This was the question:

My teacher says that 0.999... repeating equals 1. She didn't tell us why. Is she full of it, or is it true, and if so, why?

Professor Homunculus' answer:

This is one of the most commonly asked questions. Here is a short (but good) answer:

Believe it or not, your teacher is not full of it. This is why:

You know that if a=b, and b=c, then a=c, right? It's the Transitive Property of Equality, but forget the name, it makes sense. It's like saying "If two quarters equals five dimes, and five dimes equals a half-dollar, then two quarters equals a half-dollar." Well, here's and opportunity to use that principle.

1/3, expressed as a decimal, equals .333... (do the division and convince yourself).

2/3, expressed as a decimal, is .666... (ditto).

You know that 1/3 + 2/3 =1.

By the the same reasoning of "if a =b, and b=c, then a = c", you can see that if 1/3+ 2/3 =1, and 1/3+ 2/3 = .333... + .666..., then .333... + .666...=1.

Using the same reasoning, we know that .333... + .666...=1, and that .333... + .666...=.999....,
so .999... therefore equals 1.

Did it work for you? I hope so.

Hi-Ho,

Professor Homunculus

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