Hexadecimal Numbers Conversions

The hexadecimal number can be easily converted to various other numbers such as, Binary Numbers, Octal Numbers, Decimal Numbers and vice-versa. Now let’s learn about them in detail.

Hexadecimal to Decimal Conversion

Converting hexadecimal to decimal follows a similar process as before, where each digit is multiplied by the respective power of 16.

Example: Convert (A7B)₁₆ to decimal.

(A7B)₁₆ = A × 16³ + 7 × 16² + B × 16¹

⇒ (A7B)₁₆ = 10 × 4096 + 7 × 256 + 11 × 16 (convert symbols A and B to their decimal equivalents; A = 10, B = 11)

⇒ (A7B)₁₆ = 40960 + 1792 + 176

⇒ (A7B)₁₆ = 42828

Therefore, the decimal equivalent of (A7B)₁₆ is (42828)₁₀.

Decimal to Hexadecimal Conversion

To convert a decimal number to hexadecimal, we use the base number 16. The process involves dividing the number by 16 repeatedly until the quotient becomes zero. The decimal to hexadecimal number system is shown in the image added below,

Example: Convert (92)₁₀ to hexadecimal.

Solution:

Divide 92 by 16

Quotient: 5, Remainder: 12 (C in Hexadecimal)

Divide 5 by 16

Quotient: 0, Remainder: 5

Write the remainders from bottom to top

Therefore, (92)₁₀ is equivalent to (5C)₁₆ in hexadecimal.

Hexadecimal to Octal Conversion

To convert a hexadecimal number to octal, we follow a two-step process: first, convert the hexadecimal number to decimal, and then convert the decimal number to octal.

Example: 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

Octal to Hexadecimal Conversion

There is a two step process to convert an octal number into hexadecimal:

Step 1: Convert Octal to Binary

For each octal digit, replace it with its three-digit binary equivalent.

Example: Convert (345)₈ to binary.

Solution:

Step 1: Convert Octal to Binary

3 in octal is 011 in binary

4 in octal is 100 in binary

5 in octal is 101 in binary

Combine these binary equivalents: (345)₈ = (011100101)₂

Step 2: Convert Binary to Hexadecimal

Group the binary digits into sets of four, starting from the right, and convert each set to its hexadecimal equivalent.

Example 2: Convert (011100101)₂ to hexadecimal.

Solution:

0111 in binary is 7 in hexadecimal

0010 in binary is 2 in hexadecimal

1101 in binary is D in hexadecimal

Combine these hexadecimal equivalents: (011100101)2 = (72D)₁₆

Therefore, (345)₈ is equivalent to (72D)₁₆ in hexadecimal.

Hexadecimal to Binary Conversion

Converting hexadecimal to binary involves two methods: one with a conversion table and the other without a conversion table.

Method 1: Convert Hexadecimal to Binary with Conversion Table

To convert a hexadecimal number to binary using a conversion table, we follow these steps:

Example: Convert hexadecimal (4D)₁₆ to binary.

Solution:

Look up Decimal Equivalent of each digit in the conversion table.

4 in decimal is (4)₁₀, and D in decimal is (13)₁₀

Convert each decimal number to binary.

(4)₁₀ is (0100)₂, and (13)₁₀ is (1101)₂

Combine the binary numbers

(4D)₁₆ is (01001101)₂

Method 2: Convert Hexadecimal to Binary without Conversion Table

This method involves multiplying each digit by 16^(n-1) to obtain the decimal number, and then dividing by 2 until the quotient is zero.

Example: Convert hexadecimal (A2)₁₆ to binary.

Solution:

Convert (A2)₁₆ to decimal

(A)₁₆ is (10)₁₀, and (2)₁₆ is (2)₁₀

⇒ (A2)₁₆ is 10 × 16¹ + 2 × 16⁰ = 160 + 2 = 162₁₀

Convert the decimal number (162)₁₀ to binary

Divide 162 by 2: Quotient = 81, Remainder = 0

Divide 81 by 2: Quotient = 40, Remainder = 1

Continue dividing until the quotient is zero: (10100010)₂

Therefore, (A2)₁₆ is (10100010)₂ in binary

Binary to Hexadecimal Conversion

To change binary to hexadecimal, we refer to a conversion table from the previous section.

Example: Convert (10111010101)₂ to hexadecimal.

Solution:

In hexadecimal, every 4 binary digits represent one digit

Group the binary number accordingly, and find their Hexadecimal equivalent using the hexadecimal table added above.

0010 = 2, 1011 = B, 1010 = A

Combine these hexadecimal digits to get the final number.

Therefore, (10111010101)₂ is equal to (2BA)₁₆

Hexadecimal Number System is a base-16 numeral system used in diverse fields, especially in computing and digital electronics. It consists of 16 symbols, including numbers 0 to 9 and letters A to F, offering a compact way to represent binary-coded values. The hexadecimal number system is sometimes also represented as, ‘hex’.

Number Systems are various ways to use numbers to represent large numbers and information. The hexadecimal number system is introduced to students in class 9. In this article, we will learn about, the Hexadecimal Number System, Hexadecimal Number System Table, Hexadecimal Number System Examples, and Others in detail.

Before starting with the Hexadecimal Number System we first learn about the Number System.