Hexadecimal to Binary

The definition of a  Hexadecimal number is any number that has a base of 16. This means that the unit’s position in a Hexadecimal Number is represented as coefficients of 160, a hundred positions are represented as coefficients of 161, a thousand positions are represented as coefficients of 162 , and so on.

For example if we have to convert 2C3 to decimal,

(2C3)16=2× 16 × 16 + 12 × 16 + 3 * 1

(2C3)16=2 × 256 + 12 × 16 + 3

(2C3)16=512+192+3

(2C3)16=707

So 2C3 in Hexadecimal is represented as 707 in Decimal.

Now we will see how to convert some number systems into other number systems.

Hexadecimal to Binary

A binary number system is the number system in which any number represented has a base of 2. In the binary number system, a number is represented using only 2 digits, 0 and 1. 

For example, if we have to represent a number 32 in binary, it would be represented as 1111.

This is because the number in the units place of the binary Number System is a coefficient of 20, the number in the tens place of the Binary Number System is a coefficient of 21, the number in the hundreds place of the Binary Number System is a coefficient of 22 and so on.

So each number is represented in terms of 8 4 2 1.

Now we have to see how to convert a hexadecimal number into a binary number.

So as we have seen, a binary number will be represented as the coefficients of the powers of 2.

So in order to convert a Hexadecimal Number to a binary number we have to convert each digit of a Hexadecimal Number (starting from the lowest digit) to the Binary Equivalent by representing the individual digits as coefficients as the multiples of 2.

Let’s learn this process even better with the help of an example.

We are going to convert a Hexadecimal Number (6C6)16 into its Binary Equivalent.

First, we will start with 6; its Binary Equivalent would be 0110.

Then we move on to C, its Binary Equivalent is 1100.

And we already found the Binary Equivalent of 6 which is 0110.

So writing the individual digits as a single number,

(6C6)16= (011011000110)2

Now we will see another example, 

We will try to convert a Hexadecimal Number (9D2)16 into a Binary Number.

The Binary Equivalent of 9 is 1001.

The Binary Equivalent of D is 1101.

The Binary Equivalent of 2 is 0010.

So writing the digits as a single Binary Number,

100111010010

So,

(9D2)16= (100111010010)2

Now that we have very clearly understood this concept, we will look at some other important conversions.

Hexadecimal to Octal

First, we will look at how to convert a Hexadecimal Number into an Octal Number.

Octal Number System is a system in which a number has a base of 8. The Octal Number System uses the digits from 0 to 7 to represent a number. 

Now, the process to convert a Hexadecimal Number to an Octal Number is pretty easy provided you follow the steps to perfection.

First we take that Hexadecimal Number and convert it into its Binary Equivalent. Then we form groups of 3 digits from right to left. Then we convert the resulting number into the Octal Number.

Now we will see some examples to help us better understand this topic.

First we will convert (9C4)16 into an Octal Number

First, finding its Binary Equivalent,

The Binary Equivalent of 9 is 1001.

The Binary Equivalent of C is 1100.

Binary Equivalent of 4 is 0100.

So writing these individual digits as a single number, 

(9C4)16 = (100111000100)2

Now we form groups of 3 digits from the right to the left.

100, 111, 000, 100

Now converting these groups of 3 into Octal,

100=4

111=7

000=0

100=4

So, 

(9C4)16 = (4704)8

We will take another example to better understand this conversion.

We will try to convert 8DB4 into an Octal Number.

Converting this number into its Binary Equivalent,

The Binary Equivalent of 8 is 1000.

The Binary Equivalent of D is 1101.

The Binary Equivalent of B is 1011.

The Binary Equivalent of 4 is 0100.

Writing these individual digits as a single Binary Number,

(8DB4)16= (1000110110110100)2

Now to make groups of 3 and convert that into Octal. 

100, 011, 011,011, 010, 0

100=4

011=3

011=3

011=3

010=2

0=0

So (8DB4)16= (433320)2

Binary to Hexadecimal

In earlier we have already seen the description of a binary number and how it is represented. Now we will see how to convert a Binary Number to a Hexadecimal Number.

Converting a Binary Number to a Hexadecimal Number is pretty easy. You just have to take 4 binary digits as one group and convert them into their Hexadecimal Equivalent.

For example:- 

Convert (1001110100000001)2 into a Hexadecimal Number.

Taking a group of 4 Binary Digits,

1001, 1101, 0000, 0001

1001 = 9

1101 = D

0000 = 0

0000 = 1

So (1001110100000001)2 = (9D01)16

Convert (1111110110000000)2 into a Hexadecimal Number

Taking a group of 4 Binary Digits,

1111 = F

1101 = D

1000 = 8

0000 = 0

So (1111110110000000)2 = (FD80)16

Conclusion

We have described the Hexadecimal Number. We have even discussed the concept of Hexadecimal numbers in detail. Then we started with the conversions. First, we looked at the conversion of Hexadecimal Numbers to Binary numbers. Then we looked at the conversion of Hexadecimal Number to Octal Number. Finally, we looked at the conversion of Binary numbers to Hexadecimal numbers in detail and even talked about the examples to understand the concept better.