1. 1.1
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sets of
Numbers
MATH1003

2. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To be able to determine
to which set a number belongs.
MATH1003

3. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Numbers
MATH1003

4. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Uses of Numbers
• counting
• represent values (such as temperature, etc.)
• represent ratios and percentages
MATH1003

5. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Numbers and Computers
• calculation
• data storage
• graphics
• cryptography and encryption
MATH1003

6. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sets of Numbers
• Mathematicians organize numbers into sets
4
(groups of numbers)
• In this section, we look at and define 4 of those
sets
MATH1003

7. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sets of Numbers
• The 4 sets we will look at are
4
• the set of Natural numbers
• the set of Integers
• the set of Rational numbers
• the set of Real numbers
MATH1003

8. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set N - Natural Numbers
• Natural numbers are used mainly for counting and
N
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
MATH1003

9. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set N - Natural Numbers
• Natural numbers are used mainly for counting and
N
ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set 17 ispositive whole numbers and 0
of
a member of N
• These numbers do not have decimal points
17
MATH1003

10. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set N - Natural Numbers
• Natural numbers are used mainly for counting and
N
ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set of positive whole numbers and 0
-8 is not a
member of N since it
• These numbers do not
is negative have decimal points
MATH1003
X
-8

11. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set N - Natural Numbers
• Natural numbers are used mainly for counting and
N
ordering
• We say that N = {0,not a2, 3, ...}
1,
• N is the set ofit positivesince numbers and 0
1.5 is
member of N whole
has a decimal
• These numbers do not have decimal points
point
1.5
X
MATH1003

12. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set N - Natural Numbers
• Natural numbers are used mainly for counting and
N
ordering
• We say that N = {0, 1, 2, 3, ...}
• N is the set 78 ispositive whole numbers and 0
of
a member of N
• These numbers do not have decimal points
78
MATH1003

13. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• Integers are the positive whole numbers, the
Z
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
MATH1003

14. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• The set of Z includes the set of N
Z
Z N
MATH1003

15. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• The set of Z includes the set of N
Z
Z N
120 is a member of
Z
120
MATH1003

16. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• The set of Z includes the set of N
Z
Z N
-45 is a member of Z
-45
MATH1003

17. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• The set of Z includes the set of N
Z
Z -6.25 is notN
a
member of Z since
it has a decimal
point
-6.25
X
MATH1003

18. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• The set of Z includes the set of N
Z
Z 10.2 is notNa
member of Z since
it has a decimal
point
10.2
X
MATH1003

19. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Z - Integers
• The set of Z includes the set of N
Z
Z N
-98 is a member of Z
-98
MATH1003

20. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Q - Rational Numbers
• Rational numbers are numbers that result in the
Q
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
MATH1003

21. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set Q - Rational Numbers
• -2 is a rational number (-2 = -10/5)
Q
• 8 is a rational number (8 = 16/2)
• The set of N and the set of Z are part of Q
Q Z N
MATH1003

22. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set R - Real Numbers
• Then there are those numbers such as and π
R
• These decimal expressions never stop and never
repeat
MATH1003

23. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set R - Real Numbers
• = 1.41421356237...
R
• π = 3.14159265358979323846…
• These decimal expressions never stop and never
repeat
MATH1003

24. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Set R - Real Numbers
• A Real number is any number that can be written
R
in decimal notation
R Q Z N
• 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
MATH1003

25. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Which set(s) does the number belong?
R Q Z N
-98
R Q Z N
MATH1003

26. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Which set(s) does the number belong?
R Q Z N
2.56
R Q Z N
MATH1003

27. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Which set(s) does the number belong?
R Q Z N
340
R Q Z N
MATH1003

28. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Which set(s) does the number belong?
R Q Z N
-0.6
R Q Z N
MATH1003

29. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Which set(s) does the number belong?
R Q Z N
=3.31662479...
R Q Z N
MATH1003

30. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Which set(s) does the number belong?
R Q Z N
=2
R Q Z N
MATH1003

31. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
positive 100342
whole 2 N 13
numbers 45
102
MATH1003

32. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
-345
negative whole numbers, -2340912 100342
positive whole numbers, Z 0 2 N 13
and 0 -7
45
-54 102
-90872
MATH1003

33. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
a number that results
from the division of two
numbers -10000.002
200.375
1.25
-3.0075 -345
2.333333333 -2340912 100342
Q 9.1 Z 0 2 N 13
-7
-0.00045 45
-54 102
56.124124124 -90872
-3.6666666666
a rational number has
a finite amount of decimal numbers
or has a repeating pattern
MATH1003

34. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
includes the
irrationals π = 3.14159265358979...
(numbers that
go on forever -10000.002
without-89.18362...
any 200.375
repeating 1.25
-3.0075 -345
pattern)
908.17436... 2.333333333 -2340912 100342
R Q Z 0 N 13
9.1 -7 2
-0.00921... -0.00045 45
-54 102
56.124124124 -90872
= 4.358... -3.6666666666
= 2.645751311...
MATH1003

35. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider
whether a number is an Integer or a Real
-7
Integer Real
MATH1003

36. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider
whether a number is an Integer or a Real
8.96
Integer Real
MATH1003

37. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider
whether a number is an Integer or a Real
890345
Integer Real
MATH1003

38. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
For our purposes, we’ll only need to consider
whether a number is an Integer or a Real
-0.78612
Integer Real
MATH1003

39. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
What would we need to represent the
following values?
number of DVDs in a collection
Integer Real
MATH1003

40. 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

41. 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

42. 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

43. 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

44. 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
MATH1003