Irrational numbers
Math is a very logical pursuit – every number and operation in its proper place, everything making sense.
But, as with every group, there are those misfits that just don't blend in. In math, it is the irrational numbers; numbers that logically must be – even if they don't appear to follow the rules.
So take a walk on the wild side and learn about the misfits of math.
A square peg in a round hole
We owe a lot of our view of math and numbers to the Egyptians and Greeks. For those of the Greek school of Pythagoras, the real and meaningful numbers were ones that we could count with (1, 2, 3, …) or those numbers that come by dividing other numbers, such as ½ or ¼. Such numbers are called rational numbers.
But things don't always seem to go so smoothly.
Take the example of square roots.
The square root of a number is the number that gives back the given number when multiplied with itself. For example, the square root of 9 is 3 (3×3 = 9), and the square root of 4 is 2 (2×2 = 4).
That all sounds neat and pretty, but things don't always work out that simply. Take the square root of 2; 1 is too small (1×1 = 1), and 2 is obviously not right. No problem, you say, let's just split the difference and go with 1.5. The trouble is, this is too big.
Fine, take it down a notch and try 1.4. Better, but still not there.
So, keep this up and try adding another decimal, say 1.42. Still better, but we're quite not there.
So on we go, adding more digits and getting even better numbers. The frustrating thing is this; we'd keep on doing this without stopping and without a pattern forming (without a pattern, this is unacceptable as a fraction). As a matter of fact, the actual answer looks more like
This is what it means to be an irrational number – a number with decimals that don't stop, and that leaves us without a hope of ever being converted into a fraction.
Our hall of fame
One might think these irrational misfits are rare, but they crop up all over the place. A great many of the roots are irrational, but there are some famous examples.
Let them eat pi
For most people, the most famous irrational number is p (pi). p was first used to find the length around a circle (the circumference) given the distance across (the diameter). You can learn more about it here, but its value, to the first several decimal places, is given below.
3.141592653589793238462643383279502884197169
39937510582097...
Just call me 'e'
Euler's number (pronounced the same as "oiler"), usually just called 'e,' crops up all over the place, such as determining the future value of some savings accounts. Its value for the first several numbers is below.
2.71828182845904523536028747135266249775724709369995...
This number is golden
Since the times of the Greeks, there is a special number that keeps popping up called the Golden Ratio – take the square root of 5, add 1 and then divide by 2. Its value out to several decimal places is as follows.
1.6180339887499
Here are some ways it is useful.
Artists have long used the Golden Ratio to create the most beautiful image of a face – the height divided by the width is the Golden Ratio. Even today, some researchers believe this ratio helps define the most appealing personal beauty.
Some say the size of book pages and playing cards closely follow this ratio.
It also is used in art, music, and building design.
Old-school math
One last lesson. In the days before calculators, people had to actually learn about numbers and how to work with them to do calculations. In an older math textbook of mine, there is an approach to calculate square roots from scratch. The steps are simple and are as follows.
Guess an answer to the root.
Divide your given number by your guess.
Take the average of the above number with your guess – add the two and divide by 2.
The number you get won't be the answer, but it's closer to it. Keep repeating (using this as your new guess) until you get as close as you want.
This approach is called iterative; it doesn't give you the answer, but a closer value. So you keep repeating until you get as close as you want.
As an example, take our old friend, the square root of 2.
As a guess, start with 2.
Divide the two numbers, 2÷2 = 1.
Take the average. (1 + 2)÷2 = 1.5.
Do it again.
2÷1.5 = 1.3333.
(1.5 + 1.333)÷2 = 1.4165
After only two rounds, we already are getting close – and we could get closer to the real value the more we repeat doing this.
Irrational numbers reveal the wilder side of numbers. Indeed, there is something a lot deeper about numbers than your basic 1, 2, and 3. Unlocking the secrets of numbers opens a doorway to a new understanding of the world around us.
On the web
This video explains the Golden Ratio, what it is, and its history.
This shows how the Golden Ratio gets used in art.
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