MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

1.2Properties ofReal Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Goal

To be able to define and recognizethe properties

of real numbers.

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

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Properties of Real Numbers

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

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Commutative Property

We can say that

a + b = b + aa • b = b • a

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Associative Property

We can say that

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

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Identity Property

6 + 0 = 6

3 + 0 = 3

What can we observe?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Identity Property

7 • 1 = 7

4 • 1 = 4

What can we observe?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Identity Property

We can say that

a + 0 = aa • 1 = a

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Inverse Property

14 + (-14) = 0

23 + (-23) = 0

What can we observe?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Inverse Property

What can we observe?

13 • = 1113

22 • = 1122

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Inverse Property

We can say that

a + (-a) = 0

a • = 11a

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Distributive Property

We can say that

a • (b + c) = (a • b) + (a • c)

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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?

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Closure Property

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

in another Natural number?

(4 + 27) results in 31

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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)

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

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