1.3 – properties of real numbers
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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 PresentationTRANSCRIPT
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1.3 – Properties of Real Numbers
![Page 2: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/2.jpg)
Real Numbers
1.3 – Properties of Real Numbers
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1.3 – Properties of Real Numbers
Real Numbers (R)
![Page 4: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/4.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
![Page 5: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/5.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
Rational
![Page 6: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/6.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
Rational (⅓)
![Page 7: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/7.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
![Page 8: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/8.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
![Page 9: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/9.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
Integers
![Page 10: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/10.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
Integers (-6)
![Page 11: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/11.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
![Page 12: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/12.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
![Page 13: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/13.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
Whole #’s
![Page 14: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/14.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
Whole #’s (0)
![Page 15: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/15.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
![Page 16: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/16.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
![Page 17: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/17.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
Natural #’s
![Page 18: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/18.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
Natural #’s (7)
![Page 19: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/19.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (7)
![Page 20: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/20.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓)
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (1)
![Page 21: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/21.jpg)
1.3 – Properties of Real Numbers
Real Numbers (R)
(Q) Rational (⅓) Irrational
(Z) Integers (-6)
(W) Whole #’s (0)
(N) Natural #’s (1)
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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)
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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)
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Example 1
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Example 1
Name the sets of numbers to which each apply.
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Example 1
Name the sets of numbers to which each apply.
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Example 1
Name the sets of numbers to which each apply.
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q
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Example 1
Name the sets of numbers to which each apply.
(a) √ 16 = 4 - N, W, Z, Q, R
(b) -185 - Z, Q, R
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Example 2
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Example 2
Name the property used in each equation.
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Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
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Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
Commutative Addition
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Example 2
Name the property used in each equation.
(a) (5 + 7) + 8 = 8 + (5 + 7)
Commutative Addition
(b) 3(4x) = (3·4)x
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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
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Example 3
What is the additive and multiplicative inverse for -1¾?
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Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾
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Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + = 0
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Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
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Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Multiplicative: -1¾
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Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Multiplicative: -1¾ · = 1
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Example 3
What is the additive and multiplicative inverse for -1¾?
Additive: -1¾ + 1¾ = 0
Multiplicative: (-1¾)(-4/7) = 1
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1.4 – The Distributive Property
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1.4 – The Distributive Property
a(b+c)=ab+ac and (b+c)a=ba+ca
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Example 4
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Example 4
Simplify 2(5m+n)+3(2m–4n).
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+
![Page 73: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/73.jpg)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)
![Page 74: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/74.jpg)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
![Page 77: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/77.jpg)
Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m
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Example 4
Simplify 2(5m+n)+3(2m–4n).
2 (5m+n) + 3 (2m–4n)
2(5m)+2(n)+3(2m)-3(4n)
10m +
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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
![Page 80: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/80.jpg)
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 +
![Page 81: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/81.jpg)
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
![Page 82: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/82.jpg)
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 –
![Page 83: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/83.jpg)
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
![Page 84: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/84.jpg)
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
![Page 85: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/85.jpg)
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
![Page 86: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/86.jpg)
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
![Page 87: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/87.jpg)
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
![Page 88: 1.3 – Properties of Real Numbers](https://reader035.vdocuments.site/reader035/viewer/2022062801/568143a5550346895db02995/html5/thumbnails/88.jpg)
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