1.1 number systems
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Number Systems and Data Representations
Number System
Since the computer is a digital machine, all information given to it must be digitized (binary form)
Common Number Systems
Number System No. of Valid Digits Valid Digits
Base 2 Binary 2 0,1
Base 3 3 0,1,2
Base 4 4 0,1,2,3
…
Base 8 Octal 8 0,1,2,3,4,5,6,7
Base 10 Decimal 10 0,1,2,3,4,5,6,7,8,9
Base 16 Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
2
3
Number System
Using binary format in giving instructions to the computer is tedious and difficult.
Programmers usually use the decimal, octal, or hexadecimal format for clarity.
Number system is just a representation of data; thus a number in one base has its equivalent in another base.
A programmer must be familiar in converting a number
from one base to another.
4
Decimal Numbers to Other Bases
2010 = 101002 3010 = _______8 4010 = _______16
Solution:Divide with the
destination base Until quotient is Zero
Quotient Read the Remainder Upwards
20/2 10 0
10/2 5 0
5/2 2 1
2/2 1 0
1/2 0 1
5
Decimal (Base 10) to Other Bases20.687510 = 10100.10112 30.610 =
____________8
Solution:
Step 1: Convert the integral part using the previous step.
Step 2: Convert the fraction part by following the steps below.
Multiply the fraction part with the destination base until it becomes
zero
Take note of the integral part of the result then read downwards
.6875 x 2 1.375
.375 x 2 0.75
.75 x 2 1.5
.5 x 2 1.0
6
Other Bases to Base 10 (Decimal)
101002 = 2010 368 = ____10
Solution:
Exponent 4 3 2 1 0
Value of 2 raise to the exponent
16 8 4 2 1
Digit 1 0 1 0 0
Multiply the digit with 2x then add the results.
16 0 4 0 0
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Other Bases to Base 10 (Decimal)
10100.112 = 20.7510 36.58 = ____10
Solution:Exponents 4 3 2 1 0 -1 -2
Value of 2 raise to the exponent
16 8 4 2 1 0.50.25
Digit 1 0 1 0 0 1 1
Multiply the digit with 2x then add the results. Do it separately for integral and fraction part.
16 0 4 0 0 .5 .25
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Shortcut Conversion: Base 10 to Base 2 or Vice Versa
101012 = 2110 3010 = 111102
Solution:Result=Sum (Digit=1 x value)
16 0 4 0 1
Value( 2exponent) 16 8 4 2 1
Exponent 4 3 2 1 0
Binary Digit 1 0 1 0 1
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Shortcut Conversion: Base 2 to Base 8 or Vice Versa
Group the bits into 3 Examples:
11 001 1002 = 3148
5628 = _________2
BASE 2 BASE 8
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7
10
Shortcut Conversion:Base 2 to Base 16 or Vice Versa
Group the bits into 4 Examples:
1110 01012 = E516
A4C16 = _________2
BASE 2 BASE 16
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F
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Seatwork
64.210 = ______2
______8
______16
101110.1012 = ______8
______10
______16
75.128 = ______2
______10
______16
9D16 = ______2
______8
______10
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Arithmetic
Addition
3310
+ 1810
5110
34A516
+ 34A516
694A16
1101012
+ 01012
1110102
1238
+ 78
1328
Subtraction
5110
- 1810
3310
694A16
- 34A516
34A516
1110102
- 01012
1101012
1348
- 78
1258
13
Arithmetic
Multiplication
10112
x 112
10112
+ 10112
1000012
34A516
x 216
694A16
112
x 112
112
+ 112
10012
1238
+ 58
6378
14
ArithmeticDivision
10112
112 1000012
112
102
02
1002
112
112
112
02
34A516
216 694A16
616
916
816
1416
1416
A16
A16
016
58 128
15
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