Number Systems

NUMBER SYSTEMS
1
Objectives
 Understand why computers use binary (Base-2) numbering.

 Understand how to convert Base-2 numbers to Base-10 or Base-8.
 Understand how to convert Base-8 numbers to Base-10 or Base 2.
 Understand how to convert Base-16 numbers to Base-10, Base 2 or Base-8.
2
Why Binary System?
 Computers are made of a series of switches
 Each switch has two states: ON or OFF
 Each state can be represented by a number 1 for “ON” and 0 for “OFF”
3
Converting Base-2 to Base-10
(1 0 0 1 1)2
OFF
OFF
ON/OFF
ON
ON
ON
Exponent: 24 23 22 21 20
Calculation: 16 + 0 + 0 + 2 + 1
= (19)10
4
Number Systems
 Number systems include decimal, binary, octal and
hexadecimal
 Each system have four number base
Number System Base Symbol
Binary Base 2 B
Octal Base 8 O
Decimal Base 10 D
Hexadecimal Base 16 H
5
Decimal Number System
 The Decimal Number System uses base 10. It includes the digits {0, 1,2,…, 9}. The
weighted values for each position are:
Base
10^4 10^3 10^2 10^1 10^0 10^-1 10^-2 10^-3
10000 1000 100 10 1 0.1 0.01 0.001
Left of the decimal point Right of decimal point
6
Decimal Number System (cont.)
 Each digit appearing to the left of the decimal point represents a value between zero and
nine times power of ten represented by its position in the number.
 Digits appearing to the right of the decimal point represent a value between zero and
nine times an increasing negative power of ten.
 Example: the value 725.194 is represented in expansion form as follows:
 7 * 10^2 + 2 * 10^1 + 5 * 10^0 + 1 * 10^-1 + 9 * 10^-2 + 4 * 10^-3
 =7 * 100 + 2 * 10 + 5 * 1 + 1 * 0.1 + 9 * 0.01 + 4 * 0.001
 =700 + 20 + 5 + 0.1 + 0.09 + 0.004
 =725.194
7
Binary Number System
 Most modern computer system using binary logic. The computer represents
values(0,1) using two voltage levels (usually 0V for logic 0 and either +3.3 V or
+5V for logic 1).
 The Binary Number System uses base 2 includes only the digits 0 and 1
 The weighted values for each position are :

Base
2^5 2^4 2^3 2^2 2^1 2^0 2^-1 2^-2
32 16 8 4 2 1 0.5 0.25
8
Number Base Conversion
 Binary to Decimal: multiply each digit by its weighted position, and add each
of the weighted values together or use expansion form directly.
 Example the binary value 11001010 represents :
 1*2^7 + 1*2^6 + 0*2^5 + 0*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 0*2^0 =
 1 * 128 + 1 * 64 + 0 * 32 + 0 * 16 + 1 * 8 + 0 * 4 + 1 * 2 + 0 * 1 =
 128 + 64 + 0 + 0 + 8 + 0 + 2 + 0 =202
9
Number Base Conversion (cont.)
 Decimal to Binary: There are two methods, that may be used to convert from integer
number in decimal form to binary form:
• 1-Repeated Division By 2
For this method, divide the decimal number by 2,
If the remainder is 0, on the right side write down a 0.
If the remainder is 1, write down a 1.
When performing the division, the remainders which will represent the binary equivalent of
the decimal number are written beginning at the least significant digit (right) and each new
digit is written to more significant digit (the left) of the previous digit.
10
Number Base Conversion (cont.)
 Decimal to Binary (cont.)
convert the number 333 to binary.
Division Quotient Remainder Binary

333/2 166 1 1
166/2 83 0 01
83/2 41 1 101
41/2 20 1 1101
20/2 10 0 01101
10/2 5 0 001101
5/2 2 1 1001101
2/2 1 0 01001101
1/2 0 1 101001101
11
Octal Number System
 Computer scientists are often looking for shortcuts to do things
 One of the ways in which we can represent binary numbers is to use their octal
equivalents instead
 This is especially helpful when we have to do fairly complicated tasks using

numbers
• The octal numbering system includes eight base digits (0 - 7)
• After 7, the next placeholder to the right begins with a “1”
• 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13 ...
12
Octal Number System (cont.)
Number: 2 4 1
“Sixty-Fours”
“Eights”
“Ones”
Placeholder Name:
Value: 64*2 8*4 1*1
Exponential Expression: 82*2 81*4 80*1
13
Octal Number System (cont.)
Transform (44978)10 to Octal
Division Quotient Remainder Binary
44978 / 8 5622 2 2
5622 / 8 702 6 62
702/8 87 6 662
87/8 10 7 7662
10/8 1 2 27662
1/8 0 1 127662
14

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NUMBER SYSTEMS

 Understand why computers use binary (Base-2) numbering.

 Each switch has two states: ON or OFF

10000 1000 100 10 1 0.1 0.01 0.001

Left of the decimal point Right of decimal point

 Example: the value 725.194 is represented in expansion form as follows:

 7 * 10^2 + 2 * 10^1 + 5 * 10^0 + 1 * 10^-1 + 9 * 10^-2 + 4 * 10^-3

 =7 * 100 + 2 * 10 + 5 * 1 + 1 * 0.1 + 9 * 0.01 + 4 * 0.001

 =700 + 20 + 5 + 0.1 + 0.09 + 0.004

 The weighted values for each position are :

2^5 2^4 2^3 2^2 2^1 2^0 2^-1 2^-2

 Example the binary value 11001010 represents :

 1*2^7 + 1*2^6 + 0*2^5 + 0*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 0*2^0 =

For this method, divide the decimal number by 2,

If the remainder is 0, on the right side write down a 0.

If the remainder is 1, write down a 1.

Division Quotient Remainder Binary

 This is especially helpful when we have to do fairly complicated tasks using

• The octal numbering system includes eight base digits (0 - 7)

• After 7, the next placeholder to the right begins with a “1”

• 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13 ...

Value: 64*2 8*4 1*1

Exponential Expression: 82*2 81*4 80*1

Division Quotient Remainder Binary

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 12^7 + 12^6 + 02^5 + 02^4 + 12^3 + 02^2 + 12^1 + 02^0 =

Value: 642 84 1*1

Exponential Expression: 822 814 80*1