byx = cba

9.3.2 Linear equations with two variablesA linear equation with two variables has many (i.e. an innite number of) solutions.[33] For example:

Problem in words: I am 22 years older than my son. How old are we?Equivalent equation: y = x+ 22 where y is my age, x is my sons age.

This can not be worked out by itself. If I told you my sons age, then there would no longer be two unknowns(variables), and the problem becomes a linear equation with just one variable, that can be solved as described above.


Solving two linear equations with a unique solution at the point that they intersect.

To solve a linear equation with two variables (unknowns), requires two related equations. For example, if I alsorevealed that:


Now there are two related linear equations, each with two unknowns, which lets us produce a linear equation withjust one variable, by subtracting one from the other (called the elimination method):[34]

In other words, my son is aged 12, and as I am 22 years older, I must be 34. In 10 years time, my son will be 22, andI will be twice his age, 44. This problem is illustrated on the associated plot of the equations.For other ways to solve this kind of equations, see below, System of linear equations.

9.3.3 Quadratic equations

Main article: Quadratic equationA quadratic equation is one which includes a term with an exponent of 2, for example, x2 ,[35] and no term with

Quadratic equation plot of y = x2 + 3x 10 showing its roots at x = 5 and x = 2 , and that the quadratic can be rewritten asy = (x+ 5)(x 2)

higher exponent. The name derives from the Latin quadrus, meaning square.[36] In general, a quadratic equation canbe expressed in the form ax2 + bx + c = 0 ,[37] where a is not zero (if it were zero, then the equation would notbe quadratic but linear). Because of this a quadratic equation must contain the term ax2 , which is known as thequadratic term. Hence a 6= 0 , and so we may divide by a and rearrange the equation into the standard form


x2 + px+ q = 0

where p = b/a and q = c/a . Solving this, by a process known as completing the square, leads to the quadraticformula

x =bpb2 4ac

2a;

where the symbol "" indicates that both

x =b+pb2 4ac

2aand x = b

pb2 4ac2a

are solutions of the quadratic equation.Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for twolinear terms is sometimes denoted foiling). As an example of factoring:

x2 + 3x 10 = 0;which is the same thing as

(x+ 5)(x 2) = 0:It follows from the zero-product property that either x = 2 or x = 5 are the solutions, since precisely one of thefactors must be equal to zero. All quadratic equations will have two solutions in the complex number system, butneed not have any in the real number system. For example,

x2 + 1 = 0

has no real number solution since no real number squared equals 1. Sometimes a quadratic equation has a root ofmultiplicity 2, such as:

(x+ 1)2 = 0:

For this equation, 1 is a root of multiplicity 2. This means 1 appears two times, since the equation can be rewrittenin factored form as

[x (1)][x (1)] = 0:

Complex numbers

All quadratic equations have two solutions in complex numbers, a category that includes real numbers, imaginarynumbers, and sums of real and imaginary numbers. Complex numbers rst arise in the teaching of quadratic equationsand the quadratic formula. For example, the quadratic equation

x2 + x+ 1 = 0

has solutions

x =1 +p3

2and x = 1

p32

:

Sincep3 is not any real number, both of these solutions for x are complex numbers.


9.3.4 Exponential and logarithmic equations

Main article: LogarithmAn exponential equation is one which has the form ax = b for a > 0 ,[38] which has solution

The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1),(4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet orintersect it.

X = loga b =ln bln a

when b > 0 . Elementary algebraic techniques are used to rewrite a given equation in the above way before arrivingat the solution. For example, if

3 2x1 + 1 = 10

then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

2x1 = 3

whence

x 1 = log2 3


or

x = log2 3 + 1:

A logarithmic equation is an equation of the form loga(x) = b for a > 0 , which has solution

X = ab:

For example, if

4 log5(x 3) 2 = 6

then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get

log5(x 3) = 2

whence

x 3 = 52 = 25

from which we obtain

x = 28:

9.3.5 Radical equations

Radical equation showing two ways to represent the same expression. The triple bar means the equation is true for all values of x

A radical equation is one that includes a radical sign, which includes square roots, px , cube roots, 3px , and nthroots, npx . Recall that an nth root can be rewritten in exponential format, so that npx is equivalent to x 1n . Combinedwith regular exponents (powers), then 2

px3 (the square root of x cubed), can be rewritten as x 32 .[39] So a common

form of a radical equation is npxm = a (equivalent to xmn = a ) wherem and n are integers. It has real solution(s):For example, if:

(x+ 5)2/3 = 4;

then


x+ 5 = (p4)3

x+ 5 = 8x = 5 8x = 3;13

9.3.6 System of linear equations

Main article: System of linear equations

There are dierent methods to solve a system of linear equations with two variables.

Elimination method

(2,3)

x-y=-1

3x+y=

9

The solution set for the equations x y = 1 and 3x+ y = 9 is the single point (2, 3).

An example of solving a system of linear equations is by using the elimination method:


(4x+ 2y = 14

2x y = 1:

Multiplying the terms in the second equation by 2:

4x+ 2y = 14

4x 2y = 2:Adding the two equations together to get:

8x = 16

which simplies to

x = 2:

Since the fact that x = 2 is known, it is then possible to deduce that y = 3 by either of the original two equations (byusing 2 instead of x ) The full solution to this problem is then

(x = 2

y = 3:

Note that this is not the only way to solve this specic system; y could have been solved before x .

Substitution method

Another way of solving the same system of linear equations is by substitution.

(4x+ 2y = 14

2x y = 1:

An equivalent for y can be deduced by using one of the two equations. Using the second equation:

2x y = 1

Subtracting 2x from each side of the equation:

2x 2x y = 1 2xy = 1 2x

and multiplying by 1:

y = 2x 1:

Using this y value in the rst equation in the original system:


4x+ 2(2x 1) = 144x+ 4x 2 = 14

8x 2 = 14

Adding 2 on each side of the equation:

8x 2 + 2 = 14 + 28x = 16

which simplies to

x = 2

Using this value in one of the equations, the same solution as in the previous method is obtained.

(x = 2

y = 3:

Note that this is not the only way to solve this specic system; in this case as well, y could have been solved before x.

9.3.7 Other types of systems of linear equations

Inconsistent systems

In the above example, a solution exists. However, there are also systems of equations which do not have any solution.Such a system is called inconsistent. An obvious example is

(x+ y = 1

0x+ 0y = 2 :

As 02, the second equation in the system has no solution. Therefore, the system has no solution. However, not allinconsistent systems are recognized at rst sight. As an example, let us consider the system

(4x+ 2y = 12

2x y = 4 :

Multiplying by 2 both sides of the second equation, and adding it to the rst one results in

0x+ 0y = 4 ;

which has clearly no solution.

Undetermined systems

There are also systems which have innitely many solutions, in contrast to a system with a unique solution (meaning,a unique pair of values for x and y ) For example:


The equations 3x+ 2y = 6 and 3x+ 2y = 12 are parallel and cannot intersect, and is unsolvable.

(4x+ 2y = 12

2x y = 6Isolating y in the second equation:

y = 2x+ 6And using this value in the rst equation in the system:

4x+ 2(2x+ 6) = 124x 4x+ 12 = 12

12 = 12

The equality is true, but it does not provide a value for x . Indeed, one can easily verify (by just lling in some valuesof x ) that for any x there is a solution as long as y = 2x + 6 . There is an innite number of solutions for thissystem.

9.4. SEE ALSO 43

Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no commonsolution.

Over- and underdetermined systems

Systems with more variables than the number of linear equations are called underdetermined. Such a system, if ithas any solutions, does not have a unique one but rather an innitude of them. An example of such a system is

(x+ 2y = 10

y z = 2:When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, butcannot express all solutions numerically because there are an innite number of them if there are any.A system with a greater number of equations than variables is called overdetermined. If an overdetermined systemhas any solutions, necessarily some equations are linear combinations of the others.

9.4 See also History of elementary algebra


Binary operation Gaussian elimination Mathematics education Number line Polynomial

9.5 References Leonhard Euler, Elements of Algebra, 1770. English translation Tarquin Press, 2007, ISBN 978-1-899618-79-8, also online digitized editions[40] 2006,[41] 1822.

Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs. Redden, John. Elementary Algebra. Flat World Knowledge, 2011

[1] H.E. Slaught and N.J. Lennes, Elementary algebra, Publ. Allyn and Bacon, 1915, page 1 (republished by Forgotten Books)

[2] Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Pub-lisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, page 2

[3] Richard N. Aufmann, Joanne Lockwood, Introductory Algebra: An Applied Approach, Publisher Cengage Learning, 2010,ISBN 1439046042, 9781439046043, page 78

[4] William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, TheRosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71

[5] James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425,9780387985428, 221 pages, [James E. Gentle page 183]

[6] Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison,Phinney, Blakeman, & Co., 1866, page 7

[7] Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: CengageLearning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, page 6

[8] Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, Algebraic notation, in Mathematics Matters Secondary 1 Express Text-book, Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, page 68

[9] DavidAlanHerzog, Teach Yourself Visually Algebra, Publisher JohnWiley&Sons, 2008, ISBN0470185597, 9780470185599,304 pages, page 72

[10] JohnC. Peterson, TechnicalMathematicsWith Calculus, Publisher Cengage Learning, 2003, ISBN0766861899, 9780766861893,1613 pages, page 31

[11] JeromeE.Kaufmann, Karen L. Schwitters,Algebra for College Students, Publisher Cengage Learning, 2010, ISBN0538733543,9780538733540, 803 pages, page 222

[12] Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153,9789380298153, page 212

[13] George Grtzer, First Steps in LaTeX, Publisher Springer, 1999, ISBN 0817641327, 9780817641320, page 17

[14] S. Tucker Taft, Robert A. Du, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual,Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352,page 13

[15] C.Xavier, Fortran 77AndNumericalMethods, PublisherNewAge International, 1994, ISBN812240670X, 9788122406702,page 20

[16] Randal Schwartz, Brian Foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140,9781449313142, page 24

[17] MatthewA. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, ISBN1598631586,9781598631586, page 46

9.6. EXTERNAL LINKS 45

[18] Kevin C. Baird,Ruby by Example: Concepts and Code, PublisherNo Starch Press, 2007, ISBN1593271484, 9781593271480,page 72

[19] William P. Berlingho, Fernando Q. Gouva,Math through the Ages: A Gentle History for Teachers and Others, PublisherMAA, 2004, ISBN 0883857367, 9780883857366, page 75

[20] Thomas Sonnabend, Mathematics for Teachers: An Interactive Approach for Grades K-8, Publisher: Cengage Learning,2009, ISBN 0495561665, 9780495561668, 759 pages, page xvii

[21] Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Pub-lisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, page 48

[22] Lawrence S. Le, College Algebra: Barrons Ez-101 Study Keys, Publisher: Barrons Educational Series, 2005, ISBN0764129147, 9780764129148, 230 pages, page 2

[23] RonLarson, KimberlyNolting, Elementary Algebra, Publisher: Cengage Learning, 2009, ISBN0547102275, 9780547102276,622 pages, page 210

[24] Charles P. McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, ISBN 0840064217, 9780840064219,571 pages, page 49

[25] Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, PublisherKaplan Publishing, 2007, ISBN 1419552880, 9781419552885, 288 pages, page 51

[26] Mark Clark, Cynthia Annson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learn-ing, 2011, ISBN 0534419380, 9780534419387, 793 pages, page 134

[27] Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, ISBN1111567689, 9781111567682, 1163 pages, page 493

[28] Douglas Downing,Algebra the EasyWay, Publisher Barrons Educational Series, 2003, ISBN0764119729, 9780764119729,392 pages, page 20

[29] RonLarson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, ISBN0618753524, 9780618753529,857 pages, page 96

[30] http://math.stackexchange.com/a/1043755/19368[31] Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, ISBN 019914768X,

9780199147687, 144 pages, page 50[32] Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. p. 72. ISBN 0-471-50636-2.[33] Sinha, The PearsonGuide to Quantitative Aptitude for CAT 2/ePublisher: Pearson Education India, 2010, ISBN8131723666,

9788131723661, 599 pages, page 195[34] Cynthia Y. Young, Precalculus, Publisher John Wiley & Sons, 2010, ISBN 0471756849, 9780471756842, 1175 pages,

page 699[35] Mary Jane Sterling, Algebra II For Dummies, Publisher: John Wiley & Sons, 2006, ISBN 0471775819, 9780471775812,

384 pages, page 37[36] John T. Irwin, The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story, Publisher JHU Press, 1996, ISBN

0801854660, 9780801854668, 512 pages, page 372[37] Sharma/khattar, The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E, Publisher Pear-

son Education India, 2010, ISBN 8131723631, 9788131723630, 1248 pages, page 621[38] AvenChoo, LMANOLAdditionalMaths RevisionGuide 3, Publisher Pearson Education SouthAsia, 2007, ISBN9810600011,

9789810600013, page 105[39] JohnC. Peterson, TechnicalMathematicsWith Calculus, Publisher Cengage Learning, 2003, ISBN0766861899, 9780766861893,

1613 pages, page 525[40] Eulers Elements of Algebra[41] Elements of algebra Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange Google

Books

9.6 External links

Chapter 10

Proofs involving the addition of naturalnumbers

Mathematical proofs for addition of the natural numbers: additive identity, commutativity, and associativity. Theseproofs are used in the article Addition of natural numbers.

10.1 DenitionsThis article will use the Peano axioms for the denitions of addition of the natural numbers, and the successor functionS(a). In particular:For the proof of commutativity, it is useful to dene another natural number closely related to the successor function,namely 1. We dene 1 to be the successor of 0, in other words,

1 = S(0).

Note that for all natural numbers a,

10.2 Proof of associativityWe prove associativity by rst xing natural numbers a and b and applying induction on the natural number c.For the base case c = 0,

(a+b)+0 = a+b = a+(b+0)

Each equation follows by denition [A1]; the rst with a + b, the second with b.Now, for the induction. We assume the induction hypothesis, namely we assume that for some natural number c,

(a+b)+c = a+(b+c)

Then it follows,In other words, the induction hypothesis holds for S(c). Therefore, the induction on c is complete.

10.3 Proof of identity elementDenition [A1] states directly that 0 is a right identity. We prove that 0 is a left identity by induction on the naturalnumber a.

46

10.4. PROOF OF COMMUTATIVITY 47

For the base case a = 0, 0 + 0 = 0 by denition [A1]. Now we assume the induction hypothesis, that 0 + a = a. ThenThis completes the induction on a.

10.4 Proof of commutativityWe prove commutativity (a + b = b + a) by applying induction on the natural number b. First we prove the base casesb = 0 and b = S(0) = 1 (i.e. we prove that 0 and 1 commute with everything).The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which hasbeen proved above: a + 0 = a = 0 + a.Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a +1 = 1 + a. We will prove this by induction on a (an induction proof within an induction proof). Clearly, for a = 0,we have 0 + 1 = 0 + S(0) = S(0 + 0) = S(0) = 1 = 1 + 0. Now, suppose a + 1 = 1 + a. ThenThis completes the induction on a, and so we have proved the base case b = 1. Now, suppose that for all naturalnumbers a, we have a + b = b + a. We must show that for all natural numbers a, we have a + S(b) = S(b) + a. WehaveThis completes the induction on b.

10.5 See also Binary operation Proof Ring

10.6 References Edmund Landau, Foundations of Analysis, Chelsea Pub Co. ISBN 0-8218-2693-X.

48 CHAPTER 10. PROOFS INVOLVING THE ADDITION OF NATURAL NUMBERS

10.7 Text and image sources, contributors, and licenses10.7.1 Text

Additive identity Source: https://en.wikipedia.org/wiki/Additive_identity?oldid=635820081 Contributors: David Shay, Chocolateboy,Giftlite, Bobo192, Nsaa, Arthena, TheArmadillo, Salix alba, RussBot, Michael Slone, SmackBot, Elonka, Melchoir, Miquonranger03,Silly rabbit, SchftyThree, Octahedron80, MaxSem, FlyingToaster, Cydebot, Xantharius, Kilva, Mhaitham.shammaa, Hannes Eder, Gol-gofrinchian, Trusilver, SJP, ABF, Vector Potential, Paolo.dL, Oxymoron83, Lightmouse, Techman224, JackSchmidt, RegentsPark, Clue-Bot, Niceguyedc, Versus22, Addbot, CanadianLinuxUser, West.andrew.g, Luckas-bot, Piano non troppo, Mechamind90, GrouchoBot,Pinethicket, Brad Polard, Bumax5, ClueBot NG,Wcherowi, Rm1271, Jeremy112233, George8211, Rupallavi bhartiya and Anonymous:39

Additive inverse Source: https://en.wikipedia.org/wiki/Additive_inverse?oldid=667890056Contributors: TobyBartels, Ubiquity, Patrick,Uyanga, Pizza Puzzle, Schneelocke, Charles Matthews, Dysprosia, Robbot, Giftlite, Monedula, Eequor, Joyous!, Cacycle, Paul August,RoyBoy, Bobo192, Haham hanuka, Jrme, Alansohn, Burn, Alai, Drbreznjev, Oleg Alexandrov, Camw, MFH, SixWingedSeraph, Jsha-dias, Missmarple, Adjusting, Salix alba, FlaBot, Mathbot, Chobot, Waitak, Michael Slone, Meersan, Dbrs, Googl, Theda, DGaw,KnightRider~enwiki, SmackBot, RDBury, Maksim-e~enwiki, Obhave, Incnis Mrsi, Melchoir, Laaabaseball, KocjoBot~enwiki, BiT,Gilliam, Betacommand, JAn Dudk, Silly rabbit, Octahedron80, Pboyd04, UU, Cybercobra, Rigadoun, Mets501, EdC~enwiki, JForget,CRGreathouse, Makeemlighter, Dr.enh, Alaibot, Arcayne, Thijs!bot, AntiVandalBot, RobotG, Seaphoto, Mhaitham.shammaa, Edok-ter, Random user 8384993, B.S. Lawrence, Magioladitis, David Eppstein, Defenestrating Monday, Matqkks, Shucko, The AnonymousOne, J.delanoy, Bogey97, FruitMart07, VolkovBot, Bacchus87, TXiKiBoT, NPrice, Synthebot, Fischer.sebastian, Dogah, SieBot, Mal-colmxl5, Tiptoety, JackSchmidt, ClueBot, Gawaxay, Ndenison, Alexbot, Thingg, SoxBot III, Addbot, Ronhjones, Jncraton, CactusWriter,LaaknorBot, Tide rolls, Luckas-bot, Gobbleswoggler, AnomieBOT, AdjustShift, Um, Xqbot, El Caro, GrouchoBot, Intelligentsium,MacMed, Rushbugled13, Hluup, Clarkcj12, SeoMac, Specs112, JV Smithy, Rishibee, EmausBot, ZroBot, Wayne Slam, DavidHolmes0,DASHBotAV, ClueBot NG, ElphiBot, Kevkev911, DeltaCommand, Catnipeverdeen, Untouchable21-5, Comfr, DarthSivius, Soni, Lu-gia2453, Daniblue12345, Ty5ege5y54y765h56, Ginsuloft, 2431423erwq, UNALAQ123456789, Karlaghasdiabetes, Buttnuggetmcgeeand Anonymous: 118

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Brahmaguptas identity Source: https://en.wikipedia.org/wiki/Brahmagupta{}s_identity?oldid=559092695Contributors: Michael Hardy,Eric Kvaalen and Mild Bill Hiccup

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