Question

locating irrational numbers on number line

Answer 1

Representing an irrational number on a number line.

 
Learn how to represent √2 on a number line.
1. First of all draw the number line. 
2. Mark point A at "0" and B at "1". This means AB = 1 Unit.
3. Now, at B, draw BX perpendicular to AB. 
4. Cut off BC = 1 Unit.
5. Join AC. 
6. By Pythagoras theorem in right triangle ABC, we get AC = √2 Units. 
7. Now, with radius AC and centre A, mark a point on the number line. 
Let the marked point is M. M represents √2 on the number line.

Thank you

Answer 2

Constructible numbers are easy.  Construct 2–√2 by measuring the diagonal of a unit square.  Then transfer that distance onto the real number line.  No problem, right?  If you have an infinitely accurate compass then you just found the exact position of 2–√2 along a line.

So what does it mean to "have an exact place"?  If it means "able to be constructed with compass and straightedge" then, no, most irrational numbers (and all transcendental numbers) don't have an exact place.  But that's not really the standard we judge mathematics by, in the modern era.

One alternative definition is that a number has an exact place if we can provide bounds for it to an arbitrary degree of accuracy.  Given a number such as ππ we can ask "is X greater than ππ" where X is any rational number.  Find the spot where the answer changes from "yes" to "no" and that's the exact position of ππ.  (You might have to do an infinite amount of work to locate it precisely, but mathematicians are comfortable with that.  You get better approximations along the way, so practical purposes are served too.)

That style of locating a real number is, in fact, one way to construct the real numbers: by definining them as pairs of sets, where everything in set A is less than everything in set B, a "Dedekind cut".  Then we need to show that the numbers so defined obey all the rules we want the real numbers to have.  We defined all the real numbers with reference only to rational numbers, which presumably all have "exact" places on the number line.