A binary-to-decimal conversion is done to convert a binary number (base 2) to its equivalent decimal number (base 10) for simplicity in mathematical calculations.
Formula
The general formula for converting a binary into a decimal number in positional notation form can be written as:
(an-1….a3a2a1a0)2 = (an-1 × 2n-1) + … + (a2 × 22) + (a1 × 21) + (a0 × 20), here a0, a1, a2, …, an-1 are digits of a binary number.
In the positional notation method, the value of a digit in a number is determined by the weight of its position.
Steps
Now, let us convert a binary number (11001)2 to its corresponding decimal step by step.
Step 1: While converting, we consider each digit in the binary number (also known as a bit) from right to left. Here, (11001)2 has 1 as its rightmost digit, then continues with 0, 0, 1, and 1 as we move left.
Step 2: Then, we multiply each binary digit with increasing powers of 2 from right to left, such that the rightmost digit is multiplied by the greatest power of 2.
To convert a decimal binary number to its corresponding decimal, we follow the same steps for its integral part, while the fractional part is multiplied by negative powers of 2.
For example, on converting (0.0001)2 into its equivalent decimal, we get
Some binary numbers and their equivalent decimal numbers are in the following table:
Binary
Decimal
0
0
1
1
10
2
11
3
100
4
101
5
110
6
111
7
1000
8
1001
9
1010
10
1011
11
1100
12
1101
13
1110
14
1111
15
10000
16
10001
17
10010
18
10011
19
10100
20
Alternative Method (By Doubling)
Using this method, let us consider the previous binary number (11001)2 and convert it into a decimal.
Step 1: Unlike the first method, we consider each digit of the binary number from left to right. Here, (11001)2 has 1 at the leftmost.
Step 2: Now, we multiply 0 by 2 and add the leftmost digit of the binary number.
0 × 2 + 1 = 1
Step 3: Using the above result and multiplying it by 2, we get 1 × 2 = 2
On adding 2 and the second leftmost digit, we get 2 + 1 = 3
Step 4: Again, we repeat the above step until no digit is left.
On multiplying 3 by 2 and adding the third leftmost digit of (11001)2, we get
3 × 2 + 0 = 6
On multiplying 6 by 2 and adding the next digit of (11001)2, we get
6 × 2 + 0 = 12
On multiplying 12 by 2 and adding the next digit of (11001)2, we get
12 × 2 + 1 = 25
Now, no digit is left, and we have a result of 25, the converted decimal number. Thus, (11001)2 = (25)10.
Convert the binary number (101)2 into decimal using the doubling method.
Solution:
(101)2 has 1 at the leftmost, then 0, and 1. Here, 0 × 2 + 1 = 1 1 × 2 + 0 = 2 2 × 2 + 1 = 5 Now, the final result is 5 for the binary number 101. Thus, (101)2 = (5)10.
