However, we will show that it cannot in fact be represented by a decimal with a finite
number of places – i.e. it isnota rational number, and so therefore it cannot be ‘measured’
or ‘plotted’ exactly. The method of proof isby contradiction,thatis,weassumethat
√
2
can be written as a rational number and then derive acontradiction. The proof is one of
the prettiest in elementary pure mathematics and it has great physical significance, demon-
strating that there is a real physical quantity of great importance that we cannot actually
measure. It also gives good practice in elementary algebra, so try to work through it.
Assume that we can write
√
2 as a rational number:
√
2 =m
nwherem,nare integers, each cancelled down to their lowest form (i.e.mandnhave no
factors in common(12
➤
)). Then
2 =m^2
n^2or
m^2 = 2 n^2
This implies thatm^2 is even. Since the square of an odd number is oddmmust also be
even. Let’s write itm= 2 kwherekis an integer. Then:
m^2 = 4 k^2 = 2 n^2
or
n^2 = 2 k^2
By the same argument as form, this implies thatnis even.
We have therefore shown that bothmandnare even and therefore contain (at least) the
factor 2 in common. This is a contradiction since we assumed thatmandnhave no factors
in common. So we have to conclude that
√
2 cannot be written as a rational number.
Numbers such as√
2, which cannot be written in the formm/n, are calledirrational
numbers. They are clearly of more than academic interest because we cannot draw a
continuous curve without them. So, if we definitely cannot plot all points on a continuous
curve, how do we define such a curve mathematically? To define continuity precisely
we have to introduce the idea of alimitof a function at a point. This is the value of
the function as we approach closer and closer to the point indefinitely, without actually
reaching the point. It might or might not exist.
Exercises on 14.1
1.πis often approximated crudely by
22
7. Explain why this expression cannot be exactly
equal toπ.
2. Using a calculator or computer evaluate
√
2 to as many decimal places as you can.
Square your result – do you retrieve 2?Answers
22
7is rational, whereas it can be shown thatπis not.