1. Properties of Real Numbers Math 0099

2. Opposites
Two real numbers that are the same distance from the origin of the real number line are opposites of each other.
Examples of opposites:
2 and and and

3. Reciprocals
Two numbers whose product is 1 are reciprocals of each other.
Examples of Reciprocals:
and and

4. Absolute Value
The absolute value of a number is its distance from 0 on the number line. The absolute value of x is written .
Examples of absolute value:

5. Commutative Property of Addition
a + b = b + a
When adding two numbers, the order of the numbers does not matter.
Examples of the Commutative Property of Addition
2 + 3 = (-5) + 4 = 4 + (-5)

6. Commutative Property of Multiplication
a  b = b  a
When multiplying two numbers, the order of the numbers does not matter.
Examples of the Commutative Property of Multiplication
2  3 = 3  2 (-3)  24 = 24  (-3)

7. Associative Property of Addition
a + (b + c) = (a + b) + c
When three numbers are added, it makes no difference which two numbers are added first.
Examples of the Associative Property of Addition
2 + (3 + 5) = (2 + 3) + 5
(4 + 2) + 6 = 4 + (2 + 6)

8. Associative Property of Multiplication
a(bc) = (ab)c
When three numbers are multiplied, it makes no difference which two numbers are multiplied first.
Examples of the Associative Property of Multiplication
2  (3  5) = (2  3)  5
(4  2)  6 = 4  (2  6)

9. Distributive Property
a(b + c) = ab + ac
Multiplication distributes over addition.
Examples of the Distributive Property
2 (3 + 5) = (2  3) + (2  5)
(4 + 2)  6 = (4  6) + (2  6)

10. Additive Identity Property
The additive identity property states that if 0 is added to a number, the result is that number.
Example: = = 3

11. Multiplicative Identity Property
The multiplicative identity property states that if a number is multiplied by 1, the result is that number.
Example: 5  1 = 1  5 = 5

12. Additive Inverse Property
The additive inverse property states that opposites add to zero.
7 + (-7) = 0 and = 0

13. Multiplicative Inverse Property
The multiplicative inverse property states that reciprocals multiply to 1.

14. Identify which property that justifies each of the following.
4  (8  2) = (4  8)  2

15. Identify which property that justifies each of the following.
4  (8  2) = (4  8)  2
Associative Property of Multiplication

16. Identify which property that justifies each of the following.
6 + 8 = 8 + 6

17. Identify which property that justifies each of the following.
6 + 8 = 8 + 6
Commutative Property of Addition

18. Identify which property that justifies each of the following.
= 12

19. Identify which property that justifies each of the following.
= 12
Additive Identity Property

20. Identify which property that justifies each of the following.
5(2 + 9) = (5  2) + (5  9)

21. Identify which property that justifies each of the following.
5(2 + 9) = (5  2) + (5  9)
Distributive Property

22. Identify which property that justifies each of the following.
5 + (2 + 8) = (5 + 2) + 8

23. Identify which property that justifies each of the following.
5 + (2 + 8) = (5 + 2) + 8
Associative Property of Addition

24. Identify which property that justifies each of the following.

25. Identify which property that justifies each of the following.
Multiplicative Inverse Property

26. Identify which property that justifies each of the following.
5  24 = 24  5

27. Identify which property that justifies each of the following.
5  24 = 24  5
Commutative Property of Multiplication

28. Identify which property that justifies each of the following.
= 0

29. Identify which property that justifies each of the following.
= 0
Additive Inverse Property

30. Identify which property that justifies each of the following.
-34 1 = -34

31. Identify which property that justifies each of the following.
-34 1 = -34
Multiplicative Identity Property

32. Least Common Denominator
The least common denominator (LCD) is the smallest number divisible by all the denominators.
Example: The LCD of is 12 because 12 is the smallest number into which 3 and 4 will both divide.

33. Adding Two Fractions
To add two fractions you must first find the LCD. In the problem below the LCD is 12. Then rewrite the two addends as equivalent expressions with the LCD. Then add the numerators and keep the denominator.