Integer Multiplication

Student: So now that I understand addition and subtraction, are there more operations?

Mentor: Yes there are. The next operation is called multiplication. We write multiplication problems in the form a x b, or a times b. For an example we will consider 5 x 3. What this means is that 5 is being added to itself 3 times (5 + 5 + 5), but a better way to think about is that it means 5 groups of 3 units each. Therefore, to solve 5 x 3 you would count the number of units in each group.

Student: I feel a little confused.

Mentor: Alright, we will use an example you can visualize. Picture 5 separate plates, each with 3 quarters on them. How many quarters are there all together?

Student: Well, if there are 3 quarters for each of the 5 plates, then there are 15 quarters all together. So 5 x 3 = 15.

Mentor: Exactly. Now, what if we had 3 plates, each with 5 quarters?

Student: Then there would be 5 quarters for each of the 3 plates, meaning 15 quarters altogether. So 3 x 5 = 15. Does that mean that multiplication is commutative like addition?

Mentor: Yes, multiplication is commutative. For any a and b, a x b = b x a. It is also associative. If you are multiplying three numbers, then (a x b) x c = a x (b x c). Now, what do you think would happen in the case where a number was multiplied by 0 or 1? If you have trouble just try to visualize what it means.

Student: Hmm, if you had any number of plates p, each with 0 quarters, you would have 0 quarters altogether. So any number times 0 equals 0. Now, if you had 1 plate with q quarters, then there would be q quarters altogether. So any number times 1 is just the number itself.

Mentor: That is absolutely right. You thought through that very well.

Student: This isn't so hard, but what about negative numbers? I know how to add them, but how do you multiply negative numbers?

Mentor: Well, first I should explain a little more about what a negative number is. Even though the negative numbers are less than zero, this does not mean that they are less than nothing. A negative number is merely a negation of a positive number. For instance, what would happen if you traveled one mile east and then backtracked one mile west?

Student: You would end up where you started.

Mentor: Exactly, the one mile you traveled west negated the mile you traveled east. Another way to express your movement west would be to say you traveled -(one mile east), or negative one mile east. So we can think of the negative sign as an operator, just as the multiplication sign is an operator. Its operation would be to replace positive units with negative units.

Student: Where exactly is this all leading us?

Mentor: Precisely where we want to be. Remember that a positive unit plus a negative unit always equals zero, thus, the positive and negative of a number have an inverse relationship. This means that the negative of a positive is a negative and that the negative of a negative is a positive. Knowing this, we can now move on to multiplication by negative numbers.

Based off the definition I gave you earlier, what do you think 2 x (-3) equals?

Student: Well, 2 x (-3) would be 2 groups, each with (-3) units, so that would be (-6) units.

Mentor: Good, that one was tricky, but mostly straightforward. How about this one: What is (-2) x 3?

Student: That would be (-2) groups, each with 3 units. How do you count negative groups, though?

Mentor: First, let me rephrase it as negative 2 groups, each with 3 units. Now, a negative group is merely a group that would negate a positive group. What kind of group would negate a group of 3 units?

Student: A group of 3 units would be negated by a group of (-3) units, right?

Mentor: Yes, therefore (-2) x 3 = 2 x (-3), which we already know is equal to (-6) units. Now what do you think (-2) x (-3) equals?

Student: Well, that would be negative 2 groups, each with 3 negative units. So I need to know what kind of group would negate a group of 3 negative units . . . a group of 3 positive units! So (-2) x (-3) = 2 x 3 = 6. I think that's right.

Mentor: That's exactly right. So what you can see from these few examples is that a negative number times a positive number will give you a negative number and a negative number times a negative number will give you a positive number.