The document discusses properties of real numbers. It examines the commutative, associative, identity, and inverse properties through examples of addition, subtraction, multiplication and division. The commutative property states that the order of numbers does not matter for addition and multiplication. The associative property means grouping does not change the result for addition and multiplication. The identity property defines the numbers that leave other numbers unchanged when added or multiplied. And the inverse property establishes that adding or multiplying the opposite undoes the original operation.

1. 1.2
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Properties of
Real Numbers
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2. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To be able to define and recognize
the properties
of real numbers.
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3. 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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4. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Properties of Real
Numbers
We can also study numbers and see how they
“behave”. Numbers have properties as well.
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5. 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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6. 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?
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7. 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?
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8. 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?
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9. 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).
+ • XX
– ÷
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10. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Commutative Property
We can say that
a+b=b+a
a•b=b•a
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11. 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?
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12. 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?
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13. 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?
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14. 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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15. 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).
+ • XX
– ÷
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16. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Associative Property
We can say that
(a + b) + c = a + (b + c)
(a • b) • c = a • (b • c)
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17. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Identity Property
6+0=6
3+0=3
What can we observe?
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18. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Identity Property
7•1=7
4•1=4
What can we observe?
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19. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Identity Property
We can say that
a+0=a
a•1=a
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20. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Inverse Property
14 + (-14) = 0
23 + (-23) = 0
What can we observe?
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21. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Inverse Property
1
13 • =1
13
1 =1
22 •
22
What can we observe?
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22. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Inverse Property
We can say that
a + (-a) = 0
1
a• =1
a
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23. 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?
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24. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Distributive Property
We can say that
a • (b + c) = (a • b) + (a • c)
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25. 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?
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26. 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
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27. 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)
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28. 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
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29. 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
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30. 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
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31. 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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