You have heard of number systems like the whole numbers, the real numbers etc. But in the context of computer awareness, we define other types of number systems like the binary number system, the decimal system, the hexadecimal system and others. We will discuss the binary number system and others and how we can convert from one number system to the other.

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## Binary Numbers & Number Systems

Machine language is binary. This means that the machine language has binary values or two values, the combination of which represents the data. These two states are *“on”* state represented by 1 and *“off”* state, represented by* “0”*. Let us start with the more familiar number system, the one where we use the numbers 0 to 1.

### Decimal Number System

In this number system, the numbers 0 to 9 represents numbers. We call it a decimal system because when we write the decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate power of 10 depending on its position in the number.

For example

786 is written as 7 103 + 8 102 + 6 100.

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### Binary Number System

The binary number system is also a positional notation numbering system, but in this case, the base is not ten but is instead two. Each digit position in a binary number represents a power of two. When we write a binary number, each binary digit is multiplied by an appropriate power of 2 which is based on their position in the number.

For example: 101101 = 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

In the binary number system, there are only two possible values that can appear in each digit position rather than the ten that can appear in a decimal number. Only the numerals 0 and 1 are used in binary numbers.

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### Bit

The term ‘bit’ is a contraction of the words ‘binary’ and ‘digit’. It is necessary to talk about the number of bits used to store or represent the number. This merely describes the number of binary digits that would be required to write a given number or information. The number in the above example is a 6-bit number as it has 6 binary digits (0s and 1s).

In terms of the computer language or the binary language, a bit is either a *0* or a *1*. In other words, each 0 or 1 in a machine language forms a bit. A group of 8 bits like 01100001 is a byte. So a bit is the smallest unit of memory or instruction that can be given or stored on a computer.

Combination of bytes comes with various names like the kilobyte. One kilobyte is a collection of 1000 bytes. Normally 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.

### Hexadecimal Numbers

In this number system, the base used is 16. So there are 16 digits used to represent a given number. This number system is called hexadecimal number system and each digit position represents a power of 16. The following are the hexadecimal numerals.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; the number system is supplemented by letters as the base is greater than 10. To take A, B, C, D, E, and F as part of the number system is conventional and has no logical or deductive reason. Since the base numbers for any number system that has more than 9 as its base will have to be supplemented, in the hexadecimal number system, the letters A to F are used.

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### Octal Number System

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

For example, 24 = 3×81+0×80 = 308.

## Example For You

Q 1 A number system uses 10 as the base (mod). A number is 654 in it. The LSB and MSB of this number are?

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

Answer: Not fair! We do not know what LSB and MSB are. Well, let us see. 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 (Hexadecimal number system).

So we can write it as 6 103 + 5 102 + 4 100. Thus the number 6 is the most significant in a 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. In general, we can say that in the hexadecimal 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 23 in the binary number system?

A) 10111

B) 10101

C) 10001

D) 10110

Ans: A) 10111

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

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