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. 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

Suggested Quiz

9 Questions

How many symbols are there in the Hexadecimal number system?

• A

  16

• B

  15

• C

  8

• D

  7

Explanation:

The hexadecimal number system has 16 unique symbols or numbers:

• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Letters: A, B, C, D, E, F (representing 10 to 15 in decimal)

What is the hexadecimal equivalent of the decimal number 255?

• A

  FF

• B

  AF

• C

  2F

• D

  F2

Explanation:

To convert a decimal number (base-10) to hexadecimal (base-16), we repeatedly divide the decimal number by 16 and record the remainders.

• Divide 255 by 16:
  Quotient = 15, Remainder = 15.The remainder, 15, is represented as F in hexadecimal.
• Divide the quotient (15) by 16:
  Quotient = 0, Remainder = 15.Again, the remainder, 15, is represented as F in hexadecimal.

Write the remainders from bottom to top: FF

If the color red is represented as #FF0000 in hexadecimal code, and 1010 (binary) is added to it, what is the hexadecimal code of the new color formed?

• A

  #FF1010

• B

  #FF0010

• C

  #FF00A0

• D

  #FF000A

Explanation:

The binary 1010 converts to A in hexadecimal, which is added to the blue channel (00) of #FF0000, resulting in #FF000A.

Find the two's complement of (AF)₁₆.

• A

  0101 0101

• B

  0101 0001

• C

  0001 0101

• D

  1010 0001

Explanation:

Decimal	175
Binary	1010 1111
Hex	AF
1's Complement	0101 0000
2's Complement	0101 0001

What is the decimal value of (8AB)₁₆ + (B78)₁₆.

• A

  1423

• B

  5155

• C

  1251

• D

  1444

Explanation:

Covert Hex values to decimals:
8AB = 2219
B78 = 2936

Add the values
2219 + 2936 = 5155

The Logic XOR operation of (6AFD)₁₆ and (CDF)₁₆ is?

• A

  6622

• B

  1100

• C

  6EFF

• D

  66AA

Explanation:

Convert Hexadecimal NUmbers to binary:

• Convert (6AFD)₁₆: 0110 1010 1111 1101
• Convert (CDF)₁₆: 0000 1100 1101 1111

Perform the XOR operation:

• Compare each bit of (6AFD)₁₆​ and (CDF)₁₆:
  • 0110011000100010

Covert Binary back to Hexadecimal:

• 0110 = 6
• 0110 = 6
• 0010 = 2
• 0010 = 2

Hexadecimal result: 6622

Convert (5A1.C)₁₆ to Octal.

• A

  (2641.6)₈

• B

  (2461.6)₈

• C

  (3661.6)₈

• D

  (3614.6)₈

Explanation:

Convert the hexadecimal to binary:

• 5: 0101
• A: 1010
• 1: 0001
• C: 1100

So, (5A1.C)_16​ = (0101 1010 0001.1100)₂

​Convert the binary number to octal:
Group the binary digits into sets of three, starting from the decimal point. Add leading or trailing zeros if necessary. ( 010 110 100 001.110 000 )

Now convert each group to octal:

• 010 corresponds to 2
• 110 corresponds to 6
• 100 corresponds to 4
• 001 corresponds to 1
• 110 corresponds to 6
• 000 corresponds to 0

So, the octal representation is (2641.6)₈

If a hexadecimal number has n digits, what is the range of values it can represent?

• A

  0 to (16ⁿ − 1)

• B

  0 to (16ⁿ + 1)

• C

  1 to (16ⁿ - 1)

• D

  -1 to (16ⁿ − 1)

Explanation:

For unsigned hexadecimal numbers, all n digits are used to represent the magnitude, Range of Values: 0 to (16ⁿ − 1)

Find the division of (70592)_{16 ÷} (80)₁₆.

• A

  E0B

• B

  E0A

• C

  1BD

• D

  1DB

Explanation:

Convert the hexadecimal numbers to decimal

(70592)^16​:

• 7: 7 × 16⁴ = 7 × 65536 = 458752
• 0: 0 × 16³ = 0 × 4096 = 0
• 5: 5 × 16² = 5 × 256 = 1280
• 9: 9 × 16¹ = 9 × 16 = 144
• 2: 2 × 16⁰ = 2 × 1 = 2

Total: 458752 + 0 + 1280 + 144 + 2 = 460178

(80)_16​:

• 8: 8 × 16¹ = 8 × 16 = 128
• 0: 0 × 16⁰ = 0 × 1 = 0

Total: 128 + 0 = 128

Perform the division in decimal: 460178/128 ≅3595.141

3595 in decimal to hexadecimal: (E0B)₁₆

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