Pi and its irrational friends.

You may be surprised to learn that pi isn’t the only irrational number; there are many like it. Actually, there is an entire infinity of numbers like pi. Even more, an uncountable infinity. Let me explain.

Start with the natural numbers we all use to count: 1, 2, 3, 4, etc. There are, of course an infinity of such numbers, ever higher. Mathematicians call this a “countable” infinity because you know how to get from one number to the next one: just add 1, exactly as if you were counting something.

With irrational numbers it’s different. Pick any interval, say the interval between 1 and 2. This interval already contains infinitely many irrational numbers. You might think that by picking a smaller interval, say from 1 to 1.0001, you’d be able to cut down on irrational numbers. Alas, however small you pick the interval, there will always be infinitely many irrational numbers contained in it. They are packed so densely that they kind of “blend” from one to the next. There is no well defined neighbor, unlike with the natural numbers we use to count.

If you drew a line on the wall representing all numbers. You can mark 0, 1, 2, 3 etc. off by little tick marks, if you want. Then stand back a couple of feet and throw a dart at it. The probability to to hit an irrational number will be equal to 100%. It is impossible to hit a natural number, like 1 or 2, because there are waaaayyyyyy too many irrational ones around.

Pi is just one among the uncountably infinite number of its irrational brethren. It just happens to be the most famous one because of its almost ubiquitous presence in math and science. But there are other irrational numbers that at least deserve a mention.

Here is how to find another famous one: draw a square with sides of length 1 unit. This could be one inch, one centimeter, one meter, it doesn’t matter. The length of the diagonal of that square is square-root of 2 (√2) units which is just as irrational as pi. In fact, every square root of an odd number is irrational, such as √3, √5, √7 and so on, as long as it isn’t a perfect square itself (√9 = 3, since 3 * 3 = 9, so it is not irrational).

There is Euler’s number e = 718281828459045235360287471352…., also irrational. It was discovered by the Swiss mathematician Leonhard Euler in the 18th century. Since it crops up a lot in mathematics, it acquired its own symbol: e.

An irrational number contains an endless stream of digits behind the comma which never fall into an ever repeating pattern. This means that any imaginable sequence of digits is bound to occur somewhere in the stream of decimal digits. Can we conclude, then, that any irrational number contains the digits of all other rational numbers? In other words, does pi contain √2 somewhere in its endless and never repeating pattern of digits? One is almost compelled to say, “Of course! If it contains all possible combinations of digits, it must also contain the combination of digits we call √2.” Alas, that isn’t the case. The square-root of 2 is infinite itself and cannot be contained by another irrational number, even though the other number is also infinite. It just isn’t “infinite enough”.


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