The hexadecimal number system is a way of writing numbers using base 16. It uses sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, where A to F stand for the decimal numbers 10 to 15. This system is one of the methods used to represent numbers, especially in computing and digital electronics. The hexadecimal system was introduced in 1859 by Nystrom, who called it the Tonal system. It provides a simple and compact way to represent large numbers and makes it easier to convert between binary and decimal numbers.

What is Hexadecimal Number System?

The “Hexadecimal” or “Hex” number system employs the base 16 system and is a common choice for encoding large binary values due to its compact format and ease of comprehension when compared to long binary strings of 1s and 0s.

Hexadecimal numbering uses 16 (sixteen) separate digits using a mix of numbers from 0 to 15 because it is a Base-16 system. In other words, there are 16 different numerical symbols to choose from.

Base of Hexadecimal Number System

Hexadecimal is the name of the base-16 numeral system. As a result, the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 are used in this system. That means that two-digit decimal numbers 10, 11, 12, 13, 14, and 15 must be represented by a single numeral to exist in this numbering system.

Representation of Hexadecimal Number

Hexadecimal Number System is a sort of numerical representation in which the base number is 16. This indicates that there are only 16 potential digit values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Where A, B, C, D, E, and F represent the decimal values 10, 11, 12, 13, 14, and 15 in single bits.

Any digit’s value can be represented using only four bits. It requires only 4 bits to represent the value of any digit. Hexadecimal numbers are indicated by the addition of either an 0x prefix or an h suffix.

Every digit’s position has a weight that is the 16th power. Each place in the Hexadecimal system is 16 times more significant than the preceding one, hence the numeric value of a hexadecimal number is calculated by multiplying each digit by the value of the position in which the digit appears, then adding the products. As a result, it’s a weighted (or positional) number system.

Each Hexadecimal number can be represented using only 4 bits, with each group of bits having a distinct value between 0000 (for 0) and 1111 (for F = 15 = 8+4+2+1).

Hexadecimal Number System Table

The following hexadecimal number system table has an equivalent binary number of hexadecimal numbers as given below:

Digits used in Hexadecimal Number System

Unlike the decimal system, which uses 10 symbols to represent numbers, hexadecimal employs 16 symbols, with the symbols “0”–”9″ (or alternatively “a”–”f”) representing values from 0 to 9, and “A”–”F” (or alternatively “a”–”f”) representing values from 10 to 15.

Hexadecimal Number System to Binary Conversion

We can convert the hexadecimal number system integer to its binary number system equivalent by following below steps:

Step 1: Take the hexadecimal number and determine the decimal digit count.

Step 2: If it has n digits, multiply each one by \(16^{n-1}\), starting with the digit in the nth place.

Step 3: After you’ve multiplied the terms, add them together.

Step 4: The decimal number equivalent to the specified hexadecimal number is the outcome.

Step 5: This decimal number must now be converted to a binary number.

Step 6: Divide the decimal number by 2.

Step 7: Take note of the reminder.

Step 8: Repeat the previous two procedures for the quotient until it reaches zero.

Step 9: Reverse the order of the remainder.

Step 10: The result is the binary number that is required.

Solved Example of hexadecimal number system to binary conversion

Hexadecimal Number System to Decimal Conversion

The base numbers of each of the number systems are kept in mind while converting the hexadecimal number system to decimal number system. The hexadecimal number system uses both digits and characters, with characters being used to represent double digits.

There are 16 notations in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 10-15 are represented by A, B, C, D, E, and F, respectively.

To convert the hexadecimal to decimal number system, find the decimal equivalent in the conversion table, multiply as digit by 16 to the power of digit position, and then add everything together.

Solved Example of hexadecimal number system to decimal conversion

Hexadecimal Number System to Octal Conversion

We can convert the hexadecimal number system integer to its octal number system equivalent by following below steps:

Step 1: To begin the Hexadecimal to Octal Conversion, count the digits in the number.

Step 2: If n is the digit’s position from the right end, multiply each digit by \(16^{n-1}\).

Step 3: After you’ve multiplied the terms, add them together.

Step 4: The comparable decimal form is the resultant.

Step 5: 8 is divided by the decimal number. Take down the rest of the information.

Step 6: With the quotient, repeat the previous two steps until the quotient is zero.

Step 7: Reverse the order of the remainder.

Step 8: The obtained number corresponds to the desired outcome.

Solved Example of hexadecimal number system to octal conversion

Binary to Hexadecimal Conversion

Changing a binary number into a hexadecimal number is quite an easy process. Instead of converting the whole number at once, we split the binary digits into groups of four, as each group matches one hexadecimal value.

Example: Convert (11100011)₂ to hexadecimal.

Step-by-step:

1. Write the binary number in groups of four: 1110 0011
2. Now check each group in the binary-to-hex table:
   • 1110 = E
   • 0011 = 3
3. Put the two hexadecimal values together: E3

So, (11100011)₂ = (E3)₁₆

This means the binary number 11100011 is equal to E3 in hexadecimal.

Decimal to Hexadecimal Conversion

To change a decimal number into a hexadecimal number, we divide the decimal number by 16 and note down the remainder. We keep dividing the result again by 16 until the answer becomes zero. Then, we read the remainders from bottom to top to get the hexadecimal value.

Hexadecimal digits go from 0 to 9 and then A to F, where A = 10, B = 11, and so on up to F = 15.

Example: Convert 115 to hexadecimal.

Step-by-step:

1. 115 ÷ 16 = 7 remainder 3
2. Now take the quotient: 7 ÷ 16 = 0 remainder 7

Write the remainders from bottom to top:

7 3 → 73

So, 115 (decimal) = 73 (hexadecimal)

This means the decimal number 115 is written as 73 in the hexadecimal system.

Properties of the Hexadecimal Number System

The hexadecimal number system has several important features that make it widely used in computing and digital electronics. Here are its main properties:

• Base 16 System

Hexadecimal works on base 16, which means it uses sixteen different symbols to represent values. These symbols are 0–9 and A–F.

• A–F Represent Values 10 to 15

The letters from A to F are used instead of writing larger digits. For example: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

• Place Value Depends on Powers of 16

Every digit in a hexadecimal number has a place value based on powers of 16, similar to how decimal numbers use powers of 10. For instance, in the number 3A,

A stands for 10 × 16⁰

3 stands for 3 × 16¹

• Compact and Easy Representation

Hexadecimal numbers are shorter and easier to read compared to long binary strings. This is why they are often used when working with computer code and memory addresses.

• Direct Link to Binary

Each hexadecimal digit matches exactly four binary digits (bits). This makes conversion between binary and hexadecimal very simple and fast.

• Used in Computer Systems

Hex values are commonly seen in programming, colour codes (like #FF6600), networking, and microcontroller programming.

Uses of Hexadecimal Number System

The uses and applications of the Hexadecimal Number System are as follows:

• In computer programming and microprocessors, the Hexadecimal Number System is widely used. Colour descriptions on web pages are also done using a hexadecimal number. Each of the three fundamental colours (red, green, and blue) is represented by two hexadecimal digits, yielding 255 potential values and a total of over 16 million colours. Every byte is described in memory using the hexadecimal number system.
• Programmers frequently utilize the hexadecimal numbering scheme to simplify the binary numbering system. There is a linear relationship between the numbers 2 and 16 because 16 is equal to 24. This translates to four binary digits for each hexadecimal digit. To abbreviate binary and make it easier to grasp, computers utilize binary numbering systems, whereas humans use hexadecimal numbering systems.
• Media Access Control (MAC) addresses are also represented by hexadecimal numbers. MAC addresses are hexadecimal numbers with 12 digits. MM:MM:MM:SS:SS:SS or MMMM-MMSS-SSSS are the formats in use. The first six digits of the MAC address are the adapter manufacturer’s ID, while the last six numbers are the adapter’s serial number.
• Error messages are also displayed using hexadecimal numbers. Hexadecimal is used to specify the error’s memory address. This aids programmers in locating and correcting mistakes.

Advantages of Hexadecimal Number System

The advantages of the Hexadecimal Number System are as follows:

• Hexadecimal numbers have the advantage of requiring less memory to store more numbers; for example, 256 numbers may be stored in two digits, but decimal numbers can only store 100 numbers in two digits.
• Computer memory addresses are also represented using this number system. It is simple to convert from hexadecimal to binary and vice versa since it just takes four bits to represent every digit in binary.
• Input and output are easier to manage in hexadecimal format. In the fields of data science, artificial intelligence, and machine learning, there are numerous benefits.
• It facilitates life by allowing binary numbers to be grouped together, making them easier to read, write, and comprehend. It is more human-friendly, as people are accustomed to grouping numbers and objects together for ease of comprehension. Additionally, writing in fewer digits reduces the chance of an error.

Disadvantages of Hexadecimal Number System

The disadvantages of the Hexadecimal Number System are as follows:

• It facilitates life by allowing binary numbers to be grouped together, making them easier to read, write, and comprehend.
• It is more human-friendly, as people are accustomed to grouping numbers and objects together for ease of comprehension.
• Additionally, writing in fewer digits reduces the chance of an error.

Facts About the Hexadecimal Number System

The hexadecimal number system is one way to represent numbers in mathematics and computing. It uses base 16, which means it works with sixteen different symbols. These symbols are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

The letters A to F are used because we need more digits after 9. They stand for:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

In some places, a number may be shown as hexadecimal by adding a small mark or prefix, such as h or 0x (for example, 0xAF means a hexadecimal value).

Each digit in a hexadecimal number has a value that depends on its position, and each position represents a power of 16 — just like decimal numbers use powers of 10.

Solved Examples of Hexadecimal Number System

Let’s see some examples of Hexadecimal Number systems that come in exams.

Solved Example 1: Convert 2c7 from hexadecimal to decimal.

Solution:

Step 1: Write down the place value of each digit.

\(16^2\) 256 2

\(16^1\) 16 12

\(16^0\) 1 7

Step 2: Add up the place value of each digit.

256×2 + 16xc + 1×7

= 512 + 192 + 7 = 711

So 2c7 in hexadecimal is equal to 711 in decimal.

Or we could say \((2c7)_{16}\) = \((711)_{10}\)

Solved Example 2: Convert 1a7d from hexadecimal into decimal.

Solution:

Step 1: Write down the place value of each digit.

\(16^3\) 4096 1

\(16^2\) 256 10

\(16^1\) 16 7

$FC7_{16}$16^0[/latex] 1 13

Step 2: Add up the place value of each digit.

4096×1 + 256×10 + 16×7 + 1×13

4096 + 2,560 + 112 + 13 = 6781

So 1a7d in hexadecimal is equal to 6781.

Or we could say \((1a7d)_{16}\) = \((6781)_{10}\)

Solved Example 3: Evaluate: \((B A 3)_{16}\) + \((5 D E)_{16}\)

Solution:

Hexadecimal Addition:

The table for hexadecimal addition is as follows:

Addition of hexadecimal numbers can be easily carried out with the help of the above table.

We note from the table that

3 + E = 11 (11 carry)

A + D = 17 (B A 3)

17 + 1 (carry) = 18 (5 D E)

B + 5 = 10

10 + 1 (carry) = 11 (1 1 8 1)

Hence the required sum is 1181 in hexadecimal.

Solved Example 4: Convert \(100100010101111_{2}\) to a hexadecimal number.

To convert a binary number to a hexadecimal number, just divide it into groups of four digits (beginning at the right and adding leading zeros if one runs out of digits), and then reinterpret these groups of four as the hexadecimal values indicated above. Having stated that, we have

\(100100010101111_{2}=0100 1000 1010 1111\)

\(0100=4, 1000=8, 1010=A, 1111=F\)

\(100100010101111_{2}=48AF_{16}\)

To convert a binary number to an octal number, we might simply split it down into groups of three digits, and then follow the same steps as when converting a binary number to a hexadecimal number. Let’s take the same binary number and convert it to an octal number:

\(100100010101111_{2}=100 100 010 101 111\)

100=4

010=2

101=5

111=7

\(100100010101111_{2}=44257_{8}\)

It’s even easier to reverse the process. Let’s say we want to convert \(FC7_{16}\) to binary. We may read binary values for each digit of the hexadecimal integer from the table:

\(F_{16} = 1111_{2}$ $C_{16} = 1100_{2}$ $7_{16} = 0111_{2}\)

\(FC7_{16}=111111000111_{2}\)

Process of converting octal number to binary form is just the same.

Hope this article on the Hexadecimal Number System was informative. Get some practice of the same on our free Testbook App. Download Now!