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MATH1003 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001 1.1 Sets of Numbers

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Page 1: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

1.1Sets ofNumbers

Page 2: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Goal

To be able to determineto which set a number belongs.

Page 3: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Numbers

Page 4: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Uses of Numbers

• counting

• represent values (such as temperature, etc.)

• represent ratios and percentages

Page 5: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Numbers and Computers

• calculation

• data storage

• graphics

• cryptography and encryption

Page 6: Math1003 1.1 - Sets of Numbers

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

Page 7: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

Page 8: Math1003 1.1 - Sets of Numbers

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

Page 9: Math1003 1.1 - Sets of Numbers

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

1717 is a member of N

Page 10: Math1003 1.1 - Sets of Numbers

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

-8X-8 is not a

member of N since it is negative

Page 11: Math1003 1.1 - Sets of Numbers

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

1.5X1.5 is not a

member of N since it has a decimal

point

Page 12: Math1003 1.1 - Sets of Numbers

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

Page 13: Math1003 1.1 - Sets of Numbers

MATH1003

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

Page 14: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Set Z - Integers

Z• The set of Z includes the set of N

Z N

Page 15: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Set Z - Integers

Z• The set of Z includes the set of N

Z N

120120 is a member of

Z

Page 16: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Set Z - Integers

Z• The set of Z includes the set of N

Z N

-45-45 is a member of Z

Page 17: Math1003 1.1 - Sets of Numbers

MATH1003

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

Page 18: Math1003 1.1 - Sets of Numbers

MATH1003

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

Page 19: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Set Z - Integers

Z• The set of Z includes the set of N

Z N

-98-98 is a member of Z

Page 20: Math1003 1.1 - Sets of Numbers

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)

Page 21: Math1003 1.1 - Sets of Numbers

MATH1003

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

Page 22: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Set R - Real Numbers

R• Then there are those numbers such as and π

• These decimal expressions never stop and never repeat

Page 23: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Set R - Real Numbers

R• = 1.41421356237...

• π = 3.14159265358979323846…

• These decimal expressions never stop and never repeat

Page 24: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

Page 25: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

R Q Z N

-98NZQR

Which set(s) does the number belong?

Page 26: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

R Q Z N

2.56NZQR

Which set(s) does the number belong?

Page 27: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

R Q Z N

340NZQR

Which set(s) does the number belong?

Page 28: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

R Q Z N

-0.6NZQR

Which set(s) does the number belong?

Page 29: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

R Q Z N

NZQR

Which set(s) does the number belong?

=3.31662479...

Page 30: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

R Q Z N

NZQR=2

Which set(s) does the number belong?

Page 31: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

N245

102

13100342positive

whole numbers

Page 32: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

Z N245

102

13100342

-7

-90872-54

-345

0-2340912negative whole numbers,

positive whole numbers,and 0

Page 33: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

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

Page 34: Math1003 1.1 - Sets of Numbers

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)

Page 35: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

For our purposes, we’ll only need to consider whether a number is an Integer or a Real

-7Integer Real

Page 36: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

For our purposes, we’ll only need to consider whether a number is an Integer or a Real

8.96Integer Real

Page 37: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

For our purposes, we’ll only need to consider whether a number is an Integer or a Real

890345Integer Real

Page 38: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

For our purposes, we’ll only need to consider whether a number is an Integer or a Real

-0.78612Integer Real

Page 39: Math1003 1.1 - Sets of Numbers

MATH1003

10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001

What would we need to represent the following values?

number of DVDs in a collection

Integer Real

Page 40: Math1003 1.1 - Sets of Numbers

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

Page 41: Math1003 1.1 - Sets of Numbers

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

Page 42: Math1003 1.1 - Sets of Numbers

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

Page 43: Math1003 1.1 - Sets of Numbers

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

Page 44: Math1003 1.1 - Sets of Numbers

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


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