Computer Number Systems

and Data Representation.

 Lecture Outline
 Number Systems
– Binary, Octal, Hexadecimal
 Representation of characters using codes
 Representation of Numbers
– Integer, Floating Point, Binary Coded Decimal
 Program Language and Data Types

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 Data Representation?
• Representation = Measurement
 Most things in the “Real World” actually exist as
a single, continuously varying quantity Mass,
Volume, Speed, Pressure, Temperature
 Easy to measure by “representing” it using a
different thing that varies in the same way Eg.
Pressure as the height of column of mercury or as voltage
produced by a pressure transducer
 These are ANALOG measurements

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 Digital Representation
 Convert ANALOG to DIGITAL measurement by
using a scale of units
 DIGITAL measurements
– In units – a set of symbolic values - digits
– Values larger than any symbol in the set use sequence
of digits – Units, Tens, Hundreds…
– Measured in discrete or whole units
– Difficult to measure something that is not a multiple
of units in size. Eg Fractions

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 Analog vs. Digital representation

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 Data Representation
 Computers use digital representation
 Based on a binary system
(uses on/off states to represent 2 digits).
 Many different types of data.
– Examples?
 ALL data (no matter how complex)
must be represented in memory as binary
digits (bits).

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 Number systems and computers
 Computers store all data as binary digits, but we
may need to convert this to a number system we
are familiar with.

 Computer programs and data are often

represented (outside the computer) using octal
and hexadecimal number systems because they
are a short hand way of representing binary
numbers.

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 Common Number Systems

Used by Used in
System Base Symbols humans? computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa- 16 0, 1, … 9, No No
decimal A, B, … F
Quantities/Counting (1 of 3)
Hexa-
Decimal Binary Octal 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
Quantities/Counting (2 of 3)
Hexa-
Decimal Binary Octal decimal
8 1000 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 1111 17 F
Quantities/Counting (3 of 3)
Hexa-
Decimal Binary Octal decimal
16 10000 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 10111 27 17 Etc.
 Conversion Among Bases
• The possibilities:

Decimal Octal

Binary Hexadecimal

pp. 40-46
 Quick Example

2510 = 110012 = 318 = 1916

Base
Decimal to Decimal (just for fun)

Decimal Octal

Binary Hexadecimal

Next slide…
 Weight

12510 => 5 x 100 = 5

2 x 101 = 20
1 x 102 = 100
125

Base
 Binary to Decimal

Decimal Octal

Binary Hexadecimal
 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
 Example
Bit “0”

1010112 => 1 x 20 = 1
1 x 21 = 2
0 x 22 = 0
1 x 23 = 8
0 x 24 = 0
1 x 25 = 32
4310
 Octal to Decimal

Decimal Octal

Binary Hexadecimal
 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
 Example

7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810
Hexadecimal to Decimal

Decimal Octal

Binary Hexadecimal
 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
 Example

ABC16 => C x 160 = 12 x 1 = 12

B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810
 Decimal to Binary

Decimal Octal

Binary Hexadecimal
 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.
 Example
12510 = ?2 2 125
2 62 1
2 31 0
2 15 1
2 7 1
2 3 1
2 1 1
0 1

12510 = 11111012
 Octal to Binary

Decimal Octal

Binary Hexadecimal
 Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent
binary representation
 Example
7058 = ?2

7 0 5

111 000 101

7058 = 1110001012
Hexadecimal to Binary

Decimal Octal

Binary Hexadecimal
 Hexadecimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit
equivalent binary representation
 Example
10AF16 = ?2

1 0 A F

0001 0000 1010 1111

10AF16 = 00010000101011112
 Decimal to Octal

Decimal Octal

Binary Hexadecimal
 Decimal to Octal
• Technique
– Divide by 8
– Keep track of the remainder
 Example
123410 = ?8

8 1234
8 154 2
8 19 2
8 2 3
0 2

123410 = 23228
Decimal to Hexadecimal

Decimal Octal

Binary Hexadecimal
 Decimal to Hexadecimal
• Technique
– Divide by 16
– Keep track of the remainder
 Example
123410 = ?16

16 1234
16 77 2
16 4 13 = D
0 4

123410 = 4D216
 Binary to Octal

Decimal Octal

Binary Hexadecimal
 Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits
 Example
10110101112 = ?8

1 011 010 111

1 3 2 7

10110101112 = 13278
Binary to Hexadecimal

Decimal Octal

Binary Hexadecimal
 Binary to Hexadecimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits
 Example
10101110112 = ?16

10 1011 1011

2 B B

10101110112 = 2BB16
 Octal to Hexadecimal

Decimal Octal

Binary Hexadecimal
 Octal to Hexadecimal
• Technique
– Use binary as an intermediary
 Example
10768 = ?16

1 0 7 6

001 000 111 110

2 3 E

10768 = 23E16
 Hexadecimal to Octal

Decimal Octal

Binary Hexadecimal
 Hexadecimal to Octal
• Technique
– Use binary as an intermediary
 Example
1F0C16 = ?8

1 F 0 C

0001 1111 0000 1100

1 7 4 1 4

1F0C16 = 174148
 Exercise – Convert ...
Hexa-
Decimal Binary Octal decimal
33
1110101
703
1AF

Don’t use a calculator!

Skip answer Answer

 Exercise – Convert …
Answer

Hexa-
Decimal Binary Octal decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
 Common Powers (1 of 2)
• Base 10
Power Preface Symbol Value
10-12 pico p .000000000001

10-9 nano n .000000001

10-6 micro .000001

10-3 milli m .001

103 kilo k 1000

106 mega M 1000000

109 giga G 1000000000

1012 tera T 1000000000000
 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
Example
In the lab…
1. Double click on My Computer
2. Right click on C:
3. Click on Properties

/ 230 =
 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.
 Review – multiplying powers
• For common bases, add powers

ab ac = ab+c

26 210 = 216 = 65,536

or…
26 210 = 64 210 = 64k
 Binary Addition (1 of 2)
• Two 1-bit values

A B A+B
0 0 0
0 1 1
1 0 1
1 1 10
“two”
 Binary Addition (2 of 2)
• Two n-bit values
– Add individual bits
– Propagate carries
– E.g.,

1 1
10101 21
+ 11001 + 25
101110 46
 Multiplication (1 of 3)
• Decimal (just for fun)

35
x 105
175
000
35
3675
 Multiplication (2 of 3)
• Binary, two 1-bit values

A B A B
0 0 0
0 1 0
1 0 0
1 1 1
 Multiplication (3 of 3)
• Binary, two n-bit values
– As with decimal values
– E.g.,
1110
x 1011
1110
1110
0000
1110
10011010
 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
 Fractions
• Binary to decimal

10.1011 => 1 x 2-4 = 0.0625

1 x 2-3 = 0.125
0 x 2-2 = 0.0
1 x 2-1 = 0.5
0 x 20 = 0.0
1 x 21 = 2.0
2.6875
 Fractions
• Decimal to binary .14579
x 2
3.14579 0.29158
x 2
0.58316
x 2
1.16632
x 2
0.33264
x 2
0.66528
x 2
1.33056
11.001001... etc.
 Exercise – Convert ...

Hexa-
Decimal Binary Octal decimal
29.8
101.1101
3.07
C.82
Don’t use a calculator!

Skip answer Answer

 Exercise – Convert …
Answer

Hexa-
Decimal Binary Octal decimal
29.8 11101.110011… 35.63… 1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82
Thank you

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