Understanding What Irrational Numbers Are

By DR. ALFRED POSAMENTIER

In today's world, students are learning about irrational numbers much earlier than the date years back. However, all too often they are told what they are but not given enough time to appreciate what we mean by an irrational number. Let's take a little time here to understand what irrational numbers by using one of the more common ones, .

When we say that the  is irrational, what does that mean?  It may be wise to inspect the word "irrational" with regard to its meaning in English.  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 , where a and b are integers.^*

If we compute  with a calculator we will get: = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572

Notice that there is no pattern among the digits, and there is no repetition even 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 = 0.142857 142857 142857 142857 ... , which can be written as:    (a six-digit period^**). Suppose we consider the fraction 1/109 = 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623

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

1/109 = 0.0091743119266055045871559633027522935779816513761467889908256880733944954128440366972477064220183486238532110091

Suddenly it looks as though a pattern may be appearing, the 0.0091 also began the period above.

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

1/109 = 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211009174

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 already are familiar with. Such as:

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 for the more ambitious (or curious) reader that  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.

Understanding this proof may be a bit strenuous for some, but a slow and careful step-by-step consideration of it should make it understandable for most readers.