Conversion between Decimal Number and Any Number System (Especially for GATE-Geospatial 2022)

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There are many techniques which can be used to convert numbers from one base to another. We will demonstrate here the following:

1. Conversion of Any Number to Decimal Number System

A number in base with number of digits in the integer part, number of digits in the fraction part, and as the coefficients of the integer part and fraction part, can be expressed as

Examples of equivalent decimal values of the numbers in different number systems are given here.

Example Decimal to Decimal

Example Binary to Decimal

Example Hexadecimal to Decimal

2. Conversion of Decimal to Any Other Number

The conversion of decimal to other number system may be done by a method known as Dibble-Dabble method. In this method, the integer and fraction parts of the number are converted separately.

This Diagram Shows Integer and Fraction Parts of the Number

A. Conversion of Integer Part

The integer part of the decimal number is first divided by the base of the number to which the decimal number is to be converted and the remainder, if any, is noted. Then the successive quotients are also divided by the same base value until the quotient is zero and the successive remainders are noted down.

The last remainder is the most-significant digit (MSD) and the first remainder is the least-significant digit (LSD) of the converted number system.

Example Convert (58) Decimal to Binary

Convert 58 Decimal to Binary
QuotientRemainder
290 (LSD)
141
70
31
11
01 (MSD)

Hence,

Example Convert (58) Decimal to Hexadecimal

Convert 58 Decimal to Hexadecimal
QuotientRemainder
3A (LSD)
03 (MSD)

Hence,

B. Conversion of Fraction Part

The fractional part of the decimal number is continuously multiplied by the base value of the number system, to which the decimal fraction is to be converted, until either the fraction becomes zero or the desired accuracy is arrived and the successive integer part of the products are noted. Then the first integer is MSD and the last integer is the LSD of the fractional part of the converted number.

Example Convert (0.75) Decimal to Binary

Convert 0.75 Decimal to Binary
FractionInteger
0.501 (MSD)
0.001 (LSD)

Hence,

Example Convert (0.075) Decimal to Hexadecimal

Convert 0.075 Decimal- Hexadecimal
FractionInteger
0.201 (MSD)
0.203
0.203 (LSD)
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Hence,

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