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1.3 – Properties of Real Numbers. 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. - PowerPoint PPT Presentation
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1.3 – Properties of Real Numbers
Real Numbers
1.3 – Properties of 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
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
Property Addition Multiplication
Commutative a + b = b + a a·b = b·a
Associative (a+b)+c = a+(b+c) (a·b)·c = a·(b·c)
Identity a+0 = a = 0+a a·1 = a = 1·a
Inverse a+(-a) =0= -a+a a·1 =1= 1·a
a a
Distributive a(b+c)=ab+ac and (b+c)a=ba+ca
Example 2
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
1.4 – The Distributive Property
a(b+c)=ab+ac and (b+c)a=ba+ca
Example 4
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
