PROPERTIES OF REAL NUMBERS 1 ¾.215 -7PI. Subsets of real numbers – REVIEW Natural numbers numbers...
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Transcript of PROPERTIES OF REAL NUMBERS 1 ¾.215 -7PI. Subsets of real numbers – REVIEW Natural numbers numbers...
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
Rational numbers Irrational numbers
Integers
Whole numbers
Natural numbers
THE REAL 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
-1.414… .333……… -2.25
-3 -2 -1 0 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 - .1 = -.316227766017
So: - .1 < - .08 or - .08 > - .1
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 1/5 . 002 4/9
Reciprocal or multiplicative inverse
product of reciprocals equal 1
Examples:
400 4 1/5 - .002 - 4/9
1/400
Multiplicative inverse of any number a is 1/a
5/21 - 500 - 9/4
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
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
(3*4)5 = 60 3(4*5) = 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
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 =
0 =
-1 * -2 =
4
0
2
Assignment
Page 8 – 9
Problems
34 – 60 even