MATH1003 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001 1.1 Sets of Numbers
Jun 23, 2015
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1.1Sets ofNumbers
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To be able to determineto which set a number belongs.
MATH1003
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Numbers
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Uses of Numbers
• counting
• represent values (such as temperature, etc.)
• represent ratios and percentages
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Numbers and Computers
• calculation
• data storage
• graphics
• cryptography and encryption
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sets of Numbers
• Mathematicians organize numbers into sets (groups of numbers)
• In this section, we look at and define 4 of those sets 4
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4Sets of Numbers
• The 4 sets we will look at are
• the set of Natural numbers
• the set of Integers
• the set of Rational numbers
• the set of Real numbers
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NSet N - Natural Numbers
• Natural numbers are used mainly for counting and ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set of positive whole numbers and 0
• These numbers do not have decimal points
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NSet N - Natural Numbers
• Natural numbers are used mainly for counting and ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set of positive whole numbers and 0
• These numbers do not have decimal points
1717 is a member of N
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
NSet N - Natural Numbers
• Natural numbers are used mainly for counting and ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set of positive whole numbers and 0
• These numbers do not have decimal points
-8X-8 is not a
member of N since it is negative
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
NSet N - Natural Numbers
• Natural numbers are used mainly for counting and ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set of positive whole numbers and 0
• These numbers do not have decimal points
1.5X1.5 is not a
member of N since it has a decimal
point
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
NSet N - Natural Numbers
• Natural numbers are used mainly for counting and ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set of positive whole numbers and 0
• These numbers do not have decimal points
7878 is a member of N
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
Z• Integers are the positive whole numbers, the
negative whole numbers, and 0
• They do not have decimal points
• We say Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• The set of N is a part of the set of Z
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Set Z - Integers
Z• The set of Z includes the set of N
Z N
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Set Z - Integers
Z• The set of Z includes the set of N
Z N
120120 is a member of
Z
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Set Z - Integers
Z• The set of Z includes the set of N
Z N
-45-45 is a member of Z
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
Z• The set of Z includes the set of N
Z N
-6.25X-6.25 is not a
member of Z since it has a decimal
point
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
Z• The set of Z includes the set of N
Z N
10.2X10.2 is not a
member of Z since it has a decimal
point
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Set Z - Integers
Z• The set of Z includes the set of N
Z N
-98-98 is a member of Z
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Q - Rational Numbers
Q• Rational numbers are numbers that result in the
division of two numbers
• 0.5 is a rational number (0.5 = 1/2)
• 0.1 is a rational number (0.1 = 1/10)
• 1.9375 is a rational number (1.9375 = 31/16)
• -5.667 is a rational number (-5.667 = -17/3)
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10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Q - Rational Numbers
Q• -2 is a rational number (-2 = -10/5)
• 8 is a rational number (8 = 16/2)
• The set of N and the set of Z are part of Q
Q Z N
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Set R - Real Numbers
R• Then there are those numbers such as and π
• These decimal expressions never stop and never repeat
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Set R - Real Numbers
R• = 1.41421356237...
• π = 3.14159265358979323846…
• These decimal expressions never stop and never repeat
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Set R - Real Numbers
R• A Real number is any number that can be written
in decimal notation
• Since -2 can be written as -2.0, then all Integers are part of the set of Real numbers
• And if the set of Z is part of R, then the set of N is also part of R
R Q Z N
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R Q Z N
-98NZQR
Which set(s) does the number belong?
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R Q Z N
2.56NZQR
Which set(s) does the number belong?
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
R Q Z N
340NZQR
Which set(s) does the number belong?
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
R Q Z N
-0.6NZQR
Which set(s) does the number belong?
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
R Q Z N
NZQR
Which set(s) does the number belong?
=3.31662479...
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R Q Z N
NZQR=2
Which set(s) does the number belong?
MATH1003
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N245
102
13100342positive
whole numbers
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Z N245
102
13100342
-7
-90872-54
-345
0-2340912negative whole numbers,
positive whole numbers,and 0
MATH1003
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Q Z N245
102
13100342
-7
-90872-54
-345
0-2340912
1.25200.375
-0.00045
-10000.002
-3.6666666666
-3.00752.333333333
56.124124124
9.1
a number that results from the division of two
numbers
a rational number hasa finite amount of decimal numbers
or has a repeating pattern
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
R Q Z N245
102
13100342
-7
-90872-54
-345
0-2340912
1.25200.375
-0.00045
-10000.002
-3.6666666666
-3.00752.333333333
56.124124124
9.1
π = 3.14159265358979...
= 2.645751311...
= 4.358...
-89.18362...
-0.00921...
908.17436...
includes the irrationals
(numbers that go on forever without any repeating pattern)
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
-7Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
8.96Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
890345Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
-0.78612Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
What would we need to represent the following values?
number of DVDs in a collection
Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
height of a building
Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
amount of money in a bank account
Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
price of a book
Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
players on a team
Integer Real
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider whether a number is an Integer or a Real
average weight of the students in our class
Integer Real