Hexadecimal Number System

The Hexadecimal system is a base-16 number system that plays an important role in computing and digital systems. It uses sixteen symbols to represent values:

Digits (0 to 9) and the letters A to F, where A = 10, B = 11, and so on up to F = 15.

Place Value of Digits in the Hexadecimal Number System

The numbers in the hexadecimal number system have weightage in powers of 16. The power of 16 increases as the digit is shifted towards the left of the number. This is explained by the example as,

Example: (AB12)₁₆

Place value of each digit in (AB12)₁₆ is,

= A×16³ + B×16² + 1×16¹ + 2×16⁰

Conversion from Hexadecimal to Other Number Systems

Conversion of a number system means conversion from one base to another. The following are the conversions of the Hexadecimal Number System to other Number Systems:

Hexadecimal to Decimal Conversion: 

To convert a hexadecimal number to decimal (base-10), multiply each digit by its corresponding power of 16 and sum the results.

Example: To convert (8EB4)₁₆ into a decimal value

Follow the steps given below:

• Step 1: Write the decimal values of the symbols used in the Hex number i.e. from A-F
• Step 2: Multiply each digit of the Hex number with its place value. Starting from right to left i.e. LSB to MSB.
• Step 3: Add the result of multiplication and the final sum will be the decimal number.

Hexadecimal to Binary Conversion

Each hexadecimal digit corresponds to a 4-bit binary sequence. Convert each digit individually and combine.

Example: (B2E)₁₆ is to be converted to binary

Follow the steps given below:

• Step 1: Convert the Hex symbols into their equivalent decimal values.
• Step 2: Write each digit of the Hexadecimal number separately.
• Step 3: Convert each digit into an equivalent group of four binary digits.
• Step 4: Combine these groups to form the whole binary number.

Hexadecimal to Octal Conversion:

Convert hexadecimal to binary, group the binary digits into sets of three (right to left), and convert each group to its octal equivalent.

Example: (B2E)₁₆ is to be converted to hex

Follow the steps given below:

• Step 1: We need to convert the Hexadecimal number to Binary first. For that, follow the steps given in the above conversion.
• Step 2: Now to convert the binary number to an Octal number, divide the binary digits into groups of three digits starting from right to left i.e. from LSB to MSB. 
• Step 3: Add zeros before MSB to make it a proper group of three digits(if required)
• Step 4: Now convert these groups into their relevant decimal values.

Decimal to Hexadecimal Conversion

Divide the decimal number by 16 repeatedly, noting remainders, until the quotient is 0. Read remainders in reverse order, using A–F for 10–15.

Related Articles:

• Hexadecimal to Decimal Converter
• Types of Number System
• Binary Formula
• Difference Between Decimal and Binary Number Systems

Facts About Hexadecimal Numbers

• Hexadecimal is a number system with a base value of 16.
• Hexadecimal numbers use 16 symbols or digital values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• A, B, C, D, E, and F represent 10, 11, 12, 13, 14, and 15 in single-bit form.
• If you see an "0x" as a Prefix, it indicates the number is in Hexadecimal. For example, 0x3A
• The position of each digit in a Hexadecimal number has a weight of 16 to the power of its position.

Solved Examples on Hexadecimal Number System

Example 1: Convert Hexadecimal 1A5 to Decimal

Solution:

Multiply First Digit (1) by 16 squared (256)

1×16² = 256

Multiply Second Digit (A, which is 10 in decimal) by 16 to the power of 1 (16)

10×16¹ = 160

Multiply Third Digit (5) by 16 to the power of 0 (1)

5×16⁰ = 5

Adding the results,

1A5 = 1×16² +A×16¹ + 5×16⁰

⇒ 1A5 = 1×16² + 10×16¹ + 5×16⁰

⇒ 1A5 = 256 + 160 + 5 = 421

Decimal Equivalent of Hexadecimal number 1A5 is 421

Example 2: Convert Decimal 315 to Hexadecimal.

Solution:

Divide Decimal Number by 16

315÷16 = 19 with Remainder 11

The remainder (11) is represented as B in hexadecimal

Repeat the division with the quotient (19)

19÷16 = 1 with Remainder of 3

The remainder (3) is represented as 3 in hexadecimal

Hexadecimal Equivalent of Decimal Number 315 is 13B

Example 3: Convert (1F7)₁₆ to Octal.

Solution:

Step 1: Convert (1F7)₁₆ to decimal using the powers of 16:

(1F7)₁₆ = 1 × 16² + 15 × 16¹ + 7 × 16⁰

⇒ (1F7)₁₆ = 1 × 256 + 15 × 16 + 7 × 1

⇒ (1F7)₁₆ = 256 + 240 + 7

⇒ (1F7)₁₆ = (503)₁₀

Step 2: Convert the decimal number (503)₁₀ to octal by dividing it by 8 until the quotient is 0

503 ÷ 8 = 62 with a remainder of 7

62 ÷ 8 = 7 with a remainder of 6

7 ÷ 8 = 0 with a remainder of 7

Arrange the remainder from bottom to top

Therefore, (1F7)₁₆ is equivalent to (767)₈ in octal

Example 4: Convert (A7B)₁₆ to decimal.

(A7B)₁₆ = A × 16² + 7 × 16¹ + B × 16⁰

⇒ (A7B)₁₆ = 10 × 256 + 7 × 16 + 11 × 1 (convert symbols A and B to their decimal equivalents; A = 10, B = 11)

⇒ (A7B)₁₆ = 2560 + 112+ 11

⇒ (A7B)₁₆ = 2683

Therefore, the decimal equivalent of (A7B)₁₆ is (2683)₁₀.

Practice Questions on Hexadecimal Number System

Question 1: Convert the hexadecimal number 2A to binary.

Question 2: Convert the binary number 110110 to hexadecimal.

Question 3: Add the hexadecimal numbers 1F and A3. Provide the result in hexadecimal.

Question 4: Subtract the hexadecimal number B6 from D9. Provide the result in hexadecimal.

Question 5: Multiply the hexadecimal number 7E by 3. Provide the result in hexadecimal.

Answer Key

1. 101010
2. 36
3.C2
4.23
5.17A