# Lesson 8.6 of Exploding Dots

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Okay, with a very interesting
philosophical point right now,
the question at hand is, is it true that
every fraction, when converted to a decimal,
falls into a repeating pattern?
The answer turns out to be yes.
And being kind of see why right now.
Here's the examples of 6/7
we've had in the previous lesson.
And we saw that we started with number 6
We have all these different
numbers unexploding
with these different remainders
which got unexploded.
Now remainders. We've got remainders of 4.
We've got remainders of 5, remainder of 1,
remainder of 3, remainder
of 2, remainder of 6,
remainder of 4, remainder of 5,
so there are many of this stuff repeating.
As soon as the remainder stop repeating,
We know we're in a repeating pattern,
our answer will repeat as well.
So the question is, is it true, when I do
a fraction problem could be a decimal,
that I'll get repeating remainders?
Now think about the remainders.
When I'm dividing by 7,
I will never see a remainder of 7
because that would mean
there's a group of 7 I should've collected.
I'll never see a remainder of 8,
I'll never see a remainder of 9
for the only remainders I'll ever
possibly see when divided by 7
are remainder by the
1, or 2, or 3, or 4, or 5, or 6
not 7 or 8 and so on.
I might see a remainder of zero
I didn't happen to this time
but there's only
seven possible remainders
I could see when dividing by 7:
0,1,2,3,4,5, and 6.
Grand! So that means as
I do my division process,
there's only a finite number of remainders
and I'm trying for this
process for as long as I can
I must eventually repeat a remainder.
In fact we repeated the
6th right off the bat.
So with our repeating remainder,
that means I'm in a cycle.
I've got a repeating decimal pattern
Okay. Now when we did one quarter,
we actually got a repeat remainder of 0.
We got 0.25, we figured that out.
Then a remainder of 0, which led to a
remainder of 0, which led to a remainder of 0
It fell on this small repeating cycle.
So there was repeating remainders as well.
See, even 1 quarter is repeating
decimal, repeating zeros after a while.
When we did 1/3, it fell under
the same remainder right away.
remainder 3, remainder 3,
remainder 3. So when it falls repeat
remainder ones, it's in a cycle.
If I did a number like 3/13
When we draw it out
be even worse than this one
but we can think a way through it.
As I draw 3 dots that equips the 13
I know the remainders that I possibly see
could be 0,2,3,4,5,6,7,8,9,10,11, or 12
but we'll not see a remainder 13,
we'll not see a remainder 15,
we'll not see a remainder 15 and so on.
There's only a finite number of
possible remainders I will get to.
So as I play this game,
I have to eventually repeat the remainder.
I can't keep doing different remainders.
There's only 13 possible remainders.
I've got to especially hit the
same remainder like before
and as soon as I hit the same remainder,
that means I'm in a cycle.
We've just proven every fraction,
when converted to a decimal
has a repeating pattern.
It maybe a repeating zeros
or repeating pattern like this
but every fraction has
repeating decimal expansion.
Woah! This is deep. It's so deep
I'm going to write it in words.
Every fraction has a repeating
-you can see my atrocious writing-
a repeating decimal expansion.
Alright. It could be repeating zeros
like a quarter or a half or 8th or something
or it could be repeating blocks like this
but has repeating decimal expansion.
This is deep.
Because what it says now,
suppose I gave you a decimal
that didn't have a
repeating decimal expansion,
did not have a
repeating pattern of any kind,
then that means, that
number cannot be a fraction.
Now, people often call fractions rational
numbers like a ratio of 2 whoel numbers
So numbers not a fraction
will be called irrational
and I'm going to write for you,
write down for you right now an example
of an irrational number.
We can now prove at this very moment,
that irrational numbers actually do exist,
here's one. It's .10110111011110, and so on.
This number, now it has a pattern,
So there's a pattern.
You know what this pattern is.
You could figure out what the
millionth digits gonna be or
the 2 millionth digit gonna be
You know what this number actually is.
You could figure it out.
But it doesn't have a repeating pattern.
There's no repeating cycle going on.
In which case, this is some
number, it's on the number line.
It's just bigger than 1/10,
cannot be a fraction.
It had no repeating pattern
therefore it's not a fraction
This is our very first example
of an irrational number.
There's one thing that
drives me crazy as a teacher.
A lot of books say that we all know that pie,
for example, is an irrational number.
the squareroot of 2 is an irrational number.
We don't actually prove these things as hard.
Some books might prove the
square root of 2 is actually irrational.
It doesn't have repeating decimal
but not pie. Pie was very hard to prove.
that took mathematicians 2
millenia to figure out how to prove
that pie actually was an irrational number
is not at all straight board
but what I love about what we've just
done now, dots and boxes, we could see
every fraction must have
repeating decimal expansion.
Therefore, any number you write down that
does not have a repeating decimal expansion
must be an example of a
number that's not a fraction.
We have a first example here of an actual
irrational number that we could see and own for ourselves.
Brilliant! In fact, your turn.
Write down another example
of an irrational number.
Isn't this grand?