Modulo & Floor Division

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One new mathematical operation that is very important to programming is the modulo operation. In code, we typically use the percent sign for this operation, as in this example.

The modulo operation produces the remainder of a division. Remember long division? That’s where this comes from.

So, in this example, 9 % 5 returns 4, which is the is the same as saying 9 / 5, which is 1 with a remainder of 4.

In fact, many times, the modulo operation is referred to as “clock arithmetic” because of the similarities. Consider this example from Wikipedia. If the time is 9:00, and we want it to be 4 hours later, we know that it will be 1:00. That is because (9 + 4) % 12 = 1. On the clock, we see the hour hand cycle around the top, starting back over at 1.

Another way to visualize the modulo operation is by looking at a number line. In this example, we have a number line on the first line, followed by a line showing just the multiples of 4. Then, the bottom line shows the result of any value mod 4. So, if we want to find 6 % 4, we can just look for 6 in the top line, then find the result in the bottom.

Finally, we can use the modulo operation to calculate some handy facts. First off, if we take any number and perform the modulo operation with 2, we can find out if that number is even or odd. Specifically, if the result is 0, we know that it must be even since it is divisible by 0. If not, it is an odd number.

In fact, we can perform that operation to find out if a number is evenly divisible by any other number, as long as the result of the modulo operation is 0. For example, we can do 15 % 5 and find that the result is 0, so we know that 15 is evenly divisible by 5. Pretty cool, right?

We can also use that fact to determine how many extras would be in a group. So, we could do 19 % 8 to find out how many extras would be left over after dividing 19 people into groups of 8. Since the result is 3, we’d know that there are 3 people left over. If we did groups of 9, we’d do 19 % 9 and find the result is 1, so maybe that is a bit better.

Finally, as we learned previously, we can use the modulo operation to convert from 24-hour time to 12-hour time. So, we know that 1500 hours in 24-hour time is 3:00 PM in 12-hour time, since 15 % 12 = 3.

Beyond the modulo operator, we should also talk a little bit about division in programming. Depending on your programming language and the division operation you use, you may get a different result than what you are expecting.

For example, we usually think the result of 7 / 4 is 1.75. That answer is completely correct, and when dealing with floating point values in a programming language, that should be the result you receive.

However, when working with integers, or whole numbers, you might find that the result of 7 / 4 is just 1. This is because programming languages such as Java or Python will try to truncate or floor the value down to the next smallest whole number.

In Java, any division operation performed on two integers will result in a truncated result. In Python, however, the division operation will typically return a floating point value, unless you use the special operator for floor division, two slashes, which will be covered later in this chapter.

What’s really handy about this operation is how well it integrates with the modulo operation we discussed earlier. For example, let’s say we wanted to divide 14 people into equal groups of 4. How many groups could we make? We can use the truncated or floor division operation to divide 14 / 4 to get 3. We can make 3 groups.

Then, we might want to know how many people would be left over after creating the groups. For that, we could use the modulo operation to calculate 14 % 4, which is 2. So, if we made 3 groups of 4 out of 14 people, there would be 2 people left over.

Finally, we can confirm that answer by multiplying 4 by 3 to get the people in the groups, then add the remainder or the result of the modulo operation to get back to our original number of 14, confirming that everyone is accounted for.

One thing that is important to understand, however, is that each programming language deals division and modulo with negative numbers a bit differently. Java, for example, will always truncate the result, so -7 / 4 will be -1 in Java. Python, however, always calculates the floor of that operation, so it will round down the result of -7 / 4 to -2 instead. Similarly, Java and Python treat the modulo operation with negative numbers differently as well. You’ll learn about how your chosen language works later in this chapter.

Hopefully these examples help you understand the modulo operation and division in programming languages a bit more.