The Irrationality of √2

When we say that √2 is irrational, what does that mean? Irrational means not rational. Not rational means it cannot be expressed as a ratio of two integers. Not expressible as a ratio means it cannot be expressed as a common fraction.

That is, there is no fraction a/b = √2 (where a and b are integers).

If we compute √2 with a calculator we will get

√2=1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572...

Notice that there is no pattern among the digits, and there is no repetition of groups of digits. Does this mean that all rational fractions will have a period of digits? Let’s inspect a few common fractions.

1/7= 0.142857142857142857142857 · · · which can be written as 0.142857 (a six-digit period).

Suppose we consider the fraction 1/109:

1/109= 0.009174311926605504587155963302752293577981651376

146788990825688073394495412844036697247706422018348623 · · ·

 

Here we have calculated its value to more than 100 places and no period appears. Does this mean that the fraction is irrational? This would destroy our previous definition. We can try to calculate the value a bit more accurately, that is, say, to another 10 places further:

1/109=0.0091743119266055045871559633027522935779816513761467889908256880733944954128440366972477064220183486238532110091

Suddenly it looks as though a pattern may be appearing; the 0091 also began the period.

We carry out our calculation further to 220 places and notice that, in fact, a 108-digit period emerges:

1/109= 0.0091743119266055045871559633027522935779816513761

46788990825688073394495412844036697247706422018348623853211009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211009174

If we carry out the calculation to 332 places, the pattern becomes clearer:

1/109= 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211009174

We might be able to conclude (albeit without proof) that a common fraction results in a decimal equivalent that has a repeating period of digits.

Some common ones we are already familiar with, such as

                           

1/3= 0.333333333

                                                          _          

1/13= 0.0769230769230769230769230769230

To this point, we saw that a common fraction will result in a repeating decimal, sometimes with a very long period (e.g., 1/109) and sometimes with a very short period (e.g., 1/3). It would appear, from the rather flimsy evidence so far, that a fraction results in a repeating decimal and an irrational number does not. Yet this does not prove that an irrational number cannot be expressed as a fraction.

Here is a cute proof that √2 cannot be expressed as a common fraction and therefore, by definition is irrational.

Suppose a/b is a fraction in lowest terms, which means that a and b do not have a common factor.

Suppose a/b=√2. Then a²/b² = 2, or a² = 2b², which implies that a² and a are divisible by 2; written another way, a = 2r, where r is an integer.

Then 4r² = 2b², or 2r² = b².

So we have b² or b is divisible by 2.

This contradicts the beginning assumption about the fact that a and b have no common factor, so √2 cannot be expressed as a common fraction.

Understanding this proof may be a bit strenuous for somebody, but a slow and careful step-by-step presentation should make it understandable.