Irrational number and real number definition

Question

Real number - A number which can be represented uniquely by a point on axis.

And real number = rational + irrational number

By this definition, if you take an irrational number such as sqrt(2) ...it's value is non terminating non repeating (1.414....)

Doesnt this violates the def of real number? Since by this way the exact location of the point( sqrt(2) ) can never be identified.

Note: I know that sqrt(2) can be represented on axis using Pythagoras theorem and making a perpendicular of unit length on unit length on x axis. By this way also I can uniquely plot sqrt(2) on axis

But then again this would mean that the decimal expansion of sqrt(2) should terminate at a value.

Any ideas?

Thanks

Answer 1

I think you're confusing "represented" and "located". Yes, all real numbers can be represented by the points on a line. But remember that a line is an abstract thing. You can never draw or imagine a line (only represent it approximately). Once you realise these things you should be more clear about this (admittedly deep matter -- check out "continuum").

So, yes, whereas every point on a straight line can be made to correspond to one and only one real number, it does not mean we can always locate this point for all real numbers (by locate I mean using a finite sequence of steps). You may call this theology, but math sometimes often deals in such things, especially in relation to deep matters like this.

The real numbers we can locate are known as constructible numbers. But perhaps even if we allow ourselves more freedom, we may even find rectilinear lengths of any real size and use it to "locate" our points.

But don't think all is settled, or that you can finish settling this in a few hours. It's a very interesting matter that one keeps getting back to.

So how should one think rigorously of real numbers (wlog irrational numbers)? Well, they were defined because extracting the roots and logarithms of rationals does not always give a rational value. But since we know these things must correspond to some quantity, we call them irrationals (this is not very different from the invention of negative numbers and complex numbers). The special thing about irrational numbers is that the involve the "cumulation" of infinitely many operations, loosely speaking. Dedekind was first to give a rigorous definition in terms of order relations and sets (so-called Dedekind cuts).

Answer 2

One way to think about this:

Whenever we mention a number on a real number line, we technically are dividing the number line into equal parts and mentioning the value.

For example say a number 10, it means dividing the whole line into equal parts starting from the origin, and the 10th point is your number.

Similarly when we specify the point 0.35 what we actually are doing is that we are further subdividing the number line between 0 and 1, into 20 equal parts, and the 7th part is 0.35 ( 7/20). In a very similar way we are representing 0.3 ( which is 3/10...by dividing the number between 0 to 1 into 10 equal parts and the third point is 3/10)..

In a very similar way 2.4 etc etc.

But when we represent sqrt2, etc, we are able to locate the point on a number line ( using Pythagoras theorem). But there is no such whole number of partitions possible between 1 and 2, out of which one point of that partition will be sqrt(2)... Hence the decimal representation goes on till infinity and never terminates nor repeats.

Answer 3

You are correct about the square root of 2 being irrational because the decimal is nonterminating and nonrepeating. But you are incorrect about the square root of 2 being nonreal. The square root of 2 is real because it is located on a line connecting the numbers 1 and 2.