Arguments for 0.9999… Being Equal to 1

Recently I tried to explain to my 11-year-old son why 0.9999… equals 1.  The standard arguments for $0.9999... = 1$ (at least the ones I’ve seen) assume more math background than he has.  So I tried another couple of arguments, and they seemed to convince him.

The first argument.  The usual claim you get from people who aren’t yet convinced that $0.9999... = 1$ is that 0.9999… is the real number just before 1 on the number line.  Let’s suppose this is true.  What is the average of 0.9999… and 1?  Stop and think about that for a bit.

If the average exists, it must be larger than 0.9999…, and it must be smaller than 1.  But if 0.9999… is just before 1 on the number line, there can’t be such a number.  So either (1) the averaging operation doesn’t apply to 0.9999… and 1, (2) there are numbers between 0.9999… and 1 on the number line, or (3) 0.9999… equals 1.  But (1) immediately leads to the question “Why not?”, which has no obvious answer, and (2) leads to the question of what their decimal representations would be, which also has no obvious answer.  Explanation (3), that 0.9999… equals 1, starts to look more plausible.

The second argument.  Again, let’s assume that 0.9999… is the real number just before 1 on the number line.  If this is true, then what is the difference of 1 and 0.9999…?  Again, stop and think about that for a bit.

If it’s not zero, then you could just add half of that difference to 0.9999… to get a new number between 0.9999… and 1, which not only contradicts our assumption but also forces us to come up with the decimal representation of such a number.  If it is 0, then you have that $0.9999... = 1$.  And if you try to argue that you can’t subtract 0.9999… from 1, then you need to explain why that operation is not allowed for those two real numbers.  (This second argument is a lot like the first one, really.)  The most reasonable of the three options is that the difference is 0, which means that 0.9999… is actually equal to 1.

Final comments and the standard algebra argument.  Both arguments are reductio ad absurdum arguments; that is, they assume that 0.9999… is not equal to 1 and then reason to a contradiction.  The other arguments that I’ve seen are all direct arguments; i.e., they reason from basic mathematical principles to the conclusion that 0.9999… equals 1.

For example, here’s the standard argument via algebra.  We know that 0.9999… must be equal to some number, so let’s call that number x.  Multiplying by 10 yields $10x = 9.9999...$.  Subtracting the first equation from the second leaves us $9x = 9.0000...$, which implies that $x = 1$.  Thus $0.9999... = 1$.

The algebraic argument is a great one, provided you know algebra.  But for my preteen with a pre-algebra background, these two reductio ad absurdum arguments seemed to be enough to convince him.

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