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PROPERTIES
OF
REAL NUMBERS
1
¾ .215 -7
PI
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
…,-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 -3/2
-4/5
Any whole number can be written as a fraction by
placing it over the number 1
8 = 8/1
100 = 100/1
terminating decimals
¼ = .25
2/5 = .4
Repeating decimals
1/3 = .3
2/3 = .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
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
THE REAL NUMBERS
Rational numbers
Integers
Whole numbers
Natural numbers
Irrational numbers
Graphing on a number line
- 2
.3
-2 ¼
Tip: Best to put them as all decimals
Put the square root in the calculator and find its
equivalent
.333………
-1.414…
-3
-2
-1
0
-2.25
1
2
3
Ordering numbers
Use the < , >, and = symbols
Compare - .08 and - .1
Here again for square roots put them in the calculator
and get their equivalents
-.08 = -.282842712475
So:
- .1 < - .08
- .1 = -.316227766017
or
- .08 > - .1
Properties of Real Numbers
Opposite or additive inverse
sum of opposites or additive inverses is 0
Examples:
400
4 1/5
-400
- 4 1/5
- .002
. 002
Additive inverse of any number a is -a
- 4/9
4/9
Reciprocal or multiplicative inverse
product of reciprocals equal 1
Examples:
400
4 1/5
- .002
1/400
5/21
- 500
- 4/9
- 9/4
Multiplicative inverse of any number a is 1/a
Other Properties:
Addition:
Closure
a + b is a real number
Commutative
a+b=b+a
4 + 3 = 7`
3+4=7
numbers can be moved in addition
Associative
(a + b) + c = a + (b + c)
(1 + 2) + 3 = 6
3+3=6
1+ (2 + 3) = 6
1+5=6
the order in which we add the numbers
does not matter in addition
Identity
a+0=a
7+0=7
when you add nothing to a number you
still only have that number
Inverse
a + -a = 0
7 + -7 = 0
Multiplication
Closure
ab is a real number
Commutative
ab = ba
6(4) = 24
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
Identity
(3*4)5 = 60
3(4*5) = 60
12(5) = 60
3(20) = 60
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
DISTRIBUTIVE Property
Combines addition and multiplication
a(b + c) = ab + ac
2(3 + 4)
= 2(3) + 2(4)
6+8
14
ABSOLUTE VALUE
Absolute value is its distance from zero on the
number line.
Absolute value is always positive because distance is
always positive
Examples:
-4
= 4
0
= 0
-1 * -2 = 2
Assignment
Page 8 – 9
Problems
34 – 60 even