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1.2 The Real Number System

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, 0.3333,.

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1.2 The Real Number System

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  1. 1.2 The Real Number System

  2. 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, 0.3333, (0, 1, 2, 3, …) (1, 2, 3, …)

  3. (geo flashback!) *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

  4. 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 AdditionMultiplication • Closure • Commutative • Associative • Identity • Inverse a + b ab a + b = b + a ab = ba answer is unique & also in set (ab)c = a(bc) (a + b) + c = a + (b + c) you & 2 BFFs who you associate with opposite that gets you identity “move” to new place doesn’t change # a(1) = (1)a = a a + 0 = 0 + a = a a + (–a) = 0 a(b + c) = ab + ac Distributive relates + & ⨯

  5. Ex 1) What property is being illustrated? • (5•1)(2)= 5(1•2) • 0.3125(3.2) = 1 helpful: mult. identity add. associative mult. associative ?? mult. inverse

  6. 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) ∗ # What works? Closure Commutative Associative Identity Inverse “answers” are in set 1∗2=3 2∗1=3 1#2=2 2#1=2 So… NOT a Field (what won’t change #) (it is 3) (it is 1) NO (a 3 in each row) (not a 1 in each row)

  7. 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

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

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

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