# 1.3 – Properties of Real Numbers. Real Numbers 1.3 – Properties of Real Numbers.

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1.3 – Properties of Real Numbers

Real Numbers 1.3 – Properties of Real Numbers

Real Numbers (R)

1.3 – Properties of Real Numbers Real Numbers (R)

1.3 – Properties of Real Numbers Real Numbers (R) Rational

1.3 – Properties of Real Numbers Real Numbers (R) Rational (⅓)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers (-6)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s (0)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s (7)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (7)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (I) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

Example 1

Name the sets of numbers to which each apply.

Example 1 Name the sets of numbers to which each apply.

Example 1 Name the sets of numbers to which each apply.

Example 1 Name the sets of numbers to which each apply. (a) √ 16

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45 - Q

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45 - Q, R

Properties of Real Numbers PropertyAdditionMultiplication Commutativea + b = b + aa·b = b·a Associative (a+b)+c = a+(b+c) (a · b) · c = a · (b · c) Identitya+0 = a = 0+aa·1 = a = 1·a Inversea+(-a) =0= -a+aa·1 =1= 1·a a a Distributivea(b+c)=ab+ac and (b+c)a=ba+ca

Example 2

Name the property used in each equation.

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7)

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x Associative Multiplication

Example 3 What is the additive and multiplicative inverse for -1¾?

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + = 0

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾ · = 1

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: (-1¾)(- 4 / 7 ) = 1

1.4 – The Distributive Property

a(b+c)=ab+ac and (b+c)a=ba+ca

Example 4

Simplify 2(5m+n)+3(2m–4n).

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m +

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n +

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m –

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m – 10n

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