Math Lesson 1.7.5 - The Base 16 Number System

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Welcome to our Math lesson on The Base 16 Number System, this is the fifth lesson of our suite of math lessons covering the topic of Decimal Number System and Other Numbering Systems, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Base 16 Number System

We use the same approach to represent numbers and operations in the base 16 number system (otherwise known as the hexadecimal system). In this system there are 16 digits in total. In the decimal system there are only 10 digits so we complete the rest of digits in the hexadecimal system using letters of the alphabet. Thus, in addition to the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, we use A for 10, B for 11, C for 12, D for 13, E for 14 and F for 15. Everything else is same as the decimal and binary system. The hexadecimal number system is widely used in computers and programming, hence we are explaining it here.

For example, the number AC923 has the value:

We will now explain addition and subtraction in the hexadecimal number system through an example.

Example 7

Find the value of operations in the hexadecimal number system and prove the accuracy of the solutions made.

1. F95 + DA
2. 38C - 1BE

Solution 7

1. We have: Let's clarify the operations made. Thus, 5 + A = 5 + 10 = 15, which corresponds to the letter F. Then, 9 + D = 9 + 13 = 22, so we write 6 and carry one 16 (1 unit therefore) to the next position on the right. Then we have 1 + F = 1 + 15 = 16, which we write as 10 in the hexadecimal system.
   Proof:: F95 = 15 × 16² + 9 × 16¹ + 5 × 16⁰ = 15 × 256 + 9 × 16 + 5 × 1 = 3,840 + 144 + 5 = 3,989; and DA = 13 × 16¹ + 10 × 16⁰ = 208 + 10 = 218. Hence, 3,989 + 218 = 4,207.
   On the other hand, 106F = 1 × 16³ + 0 × 16² + 6 × 16¹ + 15 × 16⁰ = 4,096 + 96 + 15 = 4, 207. Hence, the two results obtained through different methods of calculations match.
2. We have Explanation: C = 12 and E = 14, so C - E is not possible as operation, therefore we have to borrow one 16 from the next position on the right. Hence, we have 16 + 12 = 28 - E = E, because E = 14. The number 8 becomes 7 as one 16's was borrowed to units. Since 7 - B is not possible, we borrow one 16 from the next position, so we have (16 + 7) - B = 23 - 11 = 12, which corresponds to the letter C in the hexadecimal system. The number 3 became 2 as one unit was borrowed to 7. So, we have 2 - 1 = 1. The difference therefore is 1CE.
   Proof:: 38C = 3 × 16² + 8 × 16¹ + 12 × 16⁰ = 3 × 256 + 8 × 16 + 12 × 1 = 768 + 128 + 12 = 908; and 1BE = 1 × 16² + 11 × 16¹ + 14 × 16⁰ = 1 × 256 + 11 × 16 + 14 × 1 = 256 + 176 + 14 = 446. Hence, 908 - 446 = 462.
   On the other hand, 1CE = 1 × 16² + 12 × 16¹ + 14 × 16⁰ = 1 × 256 + 12 × 16 + 14 × 1 = 256 + 192 + 14 = 462. Hence, the two results obtained through different methods of calculations do match and we have proved the accuracy of our calculations.

Multiplications and divisions in the hexadecimal system are completed in the same way as we explained in the other numbering systems. However, we will not deal with them here because of their complexity in calculations and it is unlikely that you will need to calculate these manually.

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7. Continuing learning arithmetic - read our next math tutorial: Numbering Systems, a Historical View

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