Main content

## Math for fun and glory

# 9.999... reasons that .999... = 1

Point Nine Repeating Equals One! 9.999... reasons in 9.999... minutes. Bonus points if you can name all 9.999... lords a-leaping. Dear YouTube, wouldn't it be nice if I could include the full script with this video? A larger character limit would not be unreasonable. Created by Vi Hart.

## Video transcript

99.9 repeating percent
of mathematicians agree, 0.9 repeating equals 1. If because I said
so works for you, you can go ahead and
do something else now. Maybe you're like,
0.9 repeating equals one, that's this 0.9
repeating-derful! Otherwise, on to reason number
2, or reason 1.9 repeating. See, it's weird, because when
we think of the number 1 or 2, in most contexts we mean
it as a natural number, like 1, 2, 3, 4, 5. In the sense then, the
next number after 1 is 2. 9.9 repeating may
equal 10, but you wouldn't say you have 9.9
repeating lords a leaping, in the same way you
wouldn't say you had 9.75 lords a leaping
plus 1/4 lord a leaping. Lords, leaping or otherwise,
come in natural numbers. So what does this
statement mean? 0.9 repeating is the same as 1? It looks pretty different, but
it equals 1 in the same way that a 1/2 equals 0.5. They have the same value, You can philosophize
over whether, if 1 is the loneliest number, 0.78
plus 0.22 is just as lonely, but there's no
mathematical doubt that they have the same value,
just as 100 years of solitude is exactly as long as 10 plus
40 times 2 years of solitude, or 99.9 repeating
years of solitude. So reason 2 is not a proof, but
a reason to stay open minded. Numbers that look different
can have the same value. Another example of this
is that, in algebra, 0 equals negative 0. Reason 3. 0.9 repeating is a decimal
number, a real number. See, if you want
0.9 repeating to be that number infinitesimally
close to 1, but not 1-- and let's face it, some
of you do-- then you're writing down the wrong number
when you write 0.9 repeating. That number infinitely
close to 1, but less than 1, is a number, but it's not 0.9
repeating or any real number. OK, let's do more 3.9
repeating-mal proof for reason 3.9 repeating. According to this
3.9 repeating-mula, 3.9 repeating is 4. First step, say 0.9
repeating equals x. Then multiply each side by 10. Third, subtract 0.9 repeating
from this side, which equals x, which we subtract
from the other side. So 9.9 repeating minus
0.9 repeating is 9. And 10x minus x is 9x. Divide by 9, and
you get 1 equals x, which you might notice
also equals 0.9 repeating. There's no tricks here. It's simple multiplication,
subtraction and division by 9, which are all allowed
because they are consistent. When something is inconsisten--
9.9 repeating --t, we just throw it out
of algebra altogether. For example, in algebra
if you try to divide by 0, you get this problem where
anything can equal anything. I mean, if you want to say
everything is equal, fine, but your algebra sucks. Normal, everyday
elementary algebra, the one they shove down
students throats as if it were the only algebra, doesn't
allow dividing by 0. So it stays consistent and
suspiciously practical. We also could have shifted
the decimal point twice, multiplying by 100 to prove
that if you have 99.9 repeating bottles of beer on the wall,
99.9 repeating bottles of beer. Take one down, pass it around,
99 bottles of beer on the wall. All right, moving
3.9 repeating-ward. Reason number 5,
there's infinite 9s. If anyone ever thinks they
have the biggest number, well, they don't, because just add 1,
or multiply by 2, or whatever, and it's even bigger. Infinity, though,
is not a number you can add 1 to to
get a bigger number. Adding 1 is an algebra thing
that you do with real numbers. Subtracting doesn't work either. Infinity bottles of
beer on the wall minus 1 is still infinity
bottles of beer. When we did this decimal
shift to multiply by 10, unlike un-infinitely
many 9s, there's no last 9 that got shifted
over to create a 0. Infinite 9s plus
another 9 is still infinite 9s, the kind
of infuriating property that makes infinity
not a real number and makes that proof work. If you're the type of person who
is discon-- 9.9 repeating --t with the idea that 9.9
repeating equals 10, you might also feel
that 1 divided by 0 should be infinity. And, as it turns out, there is
other systems or calculation besides elementary
algebra where it does. That's right, mathematicians
figured out how to divide by 0 a long time ago. But elementary algebra
can't deal with infinity. If you allow infinity in your
algebrizations, once again, you get contradictions. Infinity may not
be a real number, but it is a number,
a hyperreal number. Hyperreals, like infinity
and the infinitesimal, follow different rules. And while algebra
can't handle them, some people thought
they should be numbers, and you should be
able to use them. And so they figured
out how, and bam! you get something like calculus. Reason 6. Take the number 1, and
subtract 0.9 repeating. It's pretty clear
that it's infinite 0s, but you might be
tempted to think there's some sort of
final 1 beyond infinity. Let's write that down
as 0.0 repeating 1. Of course, if the 0
repeats infinitely, then you never get to the
1, so you might as well leave off the number. Thus the difference between
0.9 repeating and 1 is 0. There is no difference. Here's another "there's
no difference" proof. Remember how the next higher
natural number after 1 is 2? What's the next
higher real number? The game, is for any number you
claim is the next real number, I can find one that's
even closer to 1. Of the many delightful
things about real numbers is that for any two numbers,
no matter how close, there's still an infinite
amount of numbers between them, and an infinite amount
of numbers between those, and so on. There is no next higher
real number after 1. Likewise, there's no
next lower number. If 0.9 repeating and 1 were
different real numbers, there would have to be
infinite other real numbers between them. If you can't name a number
higher than 0.9 repeating, but lower than 1, it can only
be because 0.9 repeating is 1. If you don't like it,
well, go to college and learn about hyperreals,
or better yet, surreals. That's a system
where you can have a number that's infinitely
close to 1, but not 1. But even weirder,
there's infinity more numbers that are
infinitely even closer. Anyway, on to reason number
8, another common proof. Take 0.3 repeating, a
repeating decimal equal to 1/3. Multiply it by 3. Obviously, by definition, 3/3
is 1, and 0.3 repeating times 3 is 0.9 repeating, which you
might have noticed is also 1. The only assumption here is
that 0.3 repeating equals 1/3. Maybe you don't like
decimal notation in general, which
brings us to reason number 9, this sum of an
infinite series thing. 9/10 plus 9/100 plus 9/1000. And we can sum this
series and get 1. But I can see why you
might be unhappy with this. It recalls Zeno's paradoxes. How can you get across
a room, when first you have to walk halfway, and
then half of that, and so on. Or, how can you shoot an arrow
into a target, when first it needs to go halfway, but
before it can get halfway, it needs to go half of
halfway, and before that, half of half of
halfway, and half of half of half of
halfway, and so on. Therefore, it can never
start to move at all. Anyway, it's 1/2 plus 1/4 plus
1/8, dot, dot, dot, dot, dot, to get 1. Each time, you fall short of 1. So how can you ever do anything? Luckily, infinity
has got our backs. I mean, that's
like the definition of infinity, a numbers so
large, you can never get there, no matter how many steps you do,
no matter how high you count. This way of writing numbers with
this dot, dot, dot business, or with a bar over
the repeating part, is a shorthand for an
infinite series, whether it be 9/10 plus 9/100,
and so on to get 1. Or 3/10 plus 3/100,
and so on, to get 1/3. No matter how many
3s you write down, it will always be less than
1/3, but it will also always be less than infinity 3s. Infinity is what gets us
there when no real number can. The binary equivalent of 0.9
repeating is 0.1 repeating. That's exactly 1/2, plus
1/4, plus 1/8, and so on. That's how we know a dotted,
dotted, dot, dot, dot half note equals a whole note. The ultimate reason that
0.9 repeating equals 1 is because it works. It's consistent, just like 1
plus 1 equals 2 is consistent, and just like 1 divided by
0 equals infinity isn't. Mathematics is about making up
rules and seeing what happens. And it takes great creativity
to come up with good rules. The only difference
between mathematics and art is that if you don't follow
your invented rules precisely in mathematics, people
have a tendency to tell you you're wrong. Some rules give you elementary
algebra and real numbers, and these rules can't tell
the difference between 0.9 repeating and 1, just like
they can't tell the difference between 0.5 and 1/2, or
between 0 and negative 0. I hope you see now that the
view that 9.9 repeating does not equal 10 is simply un--
9.9 repeating --able. If you started this
video thinking, I h-- 7.9 repeating that 7.9
repeating is 8, I hope now, you're thinking, oh,
sweet, 4.9 repeating is 5? High 4.9 repeating! If you're having math problems,
I feel bad for you, son, I got 98.9 repeating problems,
but 0.9 repeating is 1. Here's the moral of the story. The idea of a number infinitely
close to but less than 1 is not stupid or wrong, but
wonderful, and beautiful, and interesting. The true mathematician
takes "you can't do that" as a challenge. If someone tells you
can't subtract a bigger number from a smaller number,
just invent negative numbers. If someone tells you can't
multiply a number by itself to get a negative number,
then invent imaginary numbers. If someone tells you can't
multiply two non-zero numbers together to get 0, or raise one
non-zero number to the power of another and get 0,
you should probably say, I'll do both at
once, and in 8 dimensions. And if you ignore
them telling you that numbers aren't
8-dimensional, and that inventing fake numbers
is a useless waste of time, and then actually try to figure
it out, the next thing you know, you've got
split octonions, which besides being
super awesome, just happen to be
the perfect way to describe the wave equation
of electrons and stuff.