Thank you to Curiosity Stream for sponsoring this video.
Hi there! So in a video that I just uploaded on my channel, I was looking through this exam paper.
It's from 1866 so it's very old and it was interesting to see
some of the arithmetic questions that were asked back then.
One of the stand out difficult questions or at least what I would consider difficult was this one here number 20:
What is the square root of zero point zero zero four three zero four six seven two one?
Now finding the square root of this without a calculator - which they wouldn't have had back then -
that's something that strikes me as quite difficult because I've never really had to do this in exam situations
During my small amount of research
I've found that some countries still get students to prove that they can do these things without a calculator.
The merit of having to prove such things without calculators,
even though we can do it on the calculator, I guess is up to debate.
But today I'm going to just work through this problem and see how you actually do this
if you are stranded somewhere without any help from machines.
So...
First of all, just a tiny bit of theory that might come in handy to understand the steps here.
Essentially, we're going to use an algorithm to solve the square root and it's one that you can do by hand.
It comes from the idea that we're going to approximate the square root,
and if we say that our proximation at any point is given by x,
but it's not the best approximation and we can do better, we want to find an x plus r
so that this whole value is less than the square root that we're trying to find.
So we don't want to over approximate - we want our guesses to remain below the square root.
We're essentially approaching it from below.
So every time we add on an r to our approximation, it should get closer and closer from below.
Now if we take this inequality and just do some algebra on it
we can shift the square root over to this side (←).
We can expand this side...
And then we can factorise a different part of it.
Now this last line I've written, you might wonder where it came from,
but it is actually the algorithm that we're going to use
to better our approximations to find the square root.
Let's have a look at it in practice, and then you might understand a little better where it's come from
First let's just write down our number.
And what we're going to start with is for every digit after the decimal point
we're going to group them in groups of two.
So just put a line over each group of two to group them up there. That'll just help us.
Alright, so...
To start with we want to find a number whose square is less than or equal to the first pair of numbers.
However, we're going to skip over the string of zeros here.
We're just going to put zeros here and start with our first sort of real digits here the four and the three.
If you are familiar with your squares you would know that:
six squared is 36;
seven squared is 49.
So six squared is the biggest square that is less than this one here.
So let's put down a six.
Now since six squared was 36 we're going to write that down below and we're going to subtract that from
43 so if we do that subtraction the difference will be seven.
Now, let's bring down our next set of two numbers. It's a zero and a four,
so we've got 704 here. We're going to put a line in here,
and then what we're going to do is take our previous number and double it.
So two times six is 12,
but then we've also left a little gap in here because this is going to be 120 something,
and we want to find what this digit should be such that 120 something
times that something is less than or equal to 704.
So for example, we could try 125 times five,
and see what that is equal to.
I find that trying the five first is a good place to start because you can see if you need to go higher or lower.
125 times five actually works out to be 625,
and you can do that off in the margins with a reasonably simple multiplication by hand.
So that's 625. It qualifies because it's under here (<=704),
We best try what 126 times six is just to be sure,
and 126 times six is 756. So it's over.
So actually we're going to be using five as our final answer in here.
We'll also put a five up the top to record it.
Now this might have looked kind of random but it comes from this algorithm step here
where we were taking two times our previous guess plus a new r
timesing that all by r and it has to be less than or equal to
what we're trying to find the square root of minus x squared.
so we've been minusing along the way here.
Let's just continue with this step that we went through,
we're just going to keep iterating it until we get to the end of our numbers.
Now 125 times five like I said was 625.
So we'll just subtract that from 704
What we'll get is 79,
and then we'll bring down the next two numbers which are a six and a seven.
Again, let's find this number times two,
65 times two is going to be 130, and then we'll leave a little gap here.
So we can see 1,300 and something, times that something
has to be less than or equal to this here (>=7,967).
So we could start by trying five again - 1,305 times five,
that would be 6,525. It's a little low.
So we'll keep trying we'll go a bit higher.
We'll try 1,306 times six,
and that would be 7,836.
Now that's as close as we're going to get,
so six is going to be what we'll use here and we'll record a six up there.
Subtract these two from each other, what do we get?
We get 131.
bring down our last two numbers which are a two and a one,
take this number 656 and double it.
When we do that it's 1,312.
Leave a space for a digit and then
13,120 something times that somthing
will be less than or equal to this (<=13,121).
Well, actually this is very close to this already if we just had a 1 in there,
13,121 times 1 will give us this exactly.
So 1 will be our last digit and because we don't have any
remainders left over here if we were to...
do it like we were doing before,
this would be 0. You've actually reached the end of our process
and this value here, 0.06561
is the square root of what we were trying to work out.
So in practice it actually isn't too difficult; the working is only this big.
But to get here you have to sort of know this little algorithm,
and off in the margin you have to try several attempts
at multiplying things until you get what's right.
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I'm really looking forward to watching 'The Secret Life of Chaos'.
It explores the mathematics of chaos theory and how it can explain why the universe creates order and pattern?
Thanks Curiosity Stream. Also, thank you to my patrons who support me on patreon.
And thank you for watching :)