Math1003 1.2 - Properties of Numbers

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MATH1003 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001 1.2 Properties of Real Numbers

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Transcript of Math1003 1.2 - Properties of Numbers

Page 1: Math1003 1.2 - Properties of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

1.2Properties ofReal Numbers

Page 2: Math1003 1.2 - Properties of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Goal

To be able to define and recognizethe properties

of real numbers.

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MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Properties of Real Numbers

We can observe that certain things have properties.Let’s look at paper, we know that• it is flexible• it absorbs water• it can be burned

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MATH1003

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Properties of Real Numbers

We can also study numbers and see how they “behave”. Numbers have properties as well.

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MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Commutative Property

Is 5 + 6 equal to 6 + 5?

Is 7 + 4 equal to 4 + 7?

What does that tell you about adding two numbers?

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MATH1003

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Commutative Property

Is 4 • 3 equal to 3 • 4?

Is 6 • 2 equal to 2 • 6?

What does that tell you about multiplying two numbers?

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MATH1003

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Commutative Property

Is 6 - 4 equal to 4 - 6?

Is 8 - 3 equal to 3 - 8?

What does that tell you about subtracting two numbers?

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MATH1003

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Commutative Property

Is 4 ÷ 2 equal to 2 ÷ 4?

Is 9 ÷ 3 equal to 3 ÷ 9?

What does that tell you about dividing two numbers?

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MATH1003

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Commutative Property

Given what we have observed,we can say that the commutative property

(where order does not matter)holds true for

addition and multiplication(but not for subtraction and division).

+ • – ÷X X

Page 10: Math1003 1.2 - Properties of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Commutative Property

We can say that

a + b = b + aa • b = b • a

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MATH1003

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Associative Property

Is (5 + 2) + 3 equal to 5 + (2 + 3)?

Is (4 + 1) + 6 equal to 4 + (1 + 6)?

What does that tell you about adding three numbers?

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MATH1003

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Associative Property

Is (2 • 4) • 3 equal to 2 • (4 • 3)?

Is (3 • 5) • 2 equal to 3 • (5 • 2)?

What does that tell you about multiplying three numbers?

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MATH1003

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Associative Property

Is (10 - 4) - 1 equal to 10 - (4 - 1)?

Is (2 - 5) - 6 equal to 2 - (5 - 6)?

What does that tell you about subtracting three numbers?

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MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Associative Property

Is (8 ÷ 4) ÷ 2 equal to 8 ÷ (4 ÷ 2)?

Is (6 ÷ 2) ÷ 3 equal to 6 ÷ (2 ÷ 3)?

What does that tell you about dividing three numbers?

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MATH1003

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Associative Property

Given what we have observed,we can say that the associative property

(where grouping does not matter)holds true for

addition and multiplication(but not for subtraction and division).

+ • – ÷X X

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MATH1003

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Associative Property

We can say that

(a + b) + c = a + (b + c)(a • b) • c = a • (b • c)

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MATH1003

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Identity Property

6 + 0 = 6

3 + 0 = 3

What can we observe?

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MATH1003

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Identity Property

7 • 1 = 7

4 • 1 = 4

What can we observe?

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MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Identity Property

We can say that

a + 0 = aa • 1 = a

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MATH1003

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Inverse Property

14 + (-14) = 0

23 + (-23) = 0

What can we observe?

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MATH1003

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Inverse Property

What can we observe?

13 • = 1113

22 • = 1122

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MATH1003

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Inverse Property

We can say that

a + (-a) = 0

a • = 11a

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Distributive Property

Is 5 • (2 + 3) equal to (5 • 2) + (5 • 3)?

Is 3 • (4 + 6) equal to (3 • 4) + (3 • 6)?

What can we observe?

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MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Distributive Property

We can say that

a • (b + c) = (a • b) + (a • c)

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MATH1003

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Distributive Property

Is 5 + (2 • 3) equal to (5 + 2) • (5 + 3)?

Is 3 + (4 • 6) equal to (3 + 4) • (3 + 6)?

What can we observe?

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Closure Property

If a and b are members of N,does (a + b) always result

in another Natural number?

(4 + 27) results in 31

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Closure Property

If a and b are members of N,does (a - b) always result

in another Natural number?

(5 - 7) results in -2(-2 is not a member of N)

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MATH1003

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Closure Property

If a and b are members of Z,does (a - b) always result

in another Integer?

(19 - 3) results in 16(-500 - 8) results in -508

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MATH1003

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Closure Property

If a and b are members of Z,does (a ÷ b) always result

in another Integer?

(-24 ÷ 3) results in -8(18 ÷ 5) results in 3.6

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MATH1003

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Closure Property

If a and b are members of N,does (a x b) always result

in another Natural number?

(3 x 7) results in 21(800 x 4553534) results in 3642827200

Page 31: Math1003 1.2 - Properties of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Closure Property

If a and b are members of Z,does (a x b) always result

in another Integer?

(-5 x 9) results in -45(-4635 x -3) results in 13905


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