2 1 1 real numbers
TRANSCRIPT
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 1/44
Real Numbers
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 2/44
Two Kinds of Real Numbers
� Rational Numbers
� Irrational Numbers
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 3/44
Two Kinds of Real Numbers
� Rational Numbers
� Irrational Numbers
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 4/44
Rational Numbers
� A rational number is a real
number that can be written as a
ratio of two integers.
� A rational number written in
decimal form is terminating or repeating.
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 5/44
Examples of Rational
Numbers
�16
�1/2
�3.56
�-8
�1.3333«
�- 3/4
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 6/44
Irrational Numbers
� An irrational number is a number that cannot be written as a ratio of
two integers.� Irrational numbers written as
decimals are non-terminating and
non-repeating.
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 7/44
Natural Numbers
� Natural numbers: 1,2,3«
� By any other name Counting numbers
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 8/44
Whole Numbers
� Just add 0 in the mix
� Whole numbers: 0, 1, 2, 3«
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 9/44
Integers
� Natural numbers, their opposites and zero
� Integers: «-3, -2, -1, 0,1,2,3«
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 10/44
Some of the Numbers In-
between
� Rational numbers: 2/3, -31/2 0.3333«
± Can be written as a ratio of two integers.
± Repeating decimals, terminating decimals
and fractions
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 11/44
Some of the Numbers In-
between
� Irrational Numbers
� 3. 45455455545555« has a pattern butdoesn¶t repeat. It isn¶t rational.
� It can¶t be written like a fraction.
� Square root of 2, Pi and e are alsoIRRATIONAL
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 12/44
Properties of Real Numbers
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 13/44
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 -2 -100 and 100 and 15 15
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 14/44
Reciprocals
Two numbers whose product is 1 are
reciprocals of each other.
Examples of Reciprocals:
5
4
4
5and
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 15/44
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:
x
!5 5 3
7
3
7!
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 16/44
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
Addition2 + 3 = 3 + 2 , (-5) + 4 = 4 + (-5)
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 17/44
Commutative Property of
Multiplication
a v b = b v a
When multiplying two numbers, the order
of the numbers does not matter.
Examples of the Commutative Property of
Multiplication2 v 3 = 3 v 2 (-3) v 24 = 24 v (-3)
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 18/44
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)
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 19/44
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 v (3 v 5) = (2 v 3) v 5
(4 v 2) v 6 = 4 v (2 v 6)
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 20/44
Distributive Property
a(b + c) = ab + ac
Multiplication distributes over addition.
Examples of the Distributive Property
2 (3 + 5) = (2 v 3) + (2 v 5)
(4 + 2) v 6 = (4 v 6) + (2 v 6)
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 21/44
Additive Identity Property
The additive identity property states that if
0 is added to a number, the result is that
number.
Example: 3 + 0 = 0 + 3 = 3
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 22/44
Multiplicative Identity Property
The multiplicative identity property states
that if a number is multiplied by 1, the
result is that number.
Example: 5 v 1 = 1 v 5 = 5
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 23/44
Additive Inverse Property
The additive inverse property states that
opposites add to zero.
7 + (-7) = 0 and -4 + 4 = 0
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 24/44
Multiplicative Inverse
Property
The multiplicative inverse property states
that reciprocals multiply to 1.
51
51v !
2
3
3
21v !
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 25/44
Identify which property that
justifies each of the following.
4 v (8 v 2) = (4 v 8) v 2
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 26/44
Identify which property that
justifies each of the following.
4 v (8 v 2) = (4 v 8) v 2
Associative Property of Multiplication
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 27/44
Identify which property that
justifies each of the following.
6 + 8 = 8 + 6
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 28/44
Identify which property that
justifies each of the following.
6 + 8 = 8 + 6
Commutative Property of Addition
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 29/44
Identify which property that
justifies each of the following.
12 + 0 = 12
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 30/44
Identify which property that
justifies each of the following.
12 + 0 = 12
Additive Identity Property
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 31/44
Identify which property that
justifies each of the following.
5(2 + 9) = (5 v 2) + (5 v 9)
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 32/44
Identify which property that
justifies each of the following.
5(2 + 9) = (5 v 2) + (5 v 9)
Distributive Property
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 33/44
Identify which property that
justifies each of the following.
5 + (2 + 8) = (5 + 2) + 8
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 34/44
Identify which property that
justifies each of the following.
5 + (2 + 8) = (5 + 2) + 8
Associative Property of Addition
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 35/44
Identify which property that
justifies each of the following.
5
9
9
5 1v !
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 36/44
Identify which property that
justifies each of the following.
Multiplicative Inverse Property
5
9
9
5 1v !
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 37/44
Identify which property that
justifies each of the following.
5 v 24 = 24 v 5
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 38/44
Identify which property that
justifies each of the following.
5 v 24 = 24 v 5
Commutative Property of Multiplication
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 39/44
Identify which property that
justifies each of the following.
18 + -18 = 0
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 40/44
Identify which property that
justifies each of the following.
18 + -18 = 0
Additive Inverse Property
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 41/44
Identify which property that
justifies each of the following.
-34 v1 = -34
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 42/44
Identify which property that
justifies each of the following.
-34 v1 = -34
Multiplicative Identity Property
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 43/44
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.
2
3
5
4and
8/6/2019 2 1 1 Real Numbers
http://slidepdf.com/reader/full/2-1-1-real-numbers 44/44
Adding Two Fractions
3
4
5
6
9
12
10
12
19
12 ! !
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.