Irrational numbers on the number line

Question

We can construct a right triangle with its each leg as 1 unit,then the hypotenuse would be √2 units,and then we can point √2 on the number line.
But √2 has a non-terminating and non-recurring decimal representation.We always approximate the value of √2 up to certain decimal places. What is the need for approximation,as we already know the correct lenght of √2 on the number line.

So my question is -

√2 can be plotted on the number line, and we know its exact length So how √2 has a non-terminating decimal and non-recurring decimal representation,It must have have a fixed value ,as hypotenuse of the triangle has a fixed value.

Answer 1

You can approximate the square root of $2$ to arbitrary precision using a sufficiently precise straightedge and compass. (Also note, the precision can be attained by using larger materials rather than more exacting tiny measurements.)

This is not the same as achieving the exact value of the square root of $2$.

For practical purposes such as constructing a building, what's necessary is that your construction be precise enough to satisfy aesthetic qualities and architectural integrity, i.e., the walls of your building should meet at the corners and the building should stay up. Construction foremen have no professional interest in the exact decimal value of the square root of $2$, but they might be interested in more exact means of constructing a "perfect" right angle (for instance, using electronic equipment rather than the Egyptian method with rope—although that was itself pretty good).

For the puristic (read: abstract, theoretical) sake of math itself, irrational numbers are interesting.

Remember that although geometry gives an excellent model of the real world and the relationships of shapes, lines, distances, angles, it is just a model. In the real world, there are no lines without width; there are no precise 90 degree angles; etc.

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Addendum: It's worth considering the fact that it's just as impossible to make a piece of wood whose length is an absolutely perfect double of the length of another piece of wood. In order to apply the abstraction of mathematics to the real world, it is necessary to have some concept of the scale at which you are dealing. The tool of "significant figures" is very useful in bridging the gap from abstraction to actuality. Not all decimal digits are important.

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The fact remains that if multiplication is defined as a method of manipulating decimal numbers so as to get another decimal number, there is no decimal number which can be finitely expressed (terminating decimal) that can be multiplied by itself and result in exactly $2$.

If you define multiplication geometrically in some manner involving a unit length and construction of additional lengths via similar triangles, you can construct a length which is exactly between 1 and 2 multiplicatively. This is not a decimal expression of a number, though.

Answer 2

By using a compass and a ruler we can construct a right triangle with its each leg as 1 unit

Ruler and compass are not concerned with "units", but just with straight lines and circles.

then the hypotenuse would be √2 units,and then we can point √2 on the number line.

Yes, but - per the previous point - leave out the units for now. You can construct an arbitrary isosceles right triangle, then yes, you can mark the length of its hypotenuse on the line definining one of the legs.

As √2 has non-terminating and non-recurring decimal representation, we should not be able to point it on the number line.

Why? So far, it's all been a geometric ruler-and-compass construction, which defined a few points.

Now, take the leg where the length of the hypotenuse was marked, and choose one (arbitrary) point of it to be the "unit".

• If you choose that point to be the endpoint of the leg, then that would be $1$ (rational) and the point marking the length of the hypotenuse would be $\sqrt{2}$ (irrational).

• If however you choose the unit point to be the one marking the length of the hypotenuse, then that would be $1$ (rational) and the leg would be $1 / \sqrt{2}$ (irrational).

Why would choosing the unit after the fact affect the legitimity of the construction itself? Of course, it doesn't. All that rationality tells is whether the ratio of two lengths can be expressed as the ratio of two integer numbers or not. It doesn't make either length less "measurable" than the other.

Answer 3

The question as to whether every number is a rational number is ancient and philosophically interesting - and the constructibility of the hypotenuse of an isosceles right-angled triangle in the Euclidean plane is a paradigmatic example, which shows that in Euclidean geometry there are more numbers than rationals.

The idea that the circumference of a circle has a length or that a circle has an area likewise leads to a demonstration that there are useful numbers which cannot be constructed exactly using Euclidean methods.

The understanding of the number line as consisting of the "Real Numbers" is a mathematical development which has facilitated studies in continuity and calculus. The Real Numbers are uncountable, as shown by Cantor. However it is also easy to prove (since we have only a finite alphabet) that the nameable numbers are countable. Why the difference? Well it seems important to know in advance the existence of any number we may construct (methods of construction not now confined to Euclidean methods).

Of course we cannot in practice construct any number exactly as a physical artefact - a point drawn in ink or pencil on a line has a size, a piece of wood or metal does not have a precisely flat end. The numbers form a model of reality (just as also our native geometry is not precisely Euclidean, so Euclidean geometry is a very good model for some purposes).

Then idea that only certain kinds of numbers "really count" has been present in the background of mathematics for a long time. But the broad modern understanding is that how we define our numbers depends on what we want to use them for. The rational numbers are still very important, and solving equations in integers or rationals is at the heart of things like Fermat's last theorem. There are mathematical tasks for which the rationals are unsuitable (geometry and calculus being examples) and for these we use extended number systems suitable to the purpose. It took years and considerable mathematical skill to get those number systems properly defined.

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One answer to your amended question, by the way, is that decimal expansion is only one way of naming a number. It is a big step, in fact, to suggest that only things which can be named in a certain way using a decimal expansion deserve to be called numbers.