1.1 numbers

1.1 Numbers• Classifications of Numbers

Natural numbers {1,2,3,…}

Whole numbers {0,1,2,3,…}

Integers {…-2,-1,0,1,2,…}

Rational numbers – can be expressed as where p and q are integers

-1.3, 2, 5.3147,

Irrational numbers – not rational

qp

523

137 ,

,47,5

1.1 Numbers

• The real number line:

• Real numbers:{xx is a rational or an irrational number}

-3 -2 -1 0 1 2 3

1.1 Numbers• Double negative rule:

-(-x) = x• Absolute Value of a number x: the distance

from 0 on the number line or alternatively

How is this possible if the absolute value of a number is never negative?

x0 if

0 if

xx

xx

1.1 Numbers

• 3 > -3 means 3 is to the right on the number line

• 1 < 4 means 1 is to the left on the number line

-3 -2 -1 0 1 2 3-4 4

1.2 Fundamental Operations of Algebra

• Adding numbers on the number line (-2 + -2):

-3 -2 -1 0 1 2 3-4 4

-2-2


• Adding numbers with the same sign:Add the absolute values and use the sign of both numbers

• Adding numbers with different signs:Subtract the absolute values and use the sign of the number with the larger absolute value


• Subtraction:

• To subtract signed numbers:Change the subtraction to adding the number with the opposite sign

)( yxyx

12)7(5)7(5


• Multiplication by zero:For any number x,

• Multiplying numbers with different signs:For any positive numbers x and y,

• Multiplying two negative numbers:For any positive numbers x and y,

00 x

)()()( xyyxyx

xyyx ))((


• Reciprocal or multiplicative inverse:If xy = 1, then x and y are reciprocals of each other. (example: 2 and ½ )

• Division is the same as multiplying by the reciprocal:

yyx x 1


• Division by zero:For any number x,

• Dividing numbers with different signs:For any positive numbers x and y,

• Dividing two negative numbers:For any positive numbers x and y,

undefined 0 x

)( yx

yx

yx

yx

yx


• Commutative property (addition/multiplication)

• Associative property (addition/multiplication)

baab

abba

)()(

)()(

bcacab

cbacba


• Distributive property

cabaacb

acabcba

)(

)(


• PEMDAS (Please Excuse My Dear Aunt Sally)1. Parenthesis2. Exponentiation3. Multiplication / Division

(evaluate left to right)4. Addition / Subtraction

(evaluate left to right)• Note: the fraction bar implies parenthesis

1.3 Calculators and Approximate Numbers

• Significant Digits – What’s the pattern?

Number Significant Digits

4.537 4

0.000056 2

70506 5

40.500 5


• Precision:

• Meaning of the Last Digit: 56.5 V means the number of volts is between 56.45 and 56.55

Number Precision

4.537 thousandths

56 units

56.00 hundredths

40.500 thousandths


• Rounding to a number of significant digits

OriginalNumber

Significant Digits

RoundedNumber

4.5371 1 5

4.5371 2 4.5

4.5371 3 4.54

4.5371 4 4.537


• Adding approximate numbers – only as accurate as the least precise. The following sum will be precise to the tenths position.

1253.10

1.13

123.12

1.4 Exponents

• Power Rule (a) for exponents:

• Power Rule (b) for exponents:

• Power Rule (c) for exponents: mmm baab

m

mm

b

a

b

a

nmnm aa

1.4 Exponents

• Definition of a zero exponent:

• Definition of a negative exponent:

is) at matter wha (no 10 a

n

nn

aaa

11

1.4 Exponents

• Changing from negative to positive exponents:

• This formula is not specifically in the book but is used often:

m

n

n

m

a

b

b

a

p

m

np

n

m

a

b

b

a

1.4 Exponents

• Quotient rule for exponents:

nmn

m

aa

a

1.4 Exponents

• A few tricky ones:

16222222

1622222

822222

82222

44

4

33

3

1.4 Exponents

• Formulas and non-formulas:

222222 , babababa

baba 22

nnnnnn babababa ,

)( propertyvedistributinbnanba

)( brulepowerbaba nnn

1.4 Exponents

• Examples (true or false):

222

333

1234

1234

)(

tsts

tsts

tt

ttt

1.4 Exponents

• Examples (true or false):

52

3

2

2

2

2

0

0

22

2

321

110

y

x

y

x

1.4 Exponents

• Putting it all together (example):

57

3261

3632

322

24

)8(3

23

23

yx

yx

yxxy

yxxy

1.4 Exponents

• Another example:

3

993

827

3663827

363

633

3

12

23

2

12

8

27

2

3

2

3

3

2

x

yyx

yxyx

yx

yx

xy

xy

yx

1.5 Scientific Notation

• A number is in scientific notation if :1. It is the product of a number and a 10 raised to a

power.2. The absolute value of the first number is between 1

and 10• Which of the following are in scientific notation?

– 2.45 x 102

– 12,345 x 10-5

– 0.8 x 10-12

– -5.2 x 1012


• Writing a number in scientific notation:

1. Move the decimal point to the right of the first non-zero digit.

2. Count the places you moved the decimal point.

3. The number of places that you counted in step 2 is the exponent (without the sign)

4. If your original number (without the sign) was smaller than 1, the exponent is negative. If it was bigger than 1, the exponent is positive


• Converting to scientific notation (examples):

• Converting back – just undo the process:

?

?

102.100012.

102.66200000

000,1861086.1

000,000,000,000,000,000,300,62010203.65

23

1.5 Scientific Notation• Multiplication with scientific notation (answers

given without exponents):

• Division with scientific notation:

02.1020

1010541051043

8585

000,000,80

108.108.10

10

5

4

105

104 84124

12

4

12

1.6 Roots and Radicals

• is the positive square root of a, andis the negative square root of a because

• If a is a positive number that is not a perfect square then the square root of a is irrational.

• If a is a negative number then square root of a is not a real number.

• For any real number a:

aaaa 22

and

a a

aa 2


• The nth root of a:

is the nth root of a. It is a number whose nth power equals a, so:

• n is the index or order of the radical

• Example:

aann

n a

322 because 2 32 55


• The nth root of nth powers:– If n is even, then

– If n is odd, then

• The nth root of a negative number:– If n is even, then the nth root is an imaginary number

– If n is odd, then the nth root is negative

aan n

n na

aan n

1.7 Adding and Subtracting Algebraic Expressions

• Degree of a term – sum of the exponents on the variables

• Degree of a polynomial – highest degree of any non-zero term

523 degree 5 23 ba

3 degree 100235 23 x xx


• Monomial – polynomial with one term

• Binomial - polynomial with two terms

• Trinomial – polynomial with three terms

• Polynomial in x – a term or sum of terms of the form

35x

10035 23 xx

yy 25

xxxaxn 24 3 :examplefor


• An expression is split up into terms by the +/- sign:

• Similar terms – terms with exactly the same variables with exactly the same exponents are like terms:

• When adding/subtracting polynomials we will need to combine similar terms:

35343 22 xyxx

232323 23 5 bababa


• Example:

1782

1255324

1252534

)1252()534(

2

22

22

22

xx

xxxx

xxxx

xxxx

1.8 Multiplication of Algebraic Expressions

• Multiplying a monomial and a polynomial: use the distributive property to find each product.Example:

23

22

2

2012

5434

534

xx

xxx

xx


• Multiplying two polynomials:

654

2

63x3

3

2

23

23

2

2

xxx

xxx

x

x

xx


• Multiplying binomials using FOIL (First – Inner – Outer - Last):

1. F – multiply the first 2 terms

2. O – multiply the outer 2 terms

3. I – multiply the inner 2 terms

4. L – multiply the last 2 terms

5. Combine like terms


• Squaring binomials:

• Examples:

222

222

2

2

yxyxyx

yxyxyx

11025152515

9633232222

2222

zzzzz

mmmmm


• Product of the sum and difference of 2 terms:

• Example:

22 yxyxyx

222 9333 wwww

1.9 Division of Algebraic Expressions

• Dividing a polynomial by a monomial:divide each term by the monomial

322

6

2

4

2

642

2

2

23

2

223

xyyx

yx

yx

yx

yx

yxyx


• Dividing a polynomial by a polynomial:

6

24

84

2

52

24

22854412

2

2

23

2

23

x

x

xx

xx

xx

xxxxxx


• Synthetic division:

answer is:

remainder is: -1

2

375 23

x

xxx

1131

262

37512

132 xx

2

1132

xxx

1.10 Solving Equations

• 1 – Multiply on both sides to get rid of fractions/decimals

• 2 – Use the distributive property• 3 – Combine like terms• 4 – Put variables on one side, numbers on the

other by adding/subtracting on both sides• 5 – Get “x” by itself on one side by multiplying or

dividing on both sides• 6 – Check your answers (if you have time)


• Fractions - Multiply each term on both sides by the Least Common Denominator (in this case the LCD = 4):

Multiply by 4:

Reduce Fractions:

Subtract x:

Subtract 5:

17

125

1225

342

45

4

4

32

15

4

1

x

x

xx

xx

xx


• Decimals - Multiply each term on both sides by the smallest power of 10 that gets rid of all the decimals

Multiply by 100:Cancel:Distribute:Subtract 5x:Subtract 50:Divide by 5:

16805

30505

3055010

305510

3.10005.10051.100

3.05.51.

xx

x

xx

xx

xx

xx


• Example:Clear fractions:

Combine like terms:

Get variables on one side:

Solve for x:

361

21

32 xxx

1834 xxx

187 xx

186 x

3x

1.11 Formulas and Literal Equations

• Example: d = rt; (d = 252, r = 45)

then 252 = 45tdivide both sides by 45:

5

35

45

275 t

1.11 Formulas and Literal Equations

• Example: Solve the formula for B

multiply both sides by 2:

divide both sides by h:

subtract b from both sides:

)(2 BbhA

)(21 BbhA

Bbh

A2

bh

AB 2

1.12 Applied Word Problems

• 1 – Decide what you are asked to find• 2 – Write down any other pertinent information

(use other variables, draw figures or diagrams )• 3 – Translate the problem into an equation.• 4 – Solve the equation.• 5 – Answer the question posed.• 6 – Check the solution.


• Example: The sum of 3 consecutive integers is 126. What are the integers?x = first integer, x + 1 = second integer, x + 2 = third integer

43,42,41

41

1233

12633

126)2()1(

x

x

x

xxx


• Example: Renting a car for one day costs $20 plus $.25 per mile. How much would it cost to rent the car for one day if 68 miles are driven?$20 = fixed cost, $.25 68 = variable cost

37$

17$20$

25$.6820$

1.1 numbers

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