0.9 repeating.
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Here's something that's often confusing.
0.9 repeating, by which I mean 0.999999
just going
on like that forever.
That is equal to 1.
Not just close to 1.
Not a little bit less than 1.
This is exactly equal to 1.
There's a few different ways to think
about this.
For example you might already believe that
0.3 repeating, right.
0.3333 and so on is equal to a third.
Now if I multiply all of this by 3 what
happens?
Well if I multiply this by 3 I get 0.9
repeating, right.
If I multiply 0.33333 by 3,
I get 0.99999.
But if I multiply a third by 3.
I get 1.
We could also formulate this in terms of a
series.
I can instead think about it this way.
0.9 repeating is the sum
n goes from 1 to infinity, of 9 times 10
to the negative n.
Well, why is that?
Well, what's the first term here?
What happens when I plug in 1?
That's 9 times 10 to the negative first
power.
When I plug in n equals 2, that's 9 times
10 to the negative 2nd power.
When I plug in n equals 3, that's 9 times
10 to the negative 3rd power, and so on.
Well what are these?
Alright, 9 times 10 to the negative 1st
power is 9 10ths.
9 times 10 to the
negative 2nd power is 9 100's.
9 times 10 to the negative 3rd power
is 9 1000's and it keeps on going.
Well, what's 9 10ths?
That's 0.9.
What's 9 100s?
That's 0.09.
What's 9 1000's?
That's 0.009.
And it keeps on going.
And when I add these up, well, what I end
up with is just a
0.9 from here.
This 9 gives me a 9 here.
This 9 gives me a 9 here.
The next term gives me the next 9 and so
on.
I end up with 0.9 repeating.
We can evaluate that series.
So this series, is 9 times the sum n
equals 1 to infinity of 10 to the minus n.
And I can make that look even more like a
geometric series.
Its 9 times the sum and goes from 1 to
infinity of 1
10th to the nth power.
Now how do I evaluate that?
We've got a formula for summing an
infinite series like that,
it's a geometric series whose common ratio
is between zero and one.
So that converges and its sum is 9 times
the first term, is 1 10th
divided by 1 minus 1 10th, and what is
that?
That's 9 times a 1 10th over 9 10ths.
Well, what's 1 10th over 9 10ths?
I could multiply the top and the bottom by
10.
And I get this is 9 times a 1 9th and 9
times 1 9th is 1.
And this discussion brings up an important
point.
Any time that we're writing down decimals,
I mean if I were
just make up some numbers like 0.57896 and
imagine it keeps on going.
Any time I'm writing down real numbers
like that, I'm secretly writing down a
series.
What do I even mean by dot, dot, dot?
I really mean the series.
I mean that this is the sum n goes from 1
to infinity of d sub n times
10 to the minus n.
Where these d sub n's are the digits,
alright, d sub 1 is 5, d sub 2 is 7, d sub
3 is 8, and so on.
Alright.
So any time I'm writing down a decimal
expansion, I'm secretly
writing down an infinite series, just by
adding up all the decimals.
To make it a little bit clearer, let me
just write down the first few terms.
Right,
n is equals 1 and d sub 1 is 5, means that
this is 0.5, and to 0.5, I'm adding the
n equals 2 term, which is d sub 2 is 7
times 10 to the minus 2, which is 0.07.
And then I add the n equals 3 term, which
is 0.008
and then I'll add the n equals four term,
which is 0.0009 and so on.
So all of these decimal representations of
real numbers are secretly just a series.
So when thinking about real numbers or at
least
their decimal representations, we're led
naturally to think about series.
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