Real Numbers

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1Definition

Properties

3Examples

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Definition

Real Numbers include: Integers

-3,-2,-1,0,1,2,3 Rational Numbers

Decimals that can be represented in fraction form that are either terminating or non-terminating and repeating

5/4 = 1.25 177/55 = 3.2181818… 1/3 = .33333…

Irrational Numbers Non-terminating and non-repeating decimals Π = 3.14159…, √2 = 1.41421…

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Properties

Addition is commutative a + b = b + a Order does not matter

Addition is associative a + (b + c) = (a +b) + c Grouping does not matter

0 is the additive identity a + 0 = a Adding 0 yields the same number

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Properties (Cont.)

-a is the additive inverse (negative) of a a + (-a) = 0, 12+(-12)=0 Adding a number and it’s inverse gives 0

Multiplication is commutative ab = ba, 3*4=4*3=12 Order of multiplication does not change the

result 1 is the multiplicative identity

a * 1 = a Multiplying 1 yields the same number

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Properties (Cont.)

If a ≠ 0, 1/a is the multiplicative inverse (reciprocal) of a a(1/a) = 1, 3(1/3)=1 Multiplying a non-zero number by its reciprocal

yields 1 Multiplication is distributive over addition

a(b + c) = ab + ac (a + b)c = ac + bc Multiplying a number and a sum of two numbers is

the same as multiplying each of the two numbers by the multiplier and then adding the products

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Properties (Cont.)

Trichotomy Law If a and b are real numbers, then exactly one of the

following is true: a=b, a<b, a>b Definition of Absolute Value

If a ≥ 0, then |a|=a If a <0, then |a|=-(a)

Distance on a number line d(A, B) = |B-A|

Law of the signs If a and b both have the same sign, then ab and a/b are

positive If a and b have different signs, then ab and a/b are

negative

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Examples

If p, q, r, and s denote real numbers, show that (p+q)(r+s)=pr+ps+qr+qs (p+q)(r+s) =p(r+s)+q(r+s)

=(pr+ps)+(qr+qs)= pr+ps+qr+qs

If x>0, and y<0, determine the sign of x/y + y/x Since only y is negative, both x\y and y/x will be negative

numbers A negative number increased by another negative number will

yield a “more” negative number If x<1, rewrite |x-1| without using the absolute value symbol

If x<1, then x-1<0 (negative) By part 2 of the definition of absolute value, |x-1|=-(x-1)=-x+1 or

1-x

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Examples

Let A, B, C, and D have coordinates -5, -3, 1, and 6 respectively. Find d(B,D)/\. d(B,D) = d(-3,6)

=|6-(-3)|=|6+3|=|9|=9

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Guided Practice

Do Problems on page 16, 1-40