Binary to Decimal and Decimal to Binary Conversion

Under Digital ElectronicsSimilarly decimal numbers can be converted into binary numbers also. As we have already seen that a binary number can be converted into decimal number by multiplying the numbers with certain powers of 2, the reverse operation i.e. converting a decimal number into binary number requires certain number of divisions depending upon the character of the number. In this method the decimal number is divided by 2 until the remainder reaches 1 and the dividends are qued up in reverse manner i.e. in the opposite manner of their acquiring beginning from the remainder which is 1. We can show the conversion with the help of an example which will make it easier to understand. Suppose we are converting the decimal number (87)¹⁰. Now the conversion is shown below

Binary number system can be converted into other number systems. The conversion of binary number to other system indirectly means the conversion of base or radix. The several binary conversions are:-

(a) Binary to decimal conversion and decimal to binary conversion
(b) Binary to octal conversion and octal to binary conversion
(c) Binary to hexadecimal conversion and hexadecimal to binary conversion

Decimal to Binary Conversion

Suppose we have one decimal number, 3454. This number represents the amount of 3000 + 400 + 50 + 4 only. Means 3X100 + 4X100 + 5X10 + 4. This can also be written as 3X10³ + 4X10² + 5X10¹ + 4X10⁰.

Here, 3, 4, 5 and 4 are digits of decimal system. 10³, 10², 10¹ and 10⁰ are multiplier of the respective digits. 10 is known as base of the decimal system.
Now examine this decimal number in different way.

For that we divide 3454 by base 10 and we get 345 as the quotient and 4 as the remainder.	10	3454	→ 4
Divide again 345 by 10 and we get 34 as the quotient and 5 as the remainder.	10	345	→ 5
Divide 34 by 10 and we get 3 as the quotient and 4 as the remainder.	10	34	→ 4
		3

Now if we write side by side from last quotient to first reminder we will get 3454 which is decimal equivalent of natural number three thousand four hundred fifty four.

This is a common technique of converting any natural number to any base system. In above method if we use base 2 in place of base 10, we will get binary equivalent of three thousand four hundred fifty four.

For that we divide 3454 by base 2 and we get 1727 as the quotient and 0 as the remainder.	2	3454	→ 0
Divide again 1727 by 2 and we get 863 as the quotient and 1 as the remainder.	2	1727	→ 1
Divide again 863 by 2 and we get 431 as the quotient and 1 as the remainder.	2	863	→ 1
Divide again 461 by 2 and we get 215 as the quotient and 1 as the remainder.	2	461	→ 1
Divide again 215 by 2 and we get 107 as the quotient and 1 as the remainder.	2	215	→ 1
Divide again 107 by 2 and we get 53 as the quotient and 1 as the remainder.	2	107	→ 1
Divide again 53 by 2 and we get 26 as the quotient and 1 as the remainder.	2	53	→ 1
Divide again 26 by 2 and we get 13 as the quotient and 0 as the remainder.	2	26	→ 0
Divide again 13 by 2 and we get 6 as the quotient and 1 as the remainder.	2	13	→ 1
Divide again 6 by 2 and we get 3 as the quotient and 0 as the remainder.	2	6	→ 0
Divide again 3 by 2 and we get 1 as the quotient and 1 as the remainder	2	3	→ 1
		1

For Decimal to binary conversion, we write side by side from last quotient to first reminder we will get 110101111110 which is binary equivalent of natural number three thousand four hundred fifty four. This how we do decimal to binary conversion.

Binary to Decimal Conversion

There is an inter relation among all the numbers in different number systems. And there are specific methods to transform one number of specific number system to shift it into the number in different number system. Here we are going to illustrate how a number in binary number system can be transformed into a number in decimal number system. First of all let us take any binary number Suppose we have taken (100011)₂ Now the weightage of each digit from right to left is 2⁰, 2¹, 2²…. Respectively depending on the number of digits (This is only for transforming a binary number into an equivalent decimal number). This is multiplied with the digits of the binary number respectively and the products are multiplied. The final sum is the equivalent decimal number. The procedure is illustrated below

(100011)₂ = 1*2² + 0*2⁴ + 0*2³ + 0*2² + 1*2¹ + 1*2⁰ = 32 + 0 + 0 + 0 + 2 + 1 = 35
As we have shown a decimal number 3454 can also be represented as 3X10³ + 4X10² + 5X10¹ + 4X10⁰. Here 3, 4, 5, 4 are digits used in decimal system. 10 is base of the system. 10³, 10², 10¹, and 10⁰ are multiplier of respective digits.
In binary system, we only use 1 & 0 as digits and 2 as base. 2ⁿ, 2^{n - 1}, 2^{n - 2}, ...............2², 2¹, 2⁰ are multipliers in binary system.
For binary to decimal conversion we add all products of the digits and their respective multipliers. Where the multipliers or weights of the binary digits from Least Significant Bits are 2⁰, 2¹,2²,2³, 2⁴ respectively . Now the bits are multiplied with their weights and the sum of those products is the respective decimal number.

Bit Position	11	10	9	8	7	6	5	4	3	2	1	0
Multipliers	211	210	29	28	27	26	25	24	23	22	21	20
Binary digits	1	1	0	1	0	1	1	1	1	1	1	0
Digit × Multiplier	1x211	1x210	0x29	1x28	0x27	1x26	1x25	1x24	1x23	1x22	1x21	0x20

  1x2¹¹ + 1x2¹⁰ + 0x2⁹ + 1x2⁸ + 0x2⁷ + 1x2⁶ + 1x2⁵ + 1x2⁴ + 1x2³ + 1x2² + 1x2¹ + 0x2⁰ = 3454

Now let us follow the following steps mathematically for binary number 11010.
(11010)₂ → 1 X 2⁴ + 1 X 2³ + 0 X 2² + 1 X 2¹ + 0 X 2⁰ = 16 + 8 + 2 = (26)₁₀ ∴ (11010)₂ = (26)₁₀ Hence (26)₁₀ is the required decimal number.This is how binary to decimal conversion is done.