Modular Arithmetic: Rules & Properties

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  • 0:03 Modular Arithmetic
  • 1:59 Addition in Modular Arithmetic
  • 3:17 Subtraction in Modular…
  • 5:35 Multiplication in Modular Math
  • 7:10 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Modular arithmetic is something we use everyday when we look at a clock. In this lesson, we'll look at some rules and properties of addition, subtraction, and multiplication in modular arithmetic and how to apply them.

Modular Arithmetic

Before we get to the rules and properties of modular arithmetic, let's first review the meaning of the term. Modular arithmetic, or clock arithmetic, is something we use on a daily basis. In a regular clock, such as the one pictured here, civilians tell time according to two equally spaced intervals composed of 12 hours, or integers. In modular arithmetic, whole numbers, or integers, repeat around a designated number known as the modulus.

Modular 1

In traditional clocks and watches, the modulus, or the number at which we begin again, is 12. For example, if it's 10AM, and you're meeting friends four hours later, you'll most likely say you're meeting them at 2PM, not fourteen o'clock. Now, if we were working in modulus 7, then we would start over at 7. So, 6 + 5 in modulus 7 would be 4, not 11.

Another way to look at this is with remainders. When we're working in modulus n, then any number in modulus n is equal to the remainder when that number is divided by n. Consider our modulus 7 example: 6 + 5 = 11. When we divide 11 by modulus 7, we get 1 with a remainder of 4; therefore, 6 + 5 = 4.

Notice that when dealing with modular arithmetic, there are many numbers that are equivalent for a given modulus. Consider the number 5 in modulus 12:

  1. 5 + 12 = 17
  2. 17 + 12 = 29
  3. 29 + 12 = 41

Here, 5 is equivalent to 17, 29, 41, and so on. We can express these equivalencies by saying the numbers are congruent in modulus 12, and write them using a 3-bar equal sign as shown here:

modular 2

Addition in Modular Arithmetic

You can perform many of the same operations with modular math that you can with regular math. Here are some rules for addition in modular arithmetic:

Modular 3

Based on these rules, we can either add the numbers together, and then find the sum in modulus n. Alternatively, we can first find each of the numbers in modulus n, and then add them together. Let's try working in modulus 8.

What is (4 + 7 + 6 + 8) mod 8?

  1. Add the numbers in the parentheses: 4 + 7 + 6 + 8 = 25
  2. To find 25mod8, perform the division: 25 / 8 = 3 with a remainder of 1
  3. Therefore, (4 + 7 + 6 + 8)mod8 is congruent to 25mod8, which is congruent to 1mod8

modular 4

Subtraction in Modular Arithmetic

The following rules can be used to perform subtraction in modular arithmetic:

Modular 5

Again, we have two choices. We can subtract the numbers first, and find modulus n. Or we can find each of the numbers in modulus n first, and then perform the subtraction. In this example, let's work in modulus 5.

What is (104 - 53)mod5?

  1. Subtract the numbers in the parentheses: 104 - 53 = 51. Here, (104 - 53)mod5 is congruent to 51mod5.
  2. Perform the division: 51 / 5 = 10 with a remainder of 1. So, (104 - 53)mod5 is congruent to 1mod5.
  3. Alternatively, we could first find 104mod5 and 53mod5, or 4mod5 and 3mod5 respectively, and then subtract: 4mod5 - 3mod5 is congruent to 1mod5.

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