How to Convert Binary to Decimal Examples – Step by Step

How to Convert Binary to Decimal Examples – Step by Step

The easy approach of how to convert binary to decimal :

The binary number system, also known as the base 2 number system; is used by all modern generation computers internally. The binary number system is represented by the combination of 0’s and 1’s. You can learn more about binary number system here. Now, to get the answer of how to convert binary to decimal, you have to understand the math behind it. If you have a good understanding of powers of 2, you can do any binary to decimal conversion.

power of 2's

Memorising the powers of 2 will definitely help you to find the answer of how to convert binary to decimal quite easily and quickly.

Methods of converting binary to decimal :

There are various methods available, which can be used in binary to decimal conversion. But here we will discuss a very easy approach to convert a binary number to decimal.

In this method, to convert a binary number to its decimal form, we will multiply each digit of the binary number by the base of the binary number system, i.e. by 2. Then we will start giving powers to 2 in increasing order starting from 0 and from right to left. At last, we will add the series to get the decimal equivalent of that binary number.

We take an example of (10011)2 which is a binary number. We will convert it to its decimal form using the method described above.

How to convert binary to decimal Step by Step :

  • First, we multiply all the digits in the number by 2 and add them like:

        (1*2) + (0*2) + (0*2) + (1*2) + (1*2)

  • Now we give powers to 2 starting from 0 from right to left.

        (1*24) + (0*23) + (0*22) + (1*21) + (1*20)

  • That’s all the formula part. Now we convert the powers of 2 to the numbers.

        (1*16) + (0*8) + (0*4) + (1*2) + (1*1)

  • Now summing up all gives us the answer as:

        16 + 0 + 0 + 2 + 1 = 19

So, (19)10 is the decimal equivalent of the given binary number (10011)2

Let’s see another example below:

(110111)2 = (?)10

     (1*25) + (1*24) + (0*23) + (1*22) + (1*21) + (1*20)

     = 32 + 16 +0 + 4 + 2 +1

     = 55

We write it as (55)10, which is the decimal equivalent form of (110111)2

So, we learned how to convert binary to decimal, when there is no fractional part. Now we take a binary number (1011.101)2, which has the fraction part. So, let’s learn how to convert binary fraction to decimal.

Binary fraction to decimal conversion :

The method of the binary fraction to decimal conversion is almost same as the conversion of binary to decimal without fraction. The only difference in the binary fraction to decimal conversion is the powers on 2’s.

Let’s see how binary fraction to decimal conversion works on the binary number (1011.101)2.

There are 2 parts in the binary number (1011.101)2,

  • (1011)2 – Non-fractional part
  • (.101)2 – Fractional part

The conversion of non-fractional part (1011)2 is same as we learned above. But on the fractional part (.101)2, we will start from left to right and powers of 2 increases from -1.

Binary fraction to decimal conversion step by step :

  • For the non – fractional part, (1011)2 we will follow the normal binary to decimal conversion method that we have discussed above. So, we get (11)10  which is the decimal equivalent of (1011)2.

        Step 1:  (1*2) + (0*2) + (1*2) + (1*2)

        Step 2:  (1*23) + (0*22) + (1*21) + (1*20)

        Step 3:  8 + 0 + 2 +1

        Step 4:  11

  • Now, for the fractional part (.101)2, multiply each digit of the binary fraction by 2 and also add them like:

        (1*2) + (0*2) + (1*2)

  • We give powers to 2 from left to right starting from -1, then -2, -3… and so on.

        (1*2-1) + (0*2-2) + (1*2-3)

  • Now, simplify the above step

        (1* 1/2) + 0 + (1*1/8)

 

Summing up all gives us the answer 0.625

        (1/2) + (1/8) = 5/8 = 0.625

So, we got (0.625)10, which is the decimal equivalent of the binary fraction (.101)2Therefore the decimal form of the binary number (1011.101)2 is (11.625)10.

Let’s take another example of (11111.111)2

  • (11111)2 – Non-fractional part
  • (.111)2 – Fractional part

You can convert both non-fractional and fractional part together as we have done below :

    (1*24) + (1*23) + (1*22) + (1*21) + (1*20) . (1*2-1) + (1*2-2) + (1*2-3)

    = (1*16) + (1*8) + (1*4) + (1*2) + (1*1) . (1* 1/2) + (1 * 1/4) + (1*1/8)

    = (16 + 8 + 4 + 2 + 1) . (1/2 + 1/4 + 1/8)

    = 31.875

So, (31.875)10 is the decimal form of given binary number (11111.111)2.

So, in this article we have discussed the easiest approach of how to convert binary to decimal and also how to convert binary fraction to decimal. Using this simple method you can easily convert any given binary number to its equivalent decimal form quickly.

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