Interesting numbers --- zero --- one --- complex --- root 2 --- golden ratio --- e --- pi --- googol --- infinity
Raising a number to a power is multiplying that number by itself several times. The simplest example is squaring. Three squared or 32 is 3x3=9. It's called squaring because the area of a square is the square of one of its sides. Raising to the power of three is called cubing, for a similar reason. 53=5x5x5=125. You can have higher powers. Two to the fourth or 24 is 2x2x2x2=16.
The opposite of squaring is a square root. The square root of 9 is 3. You can write this as 9½=3 or √9=3. Some numbers have obvious square roots, such as 4 or 9 or 1/25. Are all square roots rational numbers? Believe it or not, it was the Ancient Greeks who proved that √2 was not a rational number. (It worried them a lot!).
This uses algebra, as it is the easiest way to explain it.
A rational number is a number that can be written as a ratio of two numbers, as a fraction. There are always more than one way of writing the same fraction. For example, 5/10, 3/6, 25/50 are all ways of writing 1/2. But every fraction can be reduced in its simplest form. What you do is see if the top and bottom have any factors in common. For 5/10, both 5 and 10 are divisible by 5. For 25/50, both top and bottom are divisible by 25, and so on. But 1/2 have no factors in common (apart from one), so that is the simplest form. With 8/12, the common factor is 4, so the simplest form of that fraction is 2/3. For all fractions, it is possible to write the simplest form, with no common factors.
If √2 is a rational number, then you can write √2 = a/b where a and b are integers and they have no factors - it is the simplest form of the fraction.
| √2 | = a/b |
| so 2 | = a2/b2 |
| so 2b2 | = a2 |
| This means that a2 is even, and so a must be even. | |
| So we can replace a by 2c. | |
| so 2b2 | = (2c)2 |
| so 2b2 | = 4c2 |
| so b2 | = 2c2 |
| But that means that b2 is even, and so b must be even as well. | |
| So a and b are both even. | |
But that is impossible. We originally said that a and b had no common factors, and now we prove that they must both be divisible by two. So there is no pair of numbers we can chose whose ratio is √2. This form of proof is called (in Latin) reductio ad absurdum or 'reduced to the absurd'. At the start we made a statement - that we could write √2 as a fraction reduced to its simplest form, and we have now proved that we can't.
The Ancient Greeks didn't use algebra, but one of them worked this idea out. The rest of the Greek mathematicians didn't like it at all. They believed that all numbers could be written using whole numbers, or as a ratio of two whole numbers, which they felt was a beautiful idea and so true. Now we know there are different types of numbers and irrational numbers have their own beauty. Anyway truth matters more than beauty.
How did the Greeks think up the number √2 in the first place? Does it exist in the real world? Yes, it does. Imagine a floor with tiles on. The tiles are half-squares. If you look at it hard, you can see a right-angled triangle in the middle, with a square on each of its sides. Click on the tiles to make it appear. (Click on it again to make it disappear.)
The Greeks knew Pythagoras's theorem - "the square on the hypotenuse (the longest side) is equal to the sum of the squares on the other two sides." They knew that a triangle with sides of 3, 4 and 5 had a right angle (32+42=9+16=25=52) and so did a triangle with sides of 5, 12 and 13 (52+122=25+144=169=132). But this floor tile triangle has sides 1, 1 and √2. This must be true because of Pythagoras's theorem, since the longest side squared must be equal to 12+12=2. But you can also count the squares. The squares on the shorter sides have 2 tiles each, and each tile is half a square, so that makes one square. We'd expect that, since the sides are 1 long. But the big yellow square has four half tiles, so that makes two squares. So its sides must be √2. Floor tiles are definitely part of the real world, so √2 is a real number. But it's not a rational, it's an irrational number.
A fractional approximation of √2 is 99/70. For a more accurate value
√2 = 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799 (to 65 decimal places)
So how does someone come up with a value like that? Here is one way. You start with a guess - say 1.5 (since 1.52=2.25, which isn't too bad). This is fed into a formula. Imagine the original guess was n0. We want a new, better value, n1.
n1 = 0.5 x (n0 + 2 / n0)
Then you feed the new value into the right-hand side to get an even better value, n2, and so on. Try it for yourself. Click on the button to get better values of √2. You can see what it should be, above. After a few goes, you'll find that it settles down and doesn't improve. That is because you've reached the limits of this computer's accuracy.
There are plenty of these irrational roots. √3 is irrational, so is √5. (but not √4, of course.) Do we ever use these roots?
tan 45° = 1 / √2 sin 60° = (√3) / 2 golden ratio = (1 + √5) / 2
√2 has a very practical use. There is a range of paper sizes used in Britain and Europe called A0, A1, A2, etc. A4 is the size used for large sheets of writing paper, for files, etc. A0 is a square metre in area. A1 is A0 folded in half (and turned the other way up). A2 is A1 folded in half, and so on. All sheets are the same shape. This will only work if they all have their sides in proportion 1 to √2. The orange rectangle on the left has sides in that proportion. I think that it makes a good shape, and I prefer it to the golden rectangle, which is supposed to be a beautiful shape. The dotted line is where you would fold it in half. You can see that each half has the same shape (only smaller, of course).
© Jo Edkins 2007 - Return to Numbers index