Hexadecimal Number System

A hexadecimal number system is one of the types of number systems along with binary, octal, and decimal. The base number of a hexadecimal number system is 16 where it includes both numbers from 0 -9 and digits from A - F. Hexadecimal is considered as one of the most convenient ways to showcase a binary number in computers and is done by using a conversion table. Let's learn more about what the hexadecimal number system is, the uses, the conversion table and solve a few examples to understand the concept better.

The word hexadecimal can be divided into 'Hexa' and 'deci', where 'Hexa' means 6 and 'deci' means 10. The hexadecimal number system is described as a 16 digit number representation of numbers from 0 - 9 and digits from A - F. In other words, the first 9 numbers or digits are represented as numbers while the next 6 digits are represented as symbols from A - F. Hexadecimal is very similar to the decimal number system that has a base number of 9. Therefore, after 9 digits, the 10th digit is represented as a symbol - 10 as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. Hence, the 16 digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. For example: \(7B3_{16}, 6F_{16}, 4B2A_{16}\) are hexadecimal numbers.

A hexadecimal number system is also known as a positional number system as each digit has a weight of power 16. Each digit is 16 times more significant than the previous digit. Hence, when we convert any hexadecimal number to any other number system, we multiply the digits individually keeping the power of 16 in mind according to the placement of their position.

A hexadecimal number can be converted to binary, octal, and decimal by using the base number of 16. Since each digit has a weight of power 16, to convert from hexadecimal we use the sum of the product of each digit with its positional value by multiplying each digit from the left with 16⁰, 16¹, 16², 16³ and from the right 16^-1, 16^-2, 16^-3. Let us discuss each conversion method in detail.

Convert Hexadecimal to Binary

Converting hexadecimal to binary is done in two methods, one is with a conversion table and two is without a conversion table. Let us explore both methods:

Method 1: Convert Hexadecimal to Binary with Conversion Table

To convert a hexadecimal number to binary with a conversion table, we use the below-mentioned table. We first convert the hexadecimal to decimal by obtaining the equivalent of the decimal number of each digit by looking at the table. We convert that decimal number to binary by looking at the same table. Finally, to obtain the binary number we combine the digits together. Let us look at an example.

Example: Convert hexadecimal \((67F)_{16}\) to binary.

Solution: Looking at the conversion table, find the equivalent of each digit to decimal.

6 = \((6)_{10}\) , 7 = \((7)_{10}\) , F = \((15)_{10}\)

Once the decimal of each digit is obtained, looking at the conversion table convert each decimal number to binary.

\((6)_{10}\) = \((0110)_{2}\)

\((7)_{10}\) = \((0111)_{2}\)

\((F)_{10}\) = \((1111)_{2}\)

Combine all the binary numbers together to obtain the final one.

Therefore, \((67F)_{16}\) = \((011001111111)_{2}\).

Method 2: Convert Hexadecimal to Binary without Conversion Table

This method requires both multiplication and division of numbers using the respective base numbers. The hexadecimal base number is 16, the base number of a decimal number is 10, and the base of a binary number system is 2. We multiply each digit with 16^n-1, where the digit is in its nth position to obtain the decimal number. Once the digits are converted, we divide the decimal number by 2 by keeping the remainder aside and dividing the quotient by 2 until we arrive at zero. Finally, to obtain the binary number we arrange the remainders from bottom to top. Let us look at an example.

Example: Convert hexadecimal \((76)_{16}\) to binary.

Solution: Convert \((76)_{16}\) to decimal by multiplying each digit with 16^n-1. Multiply it

\((76)_{16}\) = 7 × 16^(2-1) + 6 × 16^(1-1)

\((76)_{16}\) = 7 × 16¹ + 6 × 16⁰

\((76)_{16}\) = 7 × 16 + 6 × 1

\((76)_{16}\) = 112 + 6

\((76)_{16}\) = 118

Therefore, \((76)_{16}\) = \((118)_{10}\).

Convert \((118)_{10}\) to a binary number by dividing the number by 2 until the quotient is zero.

Therefore, \((118)_{10}\) = \((1110110)_{2}\).

Hence, \((76)_{16}\) = \((1110110)_{2}\).

Convert Hexadecimal to Octal

Conversion of Hexadecimal to Octal is done in two steps i.e. first convert the hexadecimal number to decimal number and then convert it to an octal number. Let us look at an example to understand this method better.

Example: Convert \((AE5)_{16}\) to Octal.

Solution:

\((AE5)_{16}\) = A × 16² + E × 16¹ + 5 × 16⁰

\((AE5)_{16}\) = 10 × 16² + 14 × 16¹ + 5 × 16⁰ (convert the symbols to digits looking at the hexadecimal to decimal conversion table used above)

\((AE5)_{16}\) = 10 × 64 + 14 × 16 + 5 × 1

\((AE5)_{16}\) = 640 + 224 + 5

\((AE5)_{16}\) = \((869)_{10}\).

Convert the decimal number to octal by dividing it by 2 until the quotient is 0.

869/8 = 108, remainder is 5

108/8 = 13, remainder is 4

13/8 = 1, remainder is 5

1/8 = 0, remainder is 1

To obtain the final octal number, we arrange the remainder from bottom to top.

Therefore, \((AE5)_{16}\) = \((1545)_{8}\).

Convert Hexadecimal to Decimal

Converting hexadecimal to decimal is done in a similar manner as the previous two i.e. multiply each digit with the power of 16. Let us take an example.

Example: Convert \((DC24)_{16}\) to Decimal.

Solution:

\((DC24)_{16}\) = D ×16³ + C × 16² + 2 × 16¹ + 4 × 16⁰

\((DC24)_{16}\) = 13 ×16³ + 12 × 16² + 2 × 16¹ + 4 × 16⁰ (convert the symbols to digits looking at the hexadecimal to decimal conversion table used above)

\((DC24)_{16}\) = 13 × 4096 + 12 × 256 + 2 × 16 + 4 × 1

\((DC24)_{16}\) = 53248 + 3072 + 32 + 4

\((DC24)_{16}\) = 56356

Therefore, the decimal equivalent of \((DC24)_{16}\) = \((56356)_{10}\).

Conversion from other number systems as binary, octal, and decimal to hexadecimal is done in a similar manner. The conversion table used above is used here as well. Let us discuss each conversion in detail.

Convert Binary to Hexadecimal

To convert binary to hexadecimal we use the conversion table used in the previous section. Let us look at an example for a better understanding.

Example: Convert \((11001111011)_{2}\) to Hexadecimal.

Solution: For hexadecimal, the binary digits are expressed in 4. Hence, every 4 digit in binary becomes one 1 digit in hexadecimal. By looking at the conversion table we get,

0110 = 6, 0111 = 7, 1011 = B

We arrange the numbers together to get the final number.

Therefore, \((11001111011)_{2}\) = \((67B)_{16}\).

Convert Octal to Hexadecimal

To convert octal to hexadecimal, we first convert the octal number to binary number then to hexadecimal number. Octal to Binary has a different conversion table as the 3 binary digits make 1 octal digit. The conversion table is:

Example: Convert \((141)_{8}\) to hexadecimal

Solution: Looking at the table, we can convert each octal digit to binary.

1 = 001 , 4 = 100 , 1 = 001

Hence, \((141)_{8}\) = \((001100001)_{2}\)

By looking at the binary to hexadecimal conversion table we get,

0110 = 6, 0001 = 1 (the zero on the left of the last digit can be removed or added according to the requirement)

Therefore, \((141)_{8}\) = \((61)_{16}\).

Convert Decimal to Hexadecimal

The decimal to hexadecimal conversion is done by using the base number of hexadecimal that is 16 so the number needs to be divided by 16 until the quotient is zero. Let us look at an example.

Example: Convert \((150)_{10}\) to hexadecimal.

Solution: Divide 150 by 16 until the quotient is zero.

150/16 = 9, remainder is 6

9/16 = 0, remainder is 9

Write the remainder from bottom to top.

Therefore, \((150)_{10}\) = \((96)_{16}\).

With a decimal point, a hexadecimal number has two parts - integer and fraction. The integer part is on the left of the decimal point whereas the fraction part is on the right of the decimal point. Since the digits use the power of 16, the digits on the right will have a power of 16 in the decreasing order. For example: 5E.64 = 5 × 16¹ + E × 16⁰ + 6 × 16^-1 + 4 × 16^-2

Related Topics

Listed below are some interesting topics related to hexadecimal number system, take a look. 

• Hexadecimal to Binary
• Decimal to Octal 
• Octal to Decimal

What is Hexadecimal Number System?

Hexadecimal number system is also called a positional number system as each digit in the hexadecimal number has the power of 16, since the base number is 16. Unlike other number systems, the hexadecimal number system has digits from 0 - 9 and from 10 - 16 they are represented in symbols i.e 10 as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. For example \((28E)_{16}\), \((AC7)_{16}\), \((EF.6A)_{16}\) are all hexadecimal numbers.

Where is Hexadecimal Number System Used?

A hexadecimal number is most commonly used by computer system designers and programmers. This type of number system is considered very simple to use and is human-friendly as it represents the binary numbers in computers as well. Each hexadecimal digit represents 4 bits or digits in binary which is used in the conversion method as well.

What is 0xFF?

The number 0xFF is a hexadecimal number system with a base of 16 that is composed of two F numbers. The binary equivalent of F is 1111. Therefore, 0xFF in binary is 11111111.

Why is it Called Hexadecimal?

The hexadecimal number system uses decimal numbers along with 6 extra symbols. In the decimal number system, there is no numerical representation of digits beyond the number 9. Hence, symbols or alphabets are used to represent numbers from 10 - 16, as the base number of hexadecimal is 16. It is represented as 10 as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F.

What are the Four Types of Number System?

In mathematics, there are four types of number systems, they are:

• Binary number system - The base number is 2
• Octal number system - The base number is 8
• Decimal number system - The base number is 10
• Hexadecimal number system - The base number is 16

How Do You Write Hexadecimal Numbers?

Hexadecimal numbers have a power of 16 to each of its digits that are multiplied by each digit and the product is added together to get the final conversion number. 16^n-1, where n is the nth position of the digit, is multiplied to each digit starting from the left moving towards the right in decreasing order. For example: \((7D1E)_{16}\) = 7 × 16³ + D × 16² + 1 × 16¹ + E × 16⁰. While converting hexadecimal to binary or octal or decimal, we convert the symbols to digits first and then convert.