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Geometry/Irrational Numbers and Roots


Hello sir

Sir why irrational number cannot be expressed in the form of p/q, we can express any natural numbers in p/q form for eg 5=5/1 there is always 1 is present in deniminator,then irrational numbers also have nutural numbers inside the sign of root, the if 5 is expressed in the form of 5/1 , then why not root5 we can't express like root5/1.

Sir my second question, what does this root sign indicate, and why numbers which hare having root sign are called irrational.


Hi Apsara,

First and foremost, let's get the definition clear.  An irrational number is a number that cannot be expressed in the form of p/q, where p and q are both integers.  More on this in a bit.

I'll answer your second question now.  "What does this root sign indicate?"  A square root represents the number such that, if we raised it to the power of 2, we would get the number inside the root.  So if we see √5, the number is the [positive] solution the equation x=5.  Cube roots and higher roots work in a similar way.
A perfect square is a natural number (or zero) whose square root is also a natural number (or zero).  It can be shown that if a number is not a perfect square, then the square root of that number will be irrational.  This is not immediately obvious, but it is not too hard to show.  If you suppose that such a p and q exist, you will always reach a contradiction (e.g. that an odd number equals an even number).  The proof that √2 is irrational is worth looking into if you are interested.
For the moment, it is okay to accept that a square root will be an irrational number, unless the number inside the root is a perfect square (or a perfect power, in the case of higher roots).

Now that we understand what square roots are, your first question should make more sense.  √5 is not an integer, even though 5 is.  So the fraction √5/1 is not an acceptable form of p/q.  Hence √5 is irrational.

Thanks for asking,


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Azeem Hussain


I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.


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Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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