0.9 repeating.
[NOISE]
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.
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.
[SOUND]