Number System

 

Number Systems

The number system is a way to represent or express numbers. You have heard of various types of number systems such as the Ebooks and the real numbers. But in the context of computers, we define other types of number systems. They are:

• The decimal number system

• The binary number system

• The octal number system and

• The hexadecimal number system

Decimal Number System (Base 10)

In this number system, the digits 0 to 9 represents numbers. As it uses 10 digits to represent a number, it is also called the base 10 number system. Each digit has a value based on its position called place value. The value of the position increases by 10 times as we move from right to left in the number.

For example, the value of 786 is

= 7 x 102 + 8 x 101 + 6 x 100

= 700 + 80 + 6

Binary Number System (Base 2)

A computer can understand only the “on” and “off” state of a switch. These two states are represented by 1 and 0. The combination of 1 and 0 form binary numbers. These numbers represent various data. As two digits are used to represent numbers, it is called a binary or base 2 number system.

The binary number system uses positional notation. But in this case, each digit is multiplied by the appropriate power of two based on its position.

For example, (101101)2 in decimal is

= 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20

= 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1

= 32 + 8 + 4 + 1

= (45)10

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Units of measurement of data

Machine language is binary. And so it is necessary to discuss how to measure the data stored in a computer. Bit and Byte are the units to measure data.

Bit

The term ‘bit’ is a contraction of the words ‘binary’ and ‘digit’. It is the smallest unit of memory or instruction that can be given or stored on a computer. A bit is either a 0 or a 1. The number in the above example is a 6-bit number as it has 6 binary digits (0s and 1s).

Byte

A group of 8 bits like 01100001 is a byte. Combination of bytes comes with various names like the kilobyte. One kilobyte is a collection of 1000 bytes. A word or letter like ‘A’ or ‘G’ is worth 8 bits or one byte. One thousand bytes make up a kilobyte (one thousand letters approximately). 1024 kilobytes form a Megabyte (Mb) and so on.

Binary

Learn the rules of converting number systems here.

Octal Number System (Base 8)

This system uses digits 0 to 7 (i.e. 8 digits) to represent a number and the numbers are as a base of 8.

For example, (24)8 in decimal is

= 2×81+4×80

= (20)10

Hexadecimal Number System (Base 16)

In this system, 16 digits used to represent a given number. Thus it is also known as the base 16 number system. Each digit position represents a power of 16. As the base is greater than 10, the number system is supplemented by letters. Following are the hexadecimal symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

To take A, B, C, D, E, and F as part of the number system is conventional and has no logical or deductive reason.

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Number System Chart

Name

Base

Symbols

Example

Decimal 10 0,1,2,3,4,5,6,7,8,9 (2795)10

Binary 2 0,1 111000010

Octal 8 0,1,2,3,4,5,6,7 (1576)8

Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 3DB

Example For You

Q 1 A number system uses 10 as the base. The number is 654. What is the LSB and MSB of this number?

A) 6 and 4 B) 4 and 6 C) 0 and 1 D) 10 and 01.

Answer: MSB stands for the most significant Bit and the LSB stands for the Least Significant Bit. If you take a look at the number 654, we ought to represent it in a base 10 (Decimal number system).

So we can write it as 6×102 + 5×101 + 4 x100. Thus the number 6 is the most significant in the sense that it contains the bulk of the value of the number. And the number 4 is the least significant bit because it contains the least bulk of the value. Hence the answer is B that is 4 and 6. We can say that in the decimal representation, the number to the left is the MSB and the number to the right is the LSB.

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Practice Questions

Q 1: How would you represent 10111 in the decimal number system?

A) 23

B) 24

C) 25

D) 22

Ans: A) 23

Q 2: The LSB and MSB in the following number are: 1220

A) 1 & 0

B) 0 & 1

C) 10

D) 01

Ans: A) 0 & 1