WEBVTT

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Everybody knows this one.

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Turn around, then turn around again,
and you are facing the way you started.

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Flip a coin twice
and it shows the same face.

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Cancel somebody's debt,
and you have handed them money.

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Two minus signs make a plus.

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So why did one short question about
it turn into a twelve hour argument,

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on a forum full
of people learning mathematics?

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An argument that ended only
when one of the two people in it wrote,

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you are right, my apologies.

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The two sides went like this.

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One of them said the fact holds in any
system where you can add and cancel,

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so there is nothing to prove.

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The other said that the proof on offer
assumed the very thing it was trying

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to prove. You cannot get a theorem by
writing a symbol down in a suggestive way.

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They were both pointing at something real.

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There is a small, genuine theorem
hiding inside this obvious fact,

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and the argument was about where it lives.

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To find it we have to work
only from the rules.

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The person who asked was reading
the first chapter of Spivak's Calculus,

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which opens by writing down
the rules that numbers obey.

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From here on we are allowed exactly
what the rules say and nothing else.

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No obviously.

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No number line.

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Rule one: grouping does not matter.

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If you are adding three things, you may
bracket the first pair or the last pair,

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and you get the same answer.

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Rule two: there is a number called zero,
and adding it does nothing at all.

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And rule three, the one
the whole argument is about.

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For every number a, there is a number,
written minus a, that cancels it.

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a plus minus a is zero,
and minus a plus a is zero as well.

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Now read that third rule
again, slowly, in words.

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Four of those words are going to carry
this entire video. There is a number.

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It says there is a number.

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It never says there is only one.

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And notice one last thing before we go on.

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The fourth rule on that list,
that a plus b equals b plus a,

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we are never going to use.

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Three rules and a claim.

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Before we argue about the claim, we
had better be sure what its symbols mean,

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and the minus sign is where people slip.

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In this rulebook the minus sign
is not a property of a number.

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It is not negativeness. It is a job title.

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Minus a means whichever number
holds the job of cancelling a.

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Rule three hands us that number
and says what the job is:

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a plus minus a is zero.

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Let me draw it.

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Here is a, here is its canceler,

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and I will join two numbers
with a line whenever they add to zero.

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If a is five, the canceler is minus five.

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If a is minus three,
the canceler is three,

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so a symbol with a minus in front
can perfectly well be a positive number.

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And zero cancels itself.

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Now read the thing we are trying
to prove from the inside out.

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Minus a is the canceler of a.

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So minus minus a is the canceler
of the canceler of a.

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I am keeping that one grey,

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with a question mark on it,

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because at this point in the story
we do not know which number it is.

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Here is where the thread got
tangled, and it is a good tangle.

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Does a cancel minus a? Yes, at once.

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Rule three says minus a plus
a is zero, and that is exactly the job.

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So a does it, for free.

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So are we done?

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Not quite,
and the difference is a single word.

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What we have shown
is that a is a canceler of minus a.

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What the symbol names
is the canceler of minus a.

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Look at rule three once more.

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It promised that a canceler exists.

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It never promised there is only one.

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If minus a had two different cancelers,
the symbol would be ambiguous.

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It might be a. It might be the grey one.

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Two minuses making
a plus would be a coin toss.

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So here is the theorem
underneath the claim.

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Every number has exactly one canceler.

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Once you have that, minus minus
a equals a is only that theorem,

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read out at the number minus a.

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And you might think exactly one
canceler is obviously true. It is not.

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To see that exactly one
canceler is not obvious,

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let me build a number
system where it is false.

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Four numbers: zero, a, b and c.

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Its addition is given by a table,

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the same kind you learned
your sums from, only smaller.

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Check rule two.

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The zero row and the zero column hand
everything straight back, unchanged.

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Zero does nothing here.

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Check rule three.

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Every number here has a canceler.

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a and b add to zero.

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c and b add to zero.

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And zero cancels itself.

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The table is even symmetric,
so this addition commutes too.

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Now look at the row for b.

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It has two zeros in it.

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b is cancelled by a,
and b is also cancelled by c.

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In this world, b has two cancelers.

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So compute the canceler
of the canceler of a.

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The row for a has exactly one zero,
in the b column, so minus a is b.

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Now the row for b has two zeros,
so the canceler of b is a, or it is c.

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Take c.

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In this little world, minus
minus a is c, and c is not a.

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Which rule did we break?

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Not rule two.

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Not rule three.

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And commutativity is fine as well.

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Look at rule one.

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Add a and b first, then add c:
a plus b is zero, and zero plus c is c.

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Now bracket the other pair:
b plus c is zero, and a plus zero is a.

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And look at the two values we just got.

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The first was c. The second was a.

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Those are exactly the two numbers
that were fighting over the title canceler

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of b. That is not a coincidence. It
is our theorem, seen from the wrong side.

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So what does our number system
have that the toy one does not?

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One thing.

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The bracket can slide.

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Here is what that buys us.

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Suppose two numbers, a and c,
both cancel the same number b.

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So a plus b is zero, and b plus c is zero.

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I am assuming nothing else about them.

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Must a and c be the same number?

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Write all three of them
in a row: a plus b plus c.

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Rule one says this row has one
value, no matter how I bracket it.

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So let us bracket it two
different ways and compare.

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Bracket the left pair first.

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a and b cancel, so that pair is zero.

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Zero plus c is c.

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The whole row is worth c.

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Now the one motion
to remember from this whole video.

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Watch the bracket leave the left pair,
and land on the right pair instead.

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Now the same thing
happens at the other end.

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b and c cancel, so that pair is zero.

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a plus zero is a. The row is worth a.

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Same row.

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One value. So a equals c,
and that is the proof.

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Any two cancelers
of the same number are equal.

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Cancelers are unique.

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There is nothing clever in that argument.

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a and c only had to be in the same room as
b, and rule one is what puts them there.

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In the toy world the bracket cannot
slide, so a and c never have to meet,

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and nothing makes them equal.

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That is the only difference
between the two worlds.

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Let us write that down as a theorem,
because we are about to use it twice.

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If a cancels b, and c also cancels b,
then a and c are the same number.

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Rule three says a canceler exists.

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Rule one says there is only ever one.

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Before we cash it in, let us fix
what the symbol is allowed to mean.

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Minus minus a is a canceler
of minus a, any one of them.

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We are not assuming
there is only one of them,

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because that is the thing we are proving.

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Now rule three gives us two promises.

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First: a has a canceler,
and the two of them add to zero.

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Second: minus a has a canceler too,
and minus minus a is the name of one.

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Write all three
in a row, exactly as before:

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a, then the canceler of a,
then the canceler of that.

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Rule one says this row has one
value however I bracket it.

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Bracket the left pair.

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a and its canceler make zero,

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and zero plus the last slot
leaves the last slot standing.

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So the row is worth minus minus a.

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Now slide it. The bracket leaves the left
pair and lands on the right pair instead.

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And now the other two cancel,
by the promise we just wrote down.

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a plus zero is a. So the row is worth a.

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Same row, one value.

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So minus minus a equals a. That is the
theorem we came for, and it is now proved.

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Written as a single chain,
the whole proof is five lines.

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Zero does nothing, so the thing
on the left is zero plus itself.

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That is rule two.

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That zero is a plus
the canceler of a: rule three.

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And now the only
interesting line in the proof.

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Slide the bracket.

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Rule one.

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The middle and the right cancel,
by rule three again, leaving a plus zero.

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And a plus zero is a, by rule two.

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Done.

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Now notice what never
happened in that chain.

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We never swapped two terms.

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The fourth rule, that a plus b
equals b plus a, was never used.

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So this is not really
a fact about numbers.

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It holds anywhere there is a zero, an undo
for every move, and a bracket that slides.

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Rotating a cube. Shuffling a deck.

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The undo of the undo
is the move you started with.

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So back to the argument.

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Who was right?

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Both of them, describing one
theorem from opposite ends.

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The person who asked
had the shape of it exactly right.

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a cancels minus a. Cancelers are unique.

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Therefore a is the canceler of minus a,
and minus minus a is only its name.

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The objection was right too:

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you cannot get
that middle line by notation.

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Which is exactly why
the small proof earns its keep.

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It proves the middle line on the spot,
with one slide of the bracket.

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The objector saw that, and said so.

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That is a good thread.

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One more question
from the thread, and it is a good one.

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Does everything we have just done prove

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that a negative times
a negative is a positive?

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No. And seeing why is worth a minute.

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Think of the rules as a building.

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Everything today happened
on the ground floor: addition, zero,

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cancelers, and the bracket that slides.

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On this floor there is no multiplication
at all, and no such thing as a positive

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or a negative number.

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Minus a is only the canceler of a, and it
might be five, or minus five, or zero.

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To even say negative times negative you
need the next floor up: multiplication,

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and the rule that ties it to addition.

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Watch what happens there.

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Add a to minus one times a,
and rewrite the first a as one times a.

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Pull the a out.

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One plus minus one is zero,
and zero times anything is zero.

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So minus one times a cancels a.

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And because cancelers are unique,
minus one times a is minus a.

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That is our engine, borrowed.

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From there, minus a times
minus b works out to a times b,

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and that step uses today's theorem.

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But you still have not
said the word positive.

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For that you need a third floor,
the rules for which numbers count

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as positive, and only there
does the sentence mean anything.

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Today's theorem is one brick
in that wall. A load bearing one.

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But it is not the wall.

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So here is the picture to keep.

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Draw a line between two
numbers whenever they add to zero.

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Rule three says every number
gets at least one such line.

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Rule one says no number gets two.

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So the lines make a perfect
pairing, and zero is its own partner.

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Minus minus a equals

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a says exactly this:

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go to your partner, then go to your
partner's partner, and you are home.

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Turn around twice: true.

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Why it is true: one slide of a bracket.