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What are irrational numbers?
Well, they aren't crazy. Although the word can mean not logical, in mathematics, the words irrational and rational have to do with ratios. To understand what an irrational number is, first you need to know what a rational number is.
A rational number represents a quantity that can be written as a fraction. So, every 'normal' number (even 2 or 3 which can be written as 2/1 and 3/1) is rational, and all the fractions you can make with regular numbers are rational.
At first glance, that definition seems to cover all the numbers. But there are quantities that cannot be written down as rational numbers (and no fraction would be equal to those quantities). These are called irrational and we use special symbols to represent them.
How are irrational numbers used in the real world?
Pi is an irrational number. The symbol for it is and it is pronounced the same as the word 'pie'. It appears in formulas for the area or circumference of a circle (and many other places). The area of a circle is equal to the radius times the radius times This is true for any circle and every circle. It's true when you want to make car tires, drill a hole, or figure out how much dough will be needed to make a round pizza.
Pi is used in calculations with gears (circles!), wheels, pipes, coins...and pi is just one of an infinity of irrational numbers.
A basic problem with irrational numbers.
Because they cannot be written down as fractions, irrational numbers in practice are only used as approximations. An approximation for π is 3.14159. The 'real' quantity has been calculated to more than a trillion digits. But because the digits never end (nor repeat in any pattern) pi is shortened to a reasonable number of digits in actual use.
This is the best solution when calculating the area of a circle or the volume of a sphere, just use enough digits to get the accuracy you need. This is what happens when you use an irrational number on your calculator. The square root button (√), when used with 2 gives an approximate answer, but one good enough for most calculations.
Who discovered irrational numbers?
The first proof of the existence of irrational numbers is credited to Hippasus around 500 BC. However, because they arise naturally from basic geometry, they were probably discovered much earlier.
For instance, the area of a square is given as the length of one side times itself.
Area = s x s. Suppose you have a square with an area = 2 and you want to find the length of the sides? Well, you need a number that, when multiplied by itself, equals 2.
We express this number now as √2 (square root of two). This is an irrational number and cannot be written out - we use √2 as the symbol for it, just like π is a symbol and not a 'real' number. Anyone using the formulas in geometry will quickly discover these special numbers.
An interesting fact about irrational numbers.
Hippasus was a Pythagorean, a group that believed all numbers had to be rational. They felt this so strongly that when Hippasus proved irrationals existed, he was banished (or even killed, we aren't sure) for his forbidden knowledge.
It's hard to believe now that people could have had such strong feelings about mathematics, but in Ancient Greece, these matters were tied into ideas about God and perfection, and Hippasus's discovery attacked these beliefs.