Weird but interesting stuff happens when you apply infinity to certain physical things! In fact, recurring decimals are also the sums of an infinite series of fractions with a common ratio that's *less than the absolute value of 1*. They can even add up to a whole number! For example,

If the common ratio is *greater than or equal to* **1**, then the infinite series won't have a sum! Because **1 + 1 + 1 + 1 + ... = ∞**! In other words, adding **1** to a sum repeatedly forever will just make the sum bigger & bigger & bigger...it'll keep growing because there is no limit! (The same thing happens if you repeatedly add a number that's greater than **1**, except that it'll grow at a faster rate!) For a limit to exist, the common ratio must be *less than 1.* Not all infinite series have a common ratio, but most of them do.

The pattern of digits to the right of the decimal point is the clue to which power of **10** to divide the respective whole number by. For example, **0.123123123... = 123/10 ^{3} + 123/10^{6} + 123/10^{9} + ...** The respective whole number in this recurring decimal pattern is

(This is how it looks in * summation* form:)

(The variable ** n** had to be added to represent that

What you seen above this paragraph is what happens when you apply infinity to fractions with a common ratio! However, there are many different types of infinity! As a bonus for viewing this Web page, here's another type of infinity to see below!

There's an interactive Flash animation on this Web page! Click here if you don't have Flash software.

An infinite series of * ladies* followed by a

The term **first** isn't exclusionary, but the term **only** is! In other words, "Ladies First" doesn't exclude gentlemen; it just means: "Let the ladies go first"; but "Ladies Only" means: "Let nobody but ladies get in!" which DOES exclude gentlemen! If a series of ladies is infinite, then you can't ask: "What if it ends with a man?" because there is *no end*; otherwise, the series would not be infinite. Furthermore, it's technically impossible to get to infinity; not even the great **googolplex** is close to infinity! You could eventually reach the **googolplex** if you tried. (But it'll take a very, very long time if you start at **1**, going up **1 integer per second!**) All numbers, no matter how big they are, are finite. But infinity is not finite! In fact, infinity is not a number at all, it is an idea, a concept, something that never ends. The only way to get to infinity is to *never ever stop!*

Back to Index Page Back to Math Trick Menu

© Derek Cumberbatch