The Number Line 2

Up to now we have seen various types of numbers – natural, whole, fractions, decimals, rationals, irrationals & integers.

We revisit the idea of the number line which connects all these numbers under a unifying theme.

We have primarily seen numbers as representing quantities which is a very concrete idea. The Number Line makes us see numbers in a less concrete form, as points on a line. Visually it combines the arithmetic idea of numbers with the geometric idea of a line.

The line is usually imagined as a horizontal line, extending to infinity to the left & the right. 0 can be marked anywhere on this line and acts as a reference point or the origin.

1 is plotted at any convenient point on the line to the right (with reference to the observer) of 0. The length between 0 and 1 becomes a measure for number 1. All other numbers are marked on the line in relation to this measure of 1.

In theory, the number line can have any orientation as long as it is smooth & one-dimensional. Hence it can even be a curved line. For mirroring the time flow of either the 12 hour or 24 hour day, it may even be easier for children to visualise, if we represent the number line as a circular track.

But by convention it is seen as a horizontal line. Also, it is just a convention that –ve numbers fall to the left and +ve numbers fall to the right of 0.

Recent studies are pointing to the fact that the human brain seems to have a normal tendency to associate an increasing sequence of numbers from left to right. This also could be the reason for the universal adoption of a number line where positive numbers proceed from the left to the right.

2 will be marked at a point to the right of 1, at a distance equal to the distance between 0 and 1. Other whole numbers are plotted with the same logic. All the numbers marked to the right are considered +ve numbers.

Similarly, -1 is plotted to the left of 0 at s distance equal to the distance between 0 and 1. All integers are similarly plotted to the left of 0.

Hence, we can imagine the line extending from-∞on the left to+∞on the right. Though we have talked about the number line only after integers, we can introduce this idea right after children learn about numbers and ordering them in a sequence. Please refer to Chapter 5.08 "Number Line 1"

After the introduction of each type of number as quantity, students can also be introduced to the idea of plotting them on the number line. This will continuously reinforce the idea of all numbers belonging to the same family.

While introducing integers, it would be a good idea to draw the number line in a vertical manner with +ve numbers going up. This may make comparison of integers more visually satisfying.

Whole number line had obvious gaps, between numbers.

All rational numbers, both +ve and –ve can be conceptually plotted using geometrical methods. A rational number like 5/7 can be imagined at a distance of 5 segments (from 0) out of 7 segments into which the distance between 0 and 1 can be imagined to be divided.

With rational numbers, the line was thought to become continuous.

All decimal numbers can be plotted by imagining the distance between any 2 consecutive integers to be divided into 10 and powers of 10.

Decimal notation gave a way of representing all irrational numbers with an infinite series of numerals. Even if these numbers cannot be written in exact forms, with terminating decimals, their location on the number line can be pinpointed theoretically. Hence all +ve and –ve irrational numbers can be imagined to be plotted on the number line.

But discovery of irrational numbers revealed that there were gaps in the number line. Cantor proved that the gaps were more than the rest of the line! That is the irrational numbers constituted a larger infinity than rational numbers!

Rational & irrational numbers constituted Real numbers. With the decimal representation of real numbers, it was shown that the real number line was indeed “continuous” without any gaps.

The Number Line enables thinking about all numbers as linear measures and about ordering all the numbers on a single platform.

We can even say that the idea of the number line, led to the idea of a number (Cartesian) plane and eventually to the idea of Imaginary and Complex numbers.

We will now see numbers which could not be plotted on the number line.