1.2 The Real Number System

The Real Number system can be represented in a chart Real ( R ) Rational (Q)Irrational (I) Integers (Z) Whole (W) Natural (N) (…,–2, –1, 0, 1, 2, …) *Rational numbers can be written as a terminating or repeating decimal ex: 4, –5, 0.02, , (0, 1, 2, 3, …) (1, 2, 3, …)

*all rational numbers can also be constructed remember we can cut lengths in half, thirds, fourths, etc… and to get radicals we can utilize right triangles & their hypotenuses plus geometric means Binary Operations: sets up a relationship between numbers in a set ex: addition, subtraction, multiplication, division (geo flashback!)

Field Properties Field: any set of numbers (big or small) for which 5 particular properties hold for 2 binary operations Field Properties for Real Numbers Addition Multiplication 1)Closure 2)Commutative 3)Associative 4)Identity 5)Inverse answer is unique & also in set “move” to new place you & 2 BFFs who you associate with doesn’t change # opposite that gets you identity a + bab a + b = b + aab = ba (a + b) + c = a + (b + c) (ab)c = a(bc) a + 0 = 0 + a = a a(1) = (1)a = a a + (–a) = 0 Distributive relates + & ⨯ a(b + c) = ab + ac

Ex 1) What property is being illustrated? a) b) c)(51)(2)= 5(12) d)0.3125(3.2) = 1 helpful: mult. identity add. associative mult. associative ?? mult. inverse

You can test if a field is formed by examining each operation Ex 2) set S = {1, 2, 3} ∗ and # are operations (they are made up) ∗ # Closure Commutative Associative Identity Inverse ∗ # 1 ∗ 2=3 2 ∗ 1=3 1#2=2 2#1=2 “answers” are in set (what won’t change #) (it is 3) (it is 1) (a 3 in each row) (not a 1 in each row) What works? So… NOT a Field NO

Properties of Equality (geo proofs!) Reflexive Symmetric Transitive Substitution *Field properties & properties of equality used to prove theorems! a = a a = b ⇒ b = a a = b & b = c ⇒ a = c a = b, can substitute b for a

Ex 3) a, b, & c real numbers; Prove: If a – b = c, then a = c + b 1.a – b = c given 2.a + (–b) = c def. of subtraction 3.a + (–b) + b = c + b add. prop. 4.a + 0 = c + b add. inverse 5.a = c + b add. identity

Homework #102 Pg. 15 #6 – 13, 15 – 18, 21 – 23, 25, 27, 29, 31 – 34, 40 – 44