Hexadecimal Calculation

What is Hexadecimal ?

Hexadecimal is a base-16 number system that is often used in computing and digital electronics as a more human-friendly representation of binary-coded values. One hexadecimal digit represents four binary digits (bits), which is half a byte. For example, a single byte can have values ranging from 00000000 to 11111111 in binary, but this is typically represented as 00 to FF in hexadecimal.

The hexadecimal system uses sixteen distinct symbols. The first ten are the same as the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The system then uses the letters A, B, C, D, E, and F (or lowercase a, b, c, d, e, f) to represent the values ten to fifteen:

• 0-9 represent values zero to nine, as in decimal.
• A (or a) represents the value ten.
• B (or b) represents the value eleven.
• C (or c) represents the value twelve.
• D (or d) represents the value thirteen.
• E (or e) represents the value fourteen.
• F (or f) represents the value fifteen.

Each hexadecimal digit represents a four-bit binary sequence. For example:

Hexadecimal is used in computer science because it provides a more concise way to represent binary numbers. It's much easier for humans to read and write '3E8' for the binary 1111101000 (which is 1000 in decimal) than to work with the longer binary representation.

In computing, hexadecimal is commonly used to display memory addresses, define colors in HTML/CSS, represent MAC addresses, and in many other areas where a succinct representation of binary data is helpful.

How to do hex calculations?

To perform hexadecimal calculations, you should understand the basics of the hexadecimal number system. Each digit in a hexadecimal number represents a power of 16. The far right digit represents 16^0 (which is 1), the next digit to the left is 16^1 (which is 16), and it continues increasing in powers of 16.

Here's a brief guide on how to do calculations in hexadecimal:

• Hexadecimal Addition

  • When adding hexadecimal numbers, you start from the rightmost digit and work your way left, similar to decimal addition. If the sum of two digits is greater than F (15 in decimal), you carry over to the next digit.
  • Example, 1A3 + 2F7:
    • Start from the right: 3 + 7 = A (10 in decimal)
    • Move to the next digit: A (10) + F (15) = 19 in decimal, which is 13 in hexadecimal. Write down the 3 and carry over the 1 (which is 10 in decimal, since 1 hex digit is base 16).
    • Add the leftmost digit: 1 + 1 + 2 = 4 in hexadecimal (no carry because it is less than 16).
  • The result is:
  • 1A3
+ 2F7
    -----
    43A

• Hexadecimal Subtraction

  • Subtraction is done by borrowing, similar to how you subtract in decimal, but you borrow groups of 16 rather than 10.
  • Example, 2A3 - 0D4:
    • Start from the right: You cannot subtract 4 from 3, so you need to borrow. The A (10 in decimal) becomes 9 in hexadecimal, and you add 16 to the 3, making it 19 (13 in hexadecimal). 13 - 4 = F (15 in decimal).
    • Move to the next digit: 9 - D (13) is not possible, so borrow again. The 2 becomes a 1, and you add 16 to 9, making it 25 (19 in hexadecimal). 19 - D (13) = C (12 in decimal).
    • Now, subtract the leftmost digits: 1 - 0 = 1.
  • The result is:
  • 2A3
- 0D4
    -----
    1CF

• Hexadecimal Multiplication

  • Multiplication in hexadecimal is like multiplication in decimal, but you multiply based on hexadecimal values and carry over accordingly.
  • Example, 1A × B:
    • Multiply B (11 in decimal) by A (10 in decimal): 11 × 10 = 110 in decimal, which is 6E in hexadecimal (because 110 divided by 16 is 6 with a remainder of 14, which is E).
    • Multiply B by 1 (since it's the next digit, it actually represents 16 in decimal, so you're actually multiplying by 16): B × 1 = B, and add a 0 to the end because you're actually multiplying by 16: B0.
    • Now add those two results together.
  • The result is:
  • 6E
+ B0
    -----
    11E

• Hexadecimal Division

  • Division is the most complex of the operations and often requires converting to decimal to find the quotient and remainder, especially when doing it by hand. However, the concept remains the same: divide the number by the divisor and find the quotient and remainder.
  • Example, 1A3 ÷ 3:
    • Convert 1A3 to decimal: 1 × 16^2 + A (10) × 16^1 + 3 = 256 + 160 + 3 = 419 in decimal.
    • Divide 419 by 3 (in decimal, since it's easier): 419 ÷ 3 = 139 with a remainder of 2.
    • Convert the quotient back to hexadecimal: 139 in decimal is 8B in hexadecimal (8 × 16^1 + B (11) × 16^0).
  • The result of 1A3 ÷ 3 is 8B with a remainder of 2.

Hex/Decimal Conversion table