Representation of Integers

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There are various ways in which integers can be represented.

Integers on the Number Line

The number line was imagined as a line with its left extremity marked as 0 and the numbers from 1 onwards, represented sequentially and evenly spaced to the right. There is psychological evidence that the left-right orientation of the number line seems to be built into our brain.

Integers easily fit into the idea of the number line. Negative numbers could be thought of as magnitudes in the opposite direction to the positive numbers. The number line, which started from 0 and proceeded (conventionally to the right) to positive numbers in an ascending order, can also be extended from 0 in the opposite direction, which by convention, would be to the left. The number line would now look like this.

It also fits with the logic of mathematics.

Comparison of integers
 * 1) Starting from 0, a movement of +3 takes you to point &ldquo;+3&rdquo;. A movement of -3 from 0 takes you to point &ldquo;-3&rdquo;.
 * 2) Starting from +3, a movement of +2 take you to +5. A movement of -2 takes you to +1.

Extending the number line also helps in comparing numbers, whether they are positive or negative. Any number to the right is more than the number. Any number to the left is less than the number. Hence, we can arrive at the following results.

If + is taken as &ldquo;climbing up&rdquo; &amp; - as &ldquo;climbing down&rdquo;, then -3 is deeper than -2. We can interpret this as saying that -3 is less than -2 in comparison to the floor level. This could be an instance of comparing integers using a real-life situation.

Using the terms &ldquo;positive&rdquo; and &ldquo;negative&rdquo;

In the initial stages, students are likely to get confused at the use of the operators together! We can make the introduction easier by using the term &ldquo;positive&rdquo; and &ldquo;negative&rdquo; to indicate the directions.

Hence -4 +-4 could be written as N4 + N4 read out as &ldquo;Negative 4 added to Negative 4&rdquo;. It can be explained as &ldquo;Movement of 4 steps in a particular direction added to a movement of 4 steps in the same direction&rdquo;. The restatement of the problem enables us to realise that the result would be 8 steps in the same direction and hence &ldquo;N8 or -8&rdquo;! Once students get used to these ideas we can switch to the standard mathematical representation.

Representing integers with 2-sided bottle tops

A simple bottle top is an excellent manipulative to represent integers. It has 2 distinct sides which can be used to visually represent the idea of &ldquo;direction&rdquo;. The two sides can be thought of as &ldquo;open&rdquo; and &ldquo;closed&rdquo; and used to indicate the two directions. If one of the sides is taken to represent any one direction, then the other side automatically will indicate the other direction.

If the bottle top is placed on the floor with its &ldquo;open&rdquo; side up it could be called &ldquo;positive&rdquo; and the other position, &ldquo;closed&rdquo; side up, can be considered &ldquo;negative&rdquo;. Positive 3 would be 3 bottle tops with their &ldquo;open&rdquo; side up. Negative 3 would be 3 bottle tops with their &ldquo;closed&rdquo; side up.

The bottle tops can also be &ldquo;put together&rdquo; or &ldquo;taken away&rdquo;. Thus, both the interpretations of + &amp; - can be achieved. Performing integer operations using bottle tops will be explained in a further chapter.

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