 # Real Numbers 1 Definition 2 Properties 3 Examples www.themegallery.com.

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Real Numbers 1 Definition 2 Properties 3 Examples

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 = …

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

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

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

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

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

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

Guided Practice Do Problems on page 16, 1-40