The digital computer is a digital system that performs various computational tasks. The word digital implies that the information in the computer is represented by variables that take a limited number of discrete values. The count of these limited number is denoted by Base or Radix.

Base – A number system of base or radix “r” is a system that uses r distinct symbols to represent any number

The table below summarizes four number systems, their base and the distinct symbols used by these number systems to represent any number:

Number System	Base / Radix	Distinct Symbols
Binary	2	0, 1
Octal	8	0 – 7
Decimal	10	0 – 9
Hexadecimal	16	0 – 9, A, B, C, D, E, F

Note: In hexadecimal format, A=>10, B=>11, C=>12, D=>13, E=>14, F=>15

Now let’s take an example of an absolute number and a fractional number and see how to convert from one base to another.

Convert from Binary to Decimal

1010 = 1*2³ + 0*2² + 1*2¹ + 0*2⁰ = 8 + 0 + 2 + 0 = 10

1010.01 = 1*2³ + 0*2² + 1*2¹ + 0*2⁰ + 0*2^-1 + 1*2^-2 = 8 + 0 + 2 + 0 + 0 + 0.25 = 10.25

Here 2^-1 means ½ and 2^-2 means 1/2²

(1010)₂ = (10)₁₀

(1010.01)₂ = (10.25)₁₀

Convert from Binary to Octal

Since octal is represented using 3 digits (as 8 = 2³), assemble the digits in a group of 3 starting from right to left to convert from binary to octal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 3 as shown below:

1010 = 1 010 = 001 010 = 12

1010.01 = 1 010.010 = 001 010.010 = 12.2

(1010)₂ = (12)₈

(1010.01)₂ = (12.2)₈

Convert from Binary to Hexadecimal

Since hexadecimal numbers are represented using 4 digits (as 16 = 2⁴), assemble the digits in a group of 4 starting from right to left to convert from binary to hexadecimal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 4 as shown below:

1010 = 1010 = A

1010.01 = 1010.0100 = A.4

(1010)₂ = (A)₁₆

(1010.01)₂ = (A.4)₁₆

Convert from Octal to Binary

12 = 001 010 = 1010

12.2 = 001 010.0100 = 1010.01

(12)₈ = (1010)₂

(12.2)₈ = (1010.01)₂

Convert from Octal to Decimal

12 = 1*8¹ + 2*8⁰ = 8 + 2 * 1 = 10

12.2 = 1*8¹ + 2*8⁰ + 2*8^-1 = 8 + 2 * 1 + 2/8 = 8 + 2 + 0.25 = 10.25

(12)₈ = (10)₁₀

(12.2)₈ = (10.25)₁₀

Convert from Octal to Hexadecimal

Since hexadecimal numbers are represented using 4 digits (as 16 = 2⁴), assemble the digits in a group of 4 starting from right to left to convert from octal to hexadecimal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 4 as shown below:

12 = 001 010 = 00 1010 = 0000 1010 = 0A = A

12.2 = 001 010.010 = 00 1010.0100 = 0000 1010.0100 = 0A.4 = A.4

(12)₈ = (A)₁₆

(12.2)₈ = (A.4)₁₆

Convert from Decimal to Binary

To convert from decimal to binary, given number should be divided by 2, until the quotient can not be divided further.

(10)₁₀ = (1010)₂

0.25 * 2 = 0. 50 | 0 (Most Significant Digit)

0.50 * 2 = 1. 00 | 1 (MSD)

(0.25)₁₀ = (01)₂

(10.25)₁₀ = (1010.01)₂

Convert from Decimal to Octal

To convert from decimal to octal, given number should be divided by 8, until the quotient can not be divided further.

(10)₁₀ = (12)₈

0.25 * 8 = 2. 00 | 2 (Most Significant Digit)

(10.25)₁₀ = (12.2)₈

Convert from Decimal to Hexadecimal

To convert from decimal to hexadecimal, given number should be divided by 16, until the quotient can not be divided further.

(10)₁₀ = (A)₁₆

0.25 * 16 = 4. 00 | 4 (Most Significant Digit)

(10.25)₁₀ = (A.4)₁₆

Convert from Hexadecimal to Binary

A = 1010

(A)₁₆ = (1010)₂

A.4 = 1010.0100 = 1010.01

(A.4)₁₆ = (1010.01)₂

Convert from Hexadecimal to Octal

Since octal is represented using 3 digits (as 8 = 2³), assemble the digits in a group of 3 starting from right to left to convert from hexadecimal to octal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 3 as shown below:

A = 1010 = 1 010 = 001 010 = 12

A.4 = 1010.100 = 1 010.100 = 001 010.100 = 12.4

(A)₁₆ = (12)₈

(A.4)₁₆ = (12.4)₈

Convert from Hexadecimal to Decimal

A = 10*16⁰ = 10 * 1 = 10

A.4 = 10*16⁰ + 4 * 16^-1 = 10 * 1 + 4/16 = 10 + 0.25 = 10.25

(A)₁₆ = (10)₁₀

(A.4)₁₆ = (10.25)₁₀

FAQs about Number System

Q. Why we section off 3 bits in Octal and 4 bits in Hexadecimal?

A.As per definition of base given above – A number system of base or radix “r” is a system that uses r distinct symbols to represent any number. Thus, octal with a base of 8, uses 8 distinct symbols (which can be represented by 3 digits as 8=2^3) and hexadecimal with a base of 16, uses 16 distinct symbols (which can be represented by 4 digits as 16=2^4). So as octal is represented using 3 digits, we section off 3 binary numbers starting from right, and as hexadecimal number is represented using 4 digits, we section off 4 binary numbers starting from right.

Q. How to convert any non-decimal number to a number with base x?

A. To convert a number to some other base say x, the procedure is to first convert the number to decimal and then divide it by x repeatedly till the quotient is greater than or equal to x.

Above are the examples of conversions from one base to another and can be utilized to covert any number from any base to any base. Please feel free to leave your footprints in the comments section below for any clarifications to above and anything related to Number System in Digital Electronics.

Hope you find the information presented here useful and informative…!!