Ah, Mother’s Day. Admittedly, Math isn’t the first thing you think about when scrambling for that 1-800-Flowers phone number (I always forget that one) or that last box of chocolate from the drugstore. However, one of the memories that sticks in my mind most about my Mom is arguing with her about Math during dinner about .9 repeating and 1. Before you think we’re crazy, please keep in mind she’s a Math teacher, and I have some small interest in the subject myself. So here’s to you, Mom. I finally realized you were right some time ago, but don’t think I ever said so.

I didn’t really have any good arguments for my side of the case. What can I say? I was 8 or 9 and saying things like “1 minus 0.9 repeating is 0.0 repeating with a 1 at the end” made perfect sense to me. I also have to say I didn’t listen that much to Mom’s arguments, so I’m not sure she used these following proofs, but I’m pretty sure they should convince just about anyone.

First, there’s simple addition. If you agree that 1/3 = 0.3333(repeating), then:

Start with 1/3rd | 1 / 3 = | 0.33333333… |

Add another 1/3rd | 2 / 3 = | 0.66666666… |

Add a final 1/3rd | 3 / 3 = 1 = | 0.99999999… |

Ok, what about the algebraic solution?

Start with: | X = | 0.99999999… |

Multiply both sides by 10: | 10X = | 9.99999999… |

Subtract the first row from the second: | 9X = | 9 |

Now divide both sides by 9: | X = | 1 |

See how we started with X = 0.999999… and ended with X = 1? That means they’re the same!

Finally, there is the infinite geometric series, where each term is a set ratio of the previous term. In the case of 0.99999…, we can say that this is the sum of 0.9 + 0.09 + 0.009… This gives us an initial term of 0.9, and a ratio between terms of 1/10.

Geometric series sum is this formula, where A is the initial term, and R is the ratio between terms: | S = A(1-(R^N)) / (1-R) |

Since we want N to be infinite (the 9’s do go on forever) and |R| < 1, then R^N becomes 0: | S = A(1 – 0) / (1 – R) |

The initial term in the series A is 0.9, and the ratio between terms R is 0.1 | S = 0.9(1) / (1 – 0.1) |

Algebra time, see how the series sum becomes 1? | S = 0.9 / 0.9 = 1 |

I admit, that last one wouldn’t have made much sense to me at 9 years old. It just goes to show how many different ways you can prove that 0.9(repeating) is the same as 1. I hope my Mom reads this and realizes that it’s finally sunken into my brain that she was right… on at least this

*one*occasion. If you ever lose an argument with your Mom, make sure you let her know about it, too. Happy Mother’s Day to you all.

The Count

[…] Mom was right « Discrete Ideas […]

Graphing Calculators as a Primary Teaching Tool…it not only gives the right answers but also acts as their teaching guides…

[…] That’s 0.333*, where the 3s never stop, also called 0.3 repeating. And yes, 0.9 repeating does equal 1! […]

This argument is a classic, and the bottom line is that we have to accept the existence of .9 (or .3) repeating to infinity as the premise in order to agree. The problem is that .3 or .9 repeating is an approximation that approaches 1/3 or 1 depending on the level of precision that you are contemplating. Remember that infinity is a very large number, especially towards the end. I propose that .9 repeating (or any number repeating) is a human accepted shorthand to reconcile the incompatibility of base 10 numbering and dividing certain numbers by 3. We simply “know” that 1/3 is .3 repeating.

The problem with infinity is that you can never actually get there. The difference between .9 repeating and 1 is that the value of 1 doesn’t change as you contemplate it at different levels of precision. Infinity is to math and physics as Hitler is to politics. Once you go there you’ve already lost the argument.

All very true. I’m in software, and recently had to tell someone “100% reliability is like infinity, it’s a

concept, not a number”.