1. Number Systems
Location in
course textbook

Chapt. 2
ITEC 1011

Introduction to Information Technologies
Common Number Systems
System

Base Symbols

Used by
humans?

Used in
computers?

Decimal

10

0, 1, … 9

Yes

No

Binary

...
Quantities/Counting (1 of 3)
Decimal

Binary

HexaOctal decimal

0

0

0

0

1

1

1

1

2

10

2

2

3

11

3

3

4

100
...
Quantities/Counting (2 of 3)
Decimal

Binary

HexaOctal decimal

8

10

8

9

1001

11

9

10

1010

12

A

11

1011

13

...
Quantities/Counting (3 of 3)
Decimal

Binary

HexaOctal decimal

16

20

10

17

10001

21

11

18

10010

22

12

19

100...
Conversion Among Bases
• The possibilities:
Decimal

Octal

Binary

Hexadecimal
pp. 40-46

ITEC 1011

Introduction to Info...
Quick Example

2510 = 110012 = 318 = 1916
Base

ITEC 1011

Introduction to Information Technologies
Decimal to Decimal (just for fun)

Decimal

Octal

Binary

Hexadecimal
Next slide…

ITEC 1011

Introduction to Information...
Weight
12510 =>

5 x 100
2 x 101
1 x 102

=
5
= 20
= 100
125

Base

ITEC 1011

Introduction to Information Technologies
Binary to Decimal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Binary to Decimal
• Technique
– Multiply each bit by 2n, where n is the “weight”
of the bit
– The weight is the position o...
Example
Bit “0”
1010112 =>

1
1
0
1
0
1

x
x
x
x
x
x

20
21
22
23
24
25

=
=
=
=
=
=

1
2
0
8
0
32
4310

ITEC 1011

Introd...
Octal to Decimal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Octal to Decimal
• Technique
– Multiply each bit by 8n, where n is the “weight”
of the bit
– The weight is the position of...
Example

7248 =>

ITEC 1011

4 x 80 =
2 x 81 =
7 x 82 =

4
16
448
46810

Introduction to Information Technologies
Hexadecimal to Decimal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Hexadecimal to Decimal
• Technique
– Multiply each bit by 16n, where n is the
“weight” of the bit
– The weight is the posi...
Example

ABC16 =>

C x 160 = 12 x
1 =
12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810

ITEC 1011

Introduction...
Decimal to Binary
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Decimal to Binary
• Technique
– Divide by two, keep track of the remainder
– First remainder is bit 0 (LSB, least-signific...
Example
12510 = ?2

2 125
2 62
2 31
2 15
7
2
3
2
1
2
0

1
0
1
1
1
1
1

12510 = 11111012
ITEC 1011

Introduction to Informa...
Octal to Binary
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent
binary representation

ITEC 1011

Introductio...
Example
7058 = ?2

7

0

5

111 000 101

7058 = 1110001012
ITEC 1011

Introduction to Information Technologies
Hexadecimal to Binary
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Hexadecimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit
equivalent binary representation

ITEC 1011
...
Example
10AF16 = ?2

1

0

A

F

0001 0000 1010 1111

10AF16 = 00010000101011112
ITEC 1011

Introduction to Information Te...
Decimal to Octal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Decimal to Octal
• Technique
– Divide by 8
– Keep track of the remainder

ITEC 1011

Introduction to Information Technolog...
Example
123410 = ?8
8
8
8
8

1234
154
19
2
0

2
2
3
2

123410 = 23228
ITEC 1011

Introduction to Information Technologies
Decimal to Hexadecimal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Decimal to Hexadecimal
• Technique
– Divide by 16
– Keep track of the remainder

ITEC 1011

Introduction to Information Te...
Example
123410 = ?16
16
16
16

1234
77
4
0

2
13 = D
4

123410 = 4D216
ITEC 1011

Introduction to Information Technologies
Binary to Octal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits

ITEC 1011

Introduction t...
Example
10110101112 = ?8

1 011 010 111

1

3

2

7

10110101112 = 13278
ITEC 1011

Introduction to Information Technologi...
Binary to Hexadecimal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Binary to Hexadecimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits

ITEC 1011

Int...
Example
10101110112 = ?16

10 1011 1011

2

B

B

10101110112 = 2BB16
ITEC 1011

Introduction to Information Technologies
Octal to Hexadecimal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Octal to Hexadecimal
• Technique
– Use binary as an intermediary

ITEC 1011

Introduction to Information Technologies
Example
10768 = ?16
1

0

7

6

001

000

111

110

2

3

E

10768 = 23E16
ITEC 1011

Introduction to Information Technolo...
Hexadecimal to Octal
Decimal

Binary

ITEC 1011

Octal

Hexadecimal

Introduction to Information Technologies
Hexadecimal to Octal
• Technique
– Use binary as an intermediary

ITEC 1011

Introduction to Information Technologies
Example
1F0C16 = ?8
1

F

0

C

0001

1111

0000

1100

1

7

4

1

4

1F0C16 = 174148
ITEC 1011

Introduction to Informat...
Exercise – Convert ...
Decimal

Binary

Octal

Hexadecimal

33
1110101
703
1AF
Don’t use a calculator!
Skip answer
ITEC 10...
Exercise – Convert …
Answer

Decimal

Octal

33

100001

41

21

117
451
431

ITEC 1011

Binary

Hexadecimal

1110101
1110...
Common Powers (1 of 2)
• Base 10
Power

Symbol

Value

10-12

pico

p

.000000000001

10-9

nano

n

.000000001

10-6

mic...
Common Powers (2 of 2)
• Base 2
Power

Preface

Symbol

Value

210

kilo

k

1024

220

mega

M

1048576

230

Giga

G

10...
Example
In the lab…
1. Double click on My Computer
2. Right click on C:
3. Click on Properties

/ 230 =

ITEC 1011

Introd...
Exercise – Free Space
• Determine the “free space” on all drives on
a machine in the lab
Free space
Drive

Bytes

GB

A:
C...
Review – multiplying powers
• For common bases, add powers
ab × ac = ab+c

26 × 210 = 216 = 65,536
or…
26 × 210 = 64 × 210...
Binary Addition (1 of 2)
• Two 1-bit values
A
0
0
1
1

B
0
1
0
1

A+B
0
1
1
10

“two”
pp. 36-38

ITEC 1011

Introduction t...
Binary Addition (2 of 2)
• Two n-bit values
– Add individual bits
– Propagate carries
– E.g.,
1

1

10101
+ 11001
101110

...
Multiplication (1 of 3)
• Decimal (just for fun)
35
x 105
175
000
35
3675
pp. 39
ITEC 1011

Introduction to Information Te...
Multiplication (2 of 3)
• Binary, two 1-bit values
A
0
0
1
1

ITEC 1011

B
0
1
0
1

A×B
0
0
0
1

Introduction to Informati...
Multiplication (3 of 3)
• Binary, two n-bit values
– As with decimal values
– E.g.,
1110
x 1011
1110
1110
0000
1110
100110...
Fractions
• Decimal to decimal (just for fun)

3.14 =>

4 x 10-2 = 0.04
1 x 10-1 = 0.1
3 x 100 = 3
3.14

pp. 46-50
ITEC 10...
Fractions
• Binary to decimal
10.1011 =>

1
1
0
1
0
1

x
x
x
x
x
x

2-4
2-3
2-2
2-1
20
21

=
=
=
=
=
=

0.0625
0.125
0.0
0...
Fractions
• Decimal to binary
3.14579

11.001001...

.14579
x
2
0.29158
x
2
0.58316
x
2
1.16632
x
2
0.33264
x
2
0.66528
x
...
Exercise – Convert ...
Decimal

Binary

Octal

Hexadecimal

29.8
101.1101
3.07
C.82
Don’t use a calculator!
Skip answer
IT...
Exercise – Convert …
Answer

Decimal
29.8
5.8125
3.109375
12.5078125

ITEC 1011

Binary

Octal

11101.110011… 35.63…
101.1...
Thank you

Next topic
ITEC 1011

Introduction to Information Technologies
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01.number systems

  1. 1. 1. Number Systems Location in course textbook Chapt. 2 ITEC 1011 Introduction to Information Technologies
  2. 2. Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexadecimal 16 0, 1, … 9, A, B, … F No No ITEC 1011 Introduction to Information Technologies
  3. 3. Quantities/Counting (1 of 3) Decimal Binary HexaOctal decimal 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 p. 33 ITEC 1011 Introduction to Information Technologies
  4. 4. Quantities/Counting (2 of 3) Decimal Binary HexaOctal decimal 8 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 ITEC 1011 1000 1111 17 F Introduction to Information Technologies
  5. 5. Quantities/Counting (3 of 3) Decimal Binary HexaOctal decimal 16 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 20 10100 24 14 21 10101 25 15 22 10110 26 16 23 ITEC 1011 10000 10111 27 17 Introduction to Information Technologies Etc.
  6. 6. Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal pp. 40-46 ITEC 1011 Introduction to Information Technologies
  7. 7. Quick Example 2510 = 110012 = 318 = 1916 Base ITEC 1011 Introduction to Information Technologies
  8. 8. Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide… ITEC 1011 Introduction to Information Technologies
  9. 9. Weight 12510 => 5 x 100 2 x 101 1 x 102 = 5 = 20 = 100 125 Base ITEC 1011 Introduction to Information Technologies
  10. 10. Binary to Decimal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  11. 11. Binary to Decimal • Technique – Multiply each bit by 2n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results ITEC 1011 Introduction to Information Technologies
  12. 12. Example Bit “0” 1010112 => 1 1 0 1 0 1 x x x x x x 20 21 22 23 24 25 = = = = = = 1 2 0 8 0 32 4310 ITEC 1011 Introduction to Information Technologies
  13. 13. Octal to Decimal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  14. 14. Octal to Decimal • Technique – Multiply each bit by 8n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results ITEC 1011 Introduction to Information Technologies
  15. 15. Example 7248 => ITEC 1011 4 x 80 = 2 x 81 = 7 x 82 = 4 16 448 46810 Introduction to Information Technologies
  16. 16. Hexadecimal to Decimal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  17. 17. Hexadecimal to Decimal • Technique – Multiply each bit by 16n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results ITEC 1011 Introduction to Information Technologies
  18. 18. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810 ITEC 1011 Introduction to Information Technologies
  19. 19. Decimal to Binary Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  20. 20. Decimal to Binary • Technique – Divide by two, keep track of the remainder – First remainder is bit 0 (LSB, least-significant bit) – Second remainder is bit 1 – Etc. ITEC 1011 Introduction to Information Technologies
  21. 21. Example 12510 = ?2 2 125 2 62 2 31 2 15 7 2 3 2 1 2 0 1 0 1 1 1 1 1 12510 = 11111012 ITEC 1011 Introduction to Information Technologies
  22. 22. Octal to Binary Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  23. 23. Octal to Binary • Technique – Convert each octal digit to a 3-bit equivalent binary representation ITEC 1011 Introduction to Information Technologies
  24. 24. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012 ITEC 1011 Introduction to Information Technologies
  25. 25. Hexadecimal to Binary Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  26. 26. Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation ITEC 1011 Introduction to Information Technologies
  27. 27. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112 ITEC 1011 Introduction to Information Technologies
  28. 28. Decimal to Octal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  29. 29. Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder ITEC 1011 Introduction to Information Technologies
  30. 30. Example 123410 = ?8 8 8 8 8 1234 154 19 2 0 2 2 3 2 123410 = 23228 ITEC 1011 Introduction to Information Technologies
  31. 31. Decimal to Hexadecimal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  32. 32. Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder ITEC 1011 Introduction to Information Technologies
  33. 33. Example 123410 = ?16 16 16 16 1234 77 4 0 2 13 = D 4 123410 = 4D216 ITEC 1011 Introduction to Information Technologies
  34. 34. Binary to Octal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  35. 35. Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits ITEC 1011 Introduction to Information Technologies
  36. 36. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278 ITEC 1011 Introduction to Information Technologies
  37. 37. Binary to Hexadecimal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  38. 38. Binary to Hexadecimal • Technique – Group bits in fours, starting on right – Convert to hexadecimal digits ITEC 1011 Introduction to Information Technologies
  39. 39. Example 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16 ITEC 1011 Introduction to Information Technologies
  40. 40. Octal to Hexadecimal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  41. 41. Octal to Hexadecimal • Technique – Use binary as an intermediary ITEC 1011 Introduction to Information Technologies
  42. 42. Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16 ITEC 1011 Introduction to Information Technologies
  43. 43. Hexadecimal to Octal Decimal Binary ITEC 1011 Octal Hexadecimal Introduction to Information Technologies
  44. 44. Hexadecimal to Octal • Technique – Use binary as an intermediary ITEC 1011 Introduction to Information Technologies
  45. 45. Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148 ITEC 1011 Introduction to Information Technologies
  46. 46. Exercise – Convert ... Decimal Binary Octal Hexadecimal 33 1110101 703 1AF Don’t use a calculator! Skip answer ITEC 1011 Introduction to Information Technologies Answer
  47. 47. Exercise – Convert … Answer Decimal Octal 33 100001 41 21 117 451 431 ITEC 1011 Binary Hexadecimal 1110101 111000011 110101111 165 703 657 75 1C3 1AF Introduction to Information Technologies
  48. 48. Common Powers (1 of 2) • Base 10 Power Symbol Value 10-12 pico p .000000000001 10-9 nano n .000000001 10-6 micro µ .000001 10-3 milli m .001 103 kilo k 1000 106 mega M 1000000 109 giga G 1000000000 1012 ITEC 1011 Preface tera T 1000000000000 Introduction to Information Technologies
  49. 49. Common Powers (2 of 2) • Base 2 Power Preface Symbol Value 210 kilo k 1024 220 mega M 1048576 230 Giga G 1073741824 • What is the value of “k”, “M”, and “G”? • In computing, particularly w.r.t. memory, the base-2 interpretation generally applies ITEC 1011 Introduction to Information Technologies
  50. 50. Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 230 = ITEC 1011 Introduction to Information Technologies
  51. 51. Exercise – Free Space • Determine the “free space” on all drives on a machine in the lab Free space Drive Bytes GB A: C: D: E: etc. ITEC 1011 Introduction to Information Technologies
  52. 52. Review – multiplying powers • For common bases, add powers ab × ac = ab+c 26 × 210 = 216 = 65,536 or… 26 × 210 = 64 × 210 = 64k ITEC 1011 Introduction to Information Technologies
  53. 53. Binary Addition (1 of 2) • Two 1-bit values A 0 0 1 1 B 0 1 0 1 A+B 0 1 1 10 “two” pp. 36-38 ITEC 1011 Introduction to Information Technologies
  54. 54. Binary Addition (2 of 2) • Two n-bit values – Add individual bits – Propagate carries – E.g., 1 1 10101 + 11001 101110 ITEC 1011 21 + 25 46 Introduction to Information Technologies
  55. 55. Multiplication (1 of 3) • Decimal (just for fun) 35 x 105 175 000 35 3675 pp. 39 ITEC 1011 Introduction to Information Technologies
  56. 56. Multiplication (2 of 3) • Binary, two 1-bit values A 0 0 1 1 ITEC 1011 B 0 1 0 1 A×B 0 0 0 1 Introduction to Information Technologies
  57. 57. Multiplication (3 of 3) • Binary, two n-bit values – As with decimal values – E.g., 1110 x 1011 1110 1110 0000 1110 10011010 ITEC 1011 Introduction to Information Technologies
  58. 58. Fractions • Decimal to decimal (just for fun) 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14 pp. 46-50 ITEC 1011 Introduction to Information Technologies
  59. 59. Fractions • Binary to decimal 10.1011 => 1 1 0 1 0 1 x x x x x x 2-4 2-3 2-2 2-1 20 21 = = = = = = 0.0625 0.125 0.0 0.5 0.0 2.0 2.6875 pp. 46-50 ITEC 1011 Introduction to Information Technologies
  60. 60. Fractions • Decimal to binary 3.14579 11.001001... .14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. p. 50 ITEC 1011 Introduction to Information Technologies
  61. 61. Exercise – Convert ... Decimal Binary Octal Hexadecimal 29.8 101.1101 3.07 C.82 Don’t use a calculator! Skip answer ITEC 1011 Introduction to Information Technologies Answer
  62. 62. Exercise – Convert … Answer Decimal 29.8 5.8125 3.109375 12.5078125 ITEC 1011 Binary Octal 11101.110011… 35.63… 101.1101 5.64 11.000111 1100.10000010 3.07 14.404 Introduction to Information Technologies Hexadecimal 1D.CC… 5.D 3.1C C.82
  63. 63. Thank you Next topic ITEC 1011 Introduction to Information Technologies

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