Surely the most famous irrational number - that which nearly every child learns of throughout their schooling - is that which is known as "pi." This is the number which is simple in theory, yet famously impossible to state in full. Simply put, it is the number which corresponds to the ratio between a circle's circumference and its diameter.
The task of defining a number like pi is not as easy as it might seem, for as it was eventually proved, the ratio between the diameter and circumference of any circle is a number which is very close to the fraction 22/7 (in fact, this is a value which was very often attributed to pi in these early years), but which, as it turns out, is absolutely inexpressible by way of fraction or decimal. Most people are aware that the value of pi equals roughly 3.14, but the number doesn't end here; it continues on, without repetition, to infinity. This is what is meant by "irrational."
The Pythagorians and Irrational Numbers
It is said that the first person known to have proved the existence of irrational numbers was a member of the Pythagorian school/cult/social organization named Hippasus in the fifth century, BC.
The Pythagorians were famous for believing that mathematics provided a perfect representation of nature, which therefore meant that only real, positive numbers could exist. This led to certain very obvious mathematical problems, however, such as the determination of the square root of certain numbers, such as 2 or 3, which is where troubles began for Hippasus.
As the legend goes (and of course there is very little way for today's historians to know just how much of it is true), Hippasus was able to prove through relatively simple, yet clever, means, that the square root of two was an irrational number, and could not be expressed by way of fractions. Because this so clearly went against everything that the Pythagorians believed about numeracy and its connection with the natural world, they did not take kindly to such an assertion, no matter how flawless the math may have been. As a result, they took Hippasus out to sea and threw him overboard, killing him.
The Pythagorians were rather serious about their mathematics.
Common Irrational Numbers
Despite the rather harsh attempts by the Pythagorians to prevent their arrival, irrational numbers prevailed, and today there are a great many of them which are known to have very real connection to the mathematical representations of the universe.
Pi, as has already been mentioned, is surely the most famous of these, but there are others which have just as much value.
Euler's Number, often expressed simply as "e," is certainly an irrational number, and one which comes up rather frequently in advanced mathematical classes, as this number is a representation of the "natural logorithm," and is invaluable for calculating such common things as compound interest.
The Golden Ratio, often denoted as the Greek letter "phi" is also very much irrational, with a value of somewhere very close to 1.618. This ratio has been thought to represent the most aesthetically pleasing, perfect, proportion in existence, and can be defined as the lengths of the sides of a "golden rectangle," which is a rectangle where the ratio of the lengths of the short side to the long side is the same as the ratio of the long side to the short and long side put together. While this may be confusing, it does not take away from the fact that the golden ratio has been known and understood throughout history, and truly does have very real correlation with nature.
Finally, as has already been mentioned, there are a great many irrational numbers which are merely the square root of whole numbers, such as the square root of 2, 3, 61, or an infinite number of others. These are the numbers which, if found as the result of a mathematical problem, are generally just left in their square root form, as they can't be simplified any further.
Irrational numbers, though difficult to fully comprehend and absolutely impossible to fully write out on paper (but for a symbolic representation), have become integral pieces of the mathematical establishment, without which modern theories of physics, engineering, chemistry, astronomy, etc. would not be where they are today. Irrational numbers makes much of mathematics possible, and as a result, they are simply things that must be accepted and embraced.
For Further Reading:
Irrational Numbers. Math is Fun.
