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A Troubling Number for our Usual Mathematics 

Here’s a deep age-old question regularly asked by young math scholars across the globe.

Is 0.99990.9999\ldots equal to one or is it not?

This is a quantity with infinitely many 99s listed to the right of the decimal point.

Well, let’s be honest, that’s a dubious issue at the outset: we didn’t actually list infinitely many 99s to the right. We humans never can. We might have the patience to write down twenty 99s, or two hundred 99s, or we might hire a team of two thousand people to work for twenty years to list out two billion of those 99s , but we humans will never see, actually see, an infinite number of 99s written to the right of the decimal point. We have to cheat and write an ellipsis, “\ldots”, to say “imagine this going on forever.”

So right away we should realise that the question asked is one for the mind. It is really asking:

If we were somehow God-like and could list an infinite string of 99s to the right of the decimal point, would the quantity that results be equal to 11, or not?

And one might decide that since we humans are not God-like the question is meaningless—what does “the quantity that results” mean?—and so there is no point to even giving the question consideration.

That’s a valid stance to take.

But, nonetheless, this question keeps coming up. We are teased by it. And it seems we do want to try to play with it and make sense of it. Somehow it feels as though we humans, although not able to actually see and write down the infinite, can somehow envision the infinite, that we’re just a small step away from having a grasp of it.

Each approximation we humans write for 0.99990.9999\ldots is close to 11, not equal to 11, in fact, just shy of 11 each time.

0.90.9 is less than 11.

0.990.99 is less than 11.

0.999990.99999 is less than 11.

0.9999999999990.999999999999 is less than 11.

0.99999999999999999999999999999999999999999990.9999999999999999999999999999999999999999999 is less than 11.

And some people use this to argue that 0.99990.9999\ldots too must be smaller than 11, and so not equal to 11.

But let’s think a little more deeply about these numbers 0.90.9, 0.990.99, 0.9990.999, …. They’re are sequence of numbers marching rightward on the number line, heading to the number 0.99990.9999\ldots, which might or might not be 11.

I10S10A - Image01

At the same time, we see the more 99s we list after the decimal point, the closer the numbers are to the number 11. For instance, 0.990.99 is one-hundredth of a unit away from 11, and 0.9999990.999999 is one-millionth of a unit away from 11.

So in fact the numbers 0.90.9, 0.990.99, 0.9990.999, … steadily march rightward on the number line eventually entering any amount of space you might specify sitting to the left side of 11. (Want a number within a distance of 1337\frac{1}{337} from 11? 0.99990.9999 will do!)

So could the quantity 0.99990.9999\ldots be a number sitting just shy of the number 11 and not be 11? Could there be any empty space between it and 11?

No! We just argued that the sequence of 0.90.9, 0.990.99, 0.9990.999, … will march into any space we specify to the left of 11! Thus we could conclude

There can be no space between 0.99990.9999\ldots and 11 on the number line, and so 0.99990.9999\ldots must actually be 11.

Well … not quite!

There is another possibility to rule out.

Could 0.99990.9999\ldots instead be quantity sitting just to the right of 11 on the number line?

This feels unlikely since all the numbers 0.90.9, 0.990.99, 0.9990.999, … sit to the left of 11 with 11 being an upper barrier to them all.

But if we feel uneasy about a geometric argument, we can always turn to an algebraic one.

AN ALGEBRAIC ARGUMENT

If you choose to believe that 0.99990.9999\ldots is a valid arithmetical quantity (that might or might not be 11), then you probably would also choose to believe that it obeys all the usual rules of arithmetic. If so, then we can conduct a lovely swift algebra argument so show that, because of these beliefs, the quantity 0.99990.9999\ldots does indeed equal 11. The argument goes as follows.

STEP 1: Give the quantity a name.

We’ll call it FF for Fredericka:

F=0.9999F = 0.9999\ldots.

STEP 2: Multiply by ten

We obtain

10F=9.999910F = 9.9999\ldots.

STEP 3: Subtract

Notice 10F10F and FF differ only by 99 in the ones place and so 10FF10F - F is 99. That is, we have

9F=99F=9

giving

F=1F=1. .

The mathematics thus establish that 0.9999=10.9999\ldots = 1.

Lovely!

But let me be clear on what we have established here.

IF you choose to believe that 0.99990.9999\ldots is a meaningful quantity in USUAL MATHEMATICS, then you must conclude that it equals 11.

I say this because this algebraic argument can lead to philosophical woes, as we’ll see in the next lesson.

FINAL TOUGHT

Here’s a picture of 0.99990.9999\ldots in a 1101 \leftarrow 10 machine with decimals.

I10S10A - Image02

Place a dot and an antidot in the first box. This doesn’t change the number.

I10S10A - Image03

But now we can perform an explosion.

I10S10A - Image04

The number in this picture which looks like 1.19991.-1|9|9|9\ldots is still 0.99990.9999\ldots.

Now add another dot-antidot pair and perform another explosion. There is now also an annihilation.

I10S10A - Image05
I10S10A - Image06

Now we have 1.01991.0|-1|9|9\ldots is the same as 0.99990.9999\ldots.

And do this again

I10S10A - Image07
I10S10A - Image08

and again, and again, and again, forever.

It looks like we are actually showing that 0.99990.9999\ldots is the same as 1.00001.0000\ldots.

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