Multiplying Integers: Rules & Examples

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  • 0:01 What Are Integers?
  • 1:00 Multiplication
  • 2:29 Multiplying Simple Integers
  • 4:13 Multiplying More…
  • 8:39 Negative Numbers
  • 10:06 Lesson Summary
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Lesson Transcript
Instructor: Jason Furney

Jason has taught both College and High School Mathematics and holds a Master's Degree in Math Education.

In this lesson, we will discuss what numbers are included in the integers and how to multiply them. We will deal with 1-, 2-, and 3-digit numbers as well as negatives and positives. There are some definitions included as well as a quiz at the end.

What are Integers?

An integer consists of positive and negative whole numbers, as well as zero. The neat thing about integers is that they are the most common type of number that we see in our everyday lives. Because of this, it is probably important to know some basic rules that dictate how we can use them.

Adding integers is possibly one of the most familiar operations that we know how to do. Most children start their mathematical journey by counting on their fingers. Each finger when counting represents a whole number, which, in turn, represents a small part of our integers!

Sometimes, in life, we are asked to add the same integer a repeated number of times. For example, 2 + 2 + 2 + 2 + 2 = ? We can go ahead and add these up in parts, 2 + 2 = 4 + 2 = 6 + 2 = 8 + 2 = 10, or we can use another method!

Multiplication

Looking at our example of 2 + 2 + 2 + 2 + 2 = ?, we can start to look for a pattern. For example, how many 2s are in this equation? I count 5. So we could say that we are adding 2 five times. In math, we write this as 2 X 5. The 'X' in this equation represents the mathematical operator 'multiply' or 'times'. This means that we are adding 2 to itself 5 times.

2 X 5 = 2 + 2 + 2 + 2 + 2, which equals 10. Therefore, 2 X 5 = 10. Let's try some others!

3 X 2 = 3 + 3, which equals 6

4 X 6 = 4 + 4 + 4 + 4 + 4 + 4, which equals 24

10 X 4 = 10 + 10 + 10 + 10, which equals 40

Of course, we do not want to have to write out the addition counterpart to every multiplication problem, especially if we get something like 10 X 500 or 123 X 632. I don't know about you, but I don't want to have to write out 10 five hundred times! Because of this, we can use the method of multiplication to make our job simpler.

Multiplying Simple Integers

Multiplication is part memorization as much as it is understanding. So yes, we know that 2 X 5 = 2 + 2 + 2 + 2 + 2 = 10, but how can we know that 2 X 5 = 10 without the addition part? What about 3 X 6 = 18 or 7 X 7 = 49? The answer is yes! But we have to work on it. To start, let's look at a multiplication table:


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A multiplication table usually has 12 or 13 integers going across the top and 12 or 13 going down the side. Say we wanted to know what 5 X 3 is. We go across the top till we hit the number 5 and then we go down until we hit the number 3. Whatever square we reach is our answer.


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Try using the table to find what 5 X 0 is.


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Do you get 0 as your answer? What about 6 X 0 or 12 X 0? Notice something strange? Zero is the easiest integer to multiply by because anything times 0 is 0! Likewise, 0 times anything is also 0!

Notice any other strange occurrences? What about when multiplying by 1? 12 X 1 = 12, 11 X 1 = 11, 10 X 1 = 10, 9 X 1 = 9, and so on. When multiplying anything by the integer 1, you get the number that you multiplied by. One is what is known as the multiplicative identity. This means that when multiplying with 1, nothing changes!

Multiplying More Complicated Integers

As you can see, it is pretty easy to multiply by the first 13 integers (1 - 12 and 0), we can just look it up on our table until we have them memorized. What happens if we reach numbers not on our table? What if we are multiplying two 2-digit numbers together, like 12 X 13? Sure, we could write 12 + 12 + 12 + … + 12 thirteen times, but that could get daunting. Instead, we use a technique called column multiplication. Here's how it works.

First, stack the two numbers on top of each other, making sure each place value lines up.


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Second, we are going to start with the ones place of the bottom number (the 3 in this example) and multiply it to each digit in the top number, like so:


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Notice how when multiplying along the blue line we get the blue number 6, and when multiplying along the red line we get the red number 3. Once every digit of the top number has been used, we move onto the third step.

Third, we are going multiply by the tens digit of the bottom number; this number is now highlighted in green.


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This is where the catch comes in. To multiply by the tens digit, we have to throw in a 0 for a place holder. This works for every digit greater than the ones place. If multiplying by a number in the hundreds digit, we would put two zeros in. This new zero is also highlighted.


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