Integers in the Classroom - 2

< 19.11 Integers in the Classroom - 1 | Topic Index | 19.13 Integer Additions With Bottle Tops (A) >

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.

Multiplication with &ldquo;- &ldquo;
 * 1)  The + operator does not change the sign which comes with it. 
 * 2)  The – operator inverts the sign which comes with it. 

We can see that flipping a bottle top represents multiplication by &ldquo;-&ldquo;.

If the bottle top is in &ldquo;positive&rdquo; position, multiplying it by &ldquo;-&ldquo; (or flipping the bottle top) brings it to a &ldquo;negative&rdquo; position. This is same as + X - = -.

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

Multiplication with &ldquo;+&rdquo;

Multiplication with a &ldquo;+&rdquo; leaves the bottle top in the same position.

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

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

Multiplication as &ldquo;repeated put together&rdquo;

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

Let us see some examples.


 * 1) -3 X +4:     We start with a row representing &ldquo;negative&rdquo; 3. We can think of it as a row of 3 bottle tops in &ldquo;open&rdquo; position.Since we are multiplying by 4, we repeat the row of 3 bottle tops 4 timesSince it is &ldquo;positive&rdquo; 4, we do not flip the bottle tops.So we have 12 bottle tops, all in &ldquo;open&rdquo; or &ldquo;negative&rdquo; position, thus representing -12.So -3 X +4 = -12.This result can also be written down directly since &ldquo; – X + = -&ldquo; 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.
 * 1) -3 X -4     We start with a row representing &ldquo;negative&rdquo; 3. We can think of it as a row of 3 bottle tops in &ldquo;open&rdquo; position.</li>Since we are multiplying by 4, we repeat the row of 3 bottle tops 4 times</li>Since it is &ldquo;negative&rdquo; 4, we flip the bottle tops from &ldquo;open&rdquo; to &ldquo;closed&rdquo;</li>So we have 12 bottle tops, all in &ldquo;closed&rdquo; or &ldquo;positive&rdquo; position, thus representing +12.</li>So -3 X -4 = +12</li>This result can also be written down directly since &ldquo; – X - = +&ldquo; and 3 X 4 = 12</li></ol>

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

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