Number System Conversion

There are many methods or techniques which can be used to convert numbers from one base to another. 

We'll demonstrate here the following

1.           Decimal System to Binary System
2.           Binary System to Decimal System
3.           Octal System to Binary System

1. Decimal to Binary System

Steps

1. Divide the decimal number to be converted by the value of the new base.
2. Get the remainder from Step 1 as the rightmost digit (least significant digit) of new base number.
3. Divide the quotient of the previous divide by the new base.
4. Record the remainder from Step 3 as the next digit (to the left) of the new base number.

Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.

The last remainder thus obtained will be the Most Significant Digit (MSD) of the new base number.

Example :-

Decimal Number: 29₁₀

Calculating Binary Equivalent 

Steps	Operation	Result	Remainder
Step 1	29 / 2	14	1
Step 2	14 / 2	7	0
Step 3	7 / 2	3	1
Step 4	3 / 2	1	1
Step 5	1 / 2	0	1

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the Least Significant Digit (LSD) and the last remainder becomes the Most Significant Digit (MSD).

Decimal Number 29₁₀ = Binary Number 11101₂.

2. Binary System to Decimal System

Steps

1. Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system).
2. Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.
3. Sum the products calculated in Step 2. The total is the equivalent value in decimal.

Example :-

Binary Number : 11101₂

Calculating Decimal Equivalent 

Step	Binary Number	Decimal Number
Step 1	111012	((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10
Step 2	111012	(16 + 8 + 4 + 0 + 1)10
Step 3	111012	2910

Binary Number 11101₂ = Decimal Number  29₁₀

3. Octal System to Binary System

Steps

1. Convert the original number to a decimal number (base 10).
2. Convert the decimal number so obtained to the new base number.

Example :

Octal Number − 25₈

Calculating Binary Equivalent −

Step 1 − Convert to Decimal

Step	Octal Number	Decimal Number
Step 1	258	((2 × 81) + (5 × 80))10
Step 2	258	(16 + 5 )10
Step 3	258	2110

Octal Number  25₈ = Decimal Number  21₁₀

Step 2 – Convert  Decimal to Binary

Step	Operation	Result	Remainder
Step 1	21 / 2	10	1
Step 2	10 / 2	5	0
Step 3	5 / 2	2	1
Step 4	2 / 2	1	0
Step 5	1 / 2	0	1

Decimal Number 21₁₀ = Binary Number  10101₂

Octal Number 25₈ = Binary Number  10101₂