Integers in the Classroom - 2

We will now see the rules for multiplication of integers using the bottle tops. Bottle tops provide a surprisingly simple way to mirror the rules of multiplication.

Let us recall the rules of operation which we summarized in an earlier chapter.

1. The + operator does not change the sign which comes with it.

2. The – operator inverts the sign which comes with it.

Multiplication with “- “

We can see that flipping a bottle top represents multiplication by “-“.

<a name="_Hlk46217762">If the bottle top is in “positive” position, multiplying it by “-“ (or flipping the bottle top) brings it to a “negative” position. This is same as + X - = -.</a>

Conversely, if the bottle tops is in “negative” position, multiplying it by “-“ (or flipping the bottle top) brings it to the “positive” position. This is same as – X - = +.

Multiplication with “+”

Multiplication with a “+” leaves the bottle top in the same position.

If the bottle top is in “positive” position, multiplying it by “+“(or leaving the bottle top undisturbed) keeps it to in a “positive” position. This is same as + X + = +.

Conversely, if the bottle tops is in “negative” position, multiplying it by “+“(or leaving the bottle top undisturbed) keeps it in the “negative” position. This is same as – X + = -.

Multiplication as “repeated put together”

When multiplying with numbers other than 1, we also need to use the idea that multiplication can also be thought of as “repeated addition”. Since we have both + & - numbers, let us take it as “repeated put together”

Let us see some examples.

1. -3 X +4:

   1. We start with a row representing “negative” 3. We can think of it as a row of 3 bottle tops in “open” position.

   2. Since we are multiplying by 4, we repeat the row of 3 bottle tops 4 times

   3. Since it is “positive” 4, we do not flip the bottle tops.

   4. So we have 12 bottle tops, all in “open” or “negative” position, thus representing -12.

   5. So -3 X +4 = -12.

   6. This result can also be written down directly since “ – X + = -“ and 3 X 4 = 12

1. -3 X -4

   1. We start with a row representing “negative” 3. We can think of it as a row of 3 bottle tops in “open” position.

   2. Since we are multiplying by 4, we repeat the row of 3 bottle tops 4 times

   3. Since it is “negative” 4, we flip the bottle tops from “open” to “closed”

   4. So we have 12 bottle tops, all in “closed” or “positive” position, thus representing +12.

   5. So -3 X -4 = +12

   6. This result can also be written down directly since “ – X - = +“ and 3 X 4 = 12

The rules may seem complicated when there are described in writing. Once you practice them you will realize that they are very intuitive and easy.

Chapter 19.13 gives pictorial examples of this approach for all varieties of multiplication.

Integer Operations – A Summary

We can summarise these rules in the following way