The number system with base sixteen is called the Hexadecimal number system (HEX). We use a set of symbols to represent the numbers in this system. The first ten of these are the decimal digits 0, 1, 2, 3...., 9 having the same significance as in the decimal number system. The remaining six numbers are the English alphabets A, B, C, D, E and F and they are equivalent to the decimal number 10, 11, 12, 13, 14 and 15 respectively. The place value of different digit in a mixed hexadecimal number are $16_{0}, 16_{1}, 16_{2}$ and so on.
This number system gives a condensed way of representing large binary numbers stored and processed inside the computer. In this section we will be learning more about hexadecimal numbers and its operations.

The hexadecimal number chart is given below.

Hexadecimal Number Chart
Addition can be done directly with hexadecimal number by remembering that the hexadecimal digits 0 through 9 have the same value of decimal digits 0 through 9 and that hexadecimal digits A through F are equivalent to decimal numbers 10 through 15.
The following rules has to be remembered while adding to hexadecimal numbers.
  1. In any given column of an addition problem, think of the two hexadecimal digits in terms of their decimal values. For example $5_{16} =5_{10}$ and $C_{16}=12_{10}$.
  2. If the sum of these two digit is $15_{10}$ or less, bring down the corresponding hexadecimal digit .
  3. If the sum of these two digit is greater than $15_{10}$ bring down the amount of the sum that exceeds $16_{10}$ and carry a 1 to the next column.

Example for hexadecimal addition:


Example 1:
$13_{16}$

$16_{16}$
________
$29_{16}$
________


Example 2:
$59_{16}$

$22_{16}$
________
$7B_{16}$
________


Hexadecimal Numbers Problem
To subtract two hexadecimal numbers there are different methods, the simple method is that subtraction can be written as addition of the compliment of a hexadecimal number.

The following are the complimentary pairs of digits: 0 and F, 1 and E, 2 and D, 3 and C, 4 and B, 5 and A, 6 and 9, and 7 and 8. Change each digit in the number being subtracted to its compliment and add the resulting hexadecimal number.

If the final result gives you more digits than you had when you started, add the extra digit 1 to the result to get the answer.
Example 1: Find the solution of $F_{16}$ - $B_{16}$.

To find the solution add the compliment of $F_{16}$ and $B_{16}$

Compliment of $F_{16}$ = 0

Compliment of $B_{16}$ = 4

$0$ +

$4$
________
$4_{16}$
_______

Hexadecimal Numbers Example