Irrational Numbers
We saw, in the previous section, that there may be numbers on the number line that are not rationals. In this section, we are going to investigate these numbers. So far, all the numbers you have come across are of the form
So, you may ask: are there numbers which are not of this form? There are indeed such numbers.
The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, were the first to discover the numbers which were not rationals, around 400 BC. These numbers are called irrational numbers (irrationals), because they cannot be written in the form of a ratio of integers. There are many myths surrounding the discovery of irrational numbers by the Pythagorean, Hippacus of Croton. In all the myths, Hippacus has an unfortunate end, either for discovering that
Pythagoras (569 BCE – 479 BCE)
Let us formally define these numbers.
A number ‘s’ is called
You already know that there are infinitely many rationals. It turns out that there are infinitely many irrational numbers too. Some examples are:
Remark : Recall that when we use the symbol
Some of the irrational numbers listed above are familiar to you.
For example: you have already come across many of the square roots listed above and the number π.
The Pythagoreans proved that
Later in approximately 425 BC,Theodorus of Cyrene showed that
As to π, it was known to various cultures for thousands of years, it was proved to be irrational by Lambert and Legendre only in the late 1700s.
In the next section, we will discuss why 0.10110111011110... and π are irrational.
Let us return to the questions raised at the end of the previous section.
Remember the bag of rational numbers. If we now put all irrational numbers into the bag, will there be any number left on the number line? The answer is
It turns out that the collection of all rational numbers and irrational numbers together make up what we call the collection of
Therefore, a real number is
In the 1870s two German mathematicians, Cantor and Dedekind, showed that :
Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number.
R.Dedekind (1831-1916) G.Cantor (1845-1918)
Example 3: Locate
Consider a square on a numberline with each side 1 unit in length. Then you can see by the Pythagoras theorem, the length of the diagonal of the square will be =
Draw the diagonal from point "0" to point O.
Using this diagonal as the radius, we can draw an arc/circle which cuts the numberline. The point B at which the numberline and the circle intersect, represents
Drag the point O onto the numberline.
The point B represents
Example 4: Locate
Drawing a perpendicular passing through point O (end point of the hypotenuse of length
Join point 0(zero on numberline) with A which will give us a new triangle.
Using the Pythagoras theorem, we see that in the newly formed triangle:
OA =
Using a compass, with number "0" as the centre and radius OA, draw an arc which intersects the number line at the point B. (Now, drop point A on the numberline).
Then B corresponds to
In the same way, you can locate