1. PROPERTIES OF
REAL NUMBERS 1
.215 -7 PI

2. Subsets of real numbers – REVIEW
Natural numbers
numbers used for counting
1, 2, 3, 4, 5, ….
Whole numbers
the natural numbers plus zero
0, 1, 2, 3, 4, 5, …
Integers
the natural numbers ( positive integers ), zero, plus the negative integers

3. …,-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Rational numbers
numbers that can be written as fractions
decimal representations can either terminate
or repeat
Examples:
fractions: 7/5 -3/ /5
Any whole number can be written as a fraction by placing it over the number 1
8 = 8/ = 100/1

4. terminating decimals
¼ = /5 = .4
Repeating decimals
1/3 = /3 = .6
These will always have a bar over the repeating section.
Irrational numbers
Cannot be written as fractions
Decimal representations do not terminate or repeat

5. if the positive rational number is not a perfect square, then its square root is irrational
Examples:
Pi - non-repeating decimal
2 - not a perfect square

6. THE REAL NUMBERS
Rational numbers
Irrational numbers
Integers
Whole numbers
Natural numbers

7. Graphing on a number line¼
Tip: Best to put them as all decimals
Put the square root in the calculator and find its equivalent
-1.414… ………

8. Ordering numbers
Use the < , >, and = symbols
Compare and
Here again for square roots put them in the calculator and get their equivalents
.08 = =
So: < or >

9. Properties of Real Numbers
Opposite or additive inverse
sum of opposites or additive inverses is 0
Examples:
/ /9
-400
Additive inverse of any number a is -a
4/9
- 4 1/5
. 002

10. Reciprocal or multiplicative inverse
product of reciprocals equal 1
Examples:
4 1/ /9
1/400
Multiplicative inverse of any number a is 1/a
- 9/4
5/21
- 500

11. Other Properties:
Addition:
Closure a + b is a real number
Commutative a + b = b + a
4 + 3 = 7` = 7
numbers can be moved in addition
Associative (a + b) + c = a + (b + c)
(1 + 2) + 3 = (2 + 3) = 6
3 + 3 = = 6
the order in which we add the numbers
does not matter in addition

12. Identity a + 0 = a
7 + 0 = 7
when you add nothing to a number you
still only have that number
Inverse a + -a = 0
= 0

13. Multiplication
Closure ab is a real number
Commutative ab = ba
6(4) = (6) = 24
When multiplying the numbers may be
switched around, will not affect product
Associative (ab)c = a(bc)
The order in which they are multiplied
does not affect the outcome of the product

14. (3*4)5 = (4*5) = 60
12(5) = (20) = 60
Identity a * 1 = a
One times any number is the number itself
7 * 1 = 7
Inverse a * 1/a = 1
Product of reciprocals is one
7 * 1/7 = 7/7 = 1

15. DISTRIBUTIVE Property
Combines addition and multiplication
a(b + c) = ab + ac
2(3 + 4) = 2(3) + 2(4)
6 + 8 14

16. ABSOLUTE VALUE
Absolute value is its distance from zero on the number line.
Absolute value is always positive because distance is always positive
Examples: = =
-1 * -2 = 4 2

17. Assignment
Page 8 – 9
Problems
34 – 60 even