This document discusses different number systems used in computing including binary, octal, decimal, and hexadecimal. It explains that computers use the binary system because switches can be in one of two states, on or off, represented by 1 and 0. The document provides methods for converting between number bases, such as dividing a decimal number repeatedly by 2 to get the binary equivalent or using place values. Octal is introduced as a way to represent binary numbers more compactly using digits 0-7.

1. Chapter 1 1
Number Systems

2. 2
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.

3. 3
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”

4. 4
Converting Base-2 to Base-10
(1 0 1 1)
2
0
ON
OFF
ON
OFF
ON
Exponent:
Calculation: 0 0 2 1
16+ + + + =
(19)10

5. 5
• 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

6. 6
1.1 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:
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
Base
Right of decimal point
left of the decimal point

7. 7
• 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

8. 8
1.2 The Binary Number Base
Systems
• 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 :
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
Base

9. 9
1.3 Number Base Conversion
• Binary to Decimal: multiply each digit by its
weighted position, and add each of the weighted
values together or use expansion formdirectly.
• Example the binary value 1100 1010 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

10. 10
• Decimal to Binary
There are two methods, that may be used to convert
from integer number in decimal form to binaryform:
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.

11. 11
• Example: 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

12. 12
Octal 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

13. 13
• 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 ...

14. 14
Octal Placeholders
“Ones
”
“Eights”
“Sixty-
Fours”
64*2 8*4 1*1
82*2 81*4 80*1
2 4 1
Number:
Placeholder
Name:
Value:
Exponential
Expression:

15. 15
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