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Real Numbers 1 Definition 2 Properties 3 Examples

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1 Real Numbers 1 Definition 2 Properties 3 Examples

2 Definition Real Numbers include: Integers Rational Numbers
-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 = … 1/3 = … Irrational Numbers Non-terminating and non-repeating decimals Π = …, √2 = …

3 Properties Addition is commutative Addition is associative
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

4 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

5 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

6 Properties (Cont.) Trichotomy Law Definition of Absolute Value
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

7 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

8 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

9 Guided Practice Do Problems on page 16, 1-40

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