**Question Corner and Discussion Area**

I heard somewhere that there is a proof that root 2 is irrational by geometric means. Does anyone know this?The geometric proof is a somewhat more awkward version of the proof that is most commonly given for the irrationality of . Its significance is mostly historical--it was the first known proof, discovered by the Pythagoreans. This proof was taken from Euclid's

Let *ABCD* be a square (with diagonal *AC*) and consider the ratio *AC *: *AB*.
Suppose for contradiction that *AC *: *AB *= *n *: *m* for two positive integers
*m* and *n* which have no common divisor.
It can be seen (either by the Pythagorean Theorem or by comparing
areas) that and thus .
Note here that .
It follows from our assumptions that *n* is even and *m* is odd.
Since *n* is even, let *n *= 2*k*.
Then and .
But this implies that *m* is also even, a contradiction.
Thus is not a rational number.

*Asked by Robert Second, student, Greendale on October 3, 1997*:

I don't understand how either proof of root 2 is an irrational number (the geometric method and the contradiction method) works. How does proving that numbers are even prove that root 2 is irrational??Any rational number can be expressed as a fraction in lowest terms, that is, in the formThanks.

To put it another way: suppose you start with some fraction *a*/*b*. It's
certainly possible that *a* and *b* might both be even, but if they are,
you could divide both of them by two and the fraction will still represent
the same rational number. You can keep on doing this as long as they are
both even. Since no integer can be divided by 2 infinitely often and remain
an integer, this process must stop sometime, and you will end up with
a fraction where the numerator and denominator are not both even.

For example, if you start with the fraction 8/12, after twice dividing top and bottom by 2 you end up with 2/3 and 3 is odd.

Therefore, if the square root of 2 were rational, you would be able to
write it in the form *a*/*b* where *a* and *b* are not both even.
However, the proof shows that this is not possible, and therefore
the square root of 2 is not rational.

*Asked by Bryan Low, teacher, San Leandro High School on January 3, 1998*:

Why is the square root of of 2 irrational? Is there another proof besides the geometric one founded by the Pythagoreans? I have read that one on your site. I don't believe my student remeber much geometry to understand that irrational proof posted on your site.You don't need any geometry for the proof; it was just originally phrased that way because the Pythagoreans were thinking about geometry at the time.Any help will be appreciated!!!

Bryan Low

The essence of the proof is this. If the square root of two were a rational
number, you could write it as a fraction *a*/*b* in lowest terms, where *a*
and *b* were integers, not both even. (If *a* and *b* were both even, the
fraction wouldn't be in lowest terms; you could divide top and bottom by 2,
and keep doing this until one of them stops being even).

That means that, if the square root of two were a rational number, it would
be possible to find two integers *a* and *b*, not both even, such that
. But this is impossible, because the equation can be
written as
, so , so
*a* is even. That means you can write *a*=2*k* where *k* is an integer.
Now the equation becomes , so
, so *b* would have to be even as well.

Therefore, because it is impossible to find two integers *a* and *b* with
the property that *a* and *b* are not both even and ,
the square root of two cannot be a rational number.

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Go backward to How To Find The Least Common Multiple

Go up to Question Corner Index

Go forward to Which U.S. President Re-Proved the Pythagorean Theorem?

Switch to text-only version (no graphics)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page