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Exploding DotsStaircase to Infinity

Tempo de leitura: ~30 min

In ancient Greece, Achilles was one of the greatest heroes and (almost) invulnerable. On one day, he was challenged to race … by a tortoise!


Achilles knew that he could run ten times as fast as the tortoise. He felt very confident, and decided to give it a 100m head start.

And the race began. In the time it took Achilles to reach the 100m mark, the tortoise moved by m: it was now at 110m.

When Achilles arrived at 110m, the tortoise had moved by more meter(s): it was now at 111m.

When Achilles arrived at the 111m mark, the tortoise had moved by 10cm.

At every step, Achilles gets closer to the tortoise. But since the tortoise keeps moving, he never quite reaches it. And since he can’t overtake it, the tortoise wins the race!

It seems obvious that something in our argument must have gone wrong. We clearly know that Achilles would eventually overtake the tortoise. But it is difficult to pinpoint a specific error in the explanation.

It turns out that saying “and so on, forever” can be very dangerous in mathematics. Whenever something infinite is involved, things tend to behave very differently from our intuition. In this course we will explore the concept of infinity from a few different angles.

Trouble in our number system

Our number notation is incredibly powerful, and has allowed us to make amazing discoveries in mathematics, science and engineering. In Europe, mathematicians first used the Roman numeral system (numbers like I, V, X, …) before the Arabic numbers were introduced in the 12th century.

There is one important property of numbers that we usually take for granted: all numbers are unique. In other words, there are no two different numbers that are equal. 5 and 8 are different, just like 100 is different from 101, and so on.

On a clock, the number 12 (XII) is actually the same as the number 0.

Well – almost. Like for every rule, there might be some exceptions to this one. For example, here is one age-old question asked by students around the world:

Is 0.999999… equal to 1?

The “…” means that there are infinitely many 9s to the right of the decimal point. If the answer to this question is yes, it would mean that there are two completely different numbers, that are actually the same. What do you think?

We’ll answer this later in this course – but you might also think that the entire question sounds a bit dubious. There is no way we could actually write down infinitely many 9s – it would take infinitely long. We have to cheat by writing dots and using our imagination. The question should really be:

If we were somehow God-like and could write an infinite string of 9s, would the results be equal to 1?

Since we humans are not God-like, you might decide that the question is meaningless. But that seems like a very unsatisfactory solution – and new discoveries always start by asking “what if…”

As humans, we can only ever write a finite number of 9s, say ${n}:


0.9 is less than 1.
0.99 is less than 1.
0.999 is less than 1.
0.9999 is less than 1.

Each of these approximations is 1. They form a sequence of numbers that steadily marches rightward on the number line, getting closer and closer to 1 (without ever quite reaching it).

Notice that the sequence will eventually enter any amount of space you might specify on the left of 1. For example, if you want an approximation that’s within 1/${pow(x)} of 1, you could pick ${nines(x)}.

In other words, there cannot be any space between 0.999999… and 1. Since it is clearly also not bigger than 1, we can deduce that 0.999999… must actually be equal to 1.

A dot-machine explanation

If you’re not satisfied with this explanation, let’s have a look at what 0.9999… would look like in a 110 machine:

Click anywhere in the first decimal box to create a dot and anti-dot pair. This doesn’t change the number.

Now click the explode button to simplify the number using the 110 rule.

Let’s do the same thing for the second decimal box. First add a dot and anti-dot pair, and then explode.

Notice how the dot/anti-dot pair canceled out! This is called an annihilation. The resulting number is still the same as 0.9999… Keep going with the remaining cells.

If we keep doing this forever, it looks like we are actually showing that 0.9999… is the same as 1.0000…!

An algebraic explanation

If you’re still not convinced, let’s end with an algebraic argument. If you believe that 0.9999… is a valid number (that might or might not be 1), then it makes sense to assume that it also obeys all the usual rules of arithmetic.

  1. Let’s start by giving the number a name, say
    for Frederica:
    = 0.9999…
  2. Now multiply it by 10. This gives us
    = 9.9999…
  3. Subtract the equation in step 1 from the equation in step 2. Since all their decimal places are equal, they simply cancel out:
    = 9
  4. Finally, if we divide both sides by 9, we get

Amazing! But let’s be clear on what we have established here. IF you choose to believe that 0.9999… is a meaningful quantity in usual mathematics, THEN you must conclude that it equals 1. This is important, because the same algebraic argument can lead to philosophical woes – as we’ll see in the next section…