PROPERTIES

1
PROPERTIES OF REAL NUMBERS
¾
.215
-7
PI
1
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
-3/2 -4/5 Any whole number can be written
as a fraction by placing it over the number 1 8
8/1 100 100/1
4
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
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 line - 2 .3
-2 ¼ Tip Best to put them as all
decimals Put the square root in the
calculator and find its equivalent -1.414
.333 -2.25
-3 -2 -1 0 1
2 3
8

• Ordering numbers
• Use the lt , gt, and symbols
• Compare - .08 and - .1
• Here again for square roots put them in the
  calculator and get their equivalents
• .08 -.282842712475 - .1
  -.316227766017
• So - .1 lt - .08 or - .08
  gt - .1

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

- 9/4
5/21
- 500
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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 1 (2 3)
6 3 3 6 1 5
6 the order in which we add the
numbers does not matter in addition
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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
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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
14
(34)5 60 3(45) 60
12(5) 60 3(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
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DISTRIBUTIVE Property Combines addition and
multiplication a(b c) ab ac
2(3 4) 2(3) 2(4)
6 8 14
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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 0
-1 -2
4
0
2
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Assignment Page 8 9 Problems 34 60 even