Hi, I'm Toby and welcome to the Joy of Mathematics, I'm so glad you could join me today.
I'm going to run some ideas across the bottom of the screen and these are ideas which we will come to understand today.
It's another beautiful day outside today, although it is a little windy.
I've gone ahead and already drawn a tree for us
but a tree is a little bit like an iceberg in that what you can see on the surface is only a small part of
the story.
What is going to be really important to us today is what we can see under ground, a
tree could not exist without its roots.
Now last time I saw you we were talking about logarithms and how they can be used to rewrite
exponents, one particularly popular exponent is the square, a number times itself.
We could have 3 times 3 is equal to 9
4 times 4 which is 16
or even 5 times 5
which is 25.
These are all squares and could be written as
3 squared
4 squared
or
5 squared but what if some leaves carried by the wind were to come across and land
right here
and right here so
that all we could see was the result.
We knew that something squared was equal to 16, but we couldn't quite remember what.
Well, what we would need to do would be to find the square root of
16.
Roots are what allow a tree to grow
the better the roots the more nutrients they can collect and the more support they can provide.
Just like in life having support can make it feel easier to achieve great things,
although support can often be hard to come by or difficult to recognize.
Just drawing in our roots here.
You can see these roots would be very good at providing some moral and
structural support to the tree.
We also forgot to add the apple to our tree before so we can go ahead and sneak that in now.
Now in these square roots there are 16 units of dirt, that's the area of the square.
The area of a square is
calculated from its base times its height and
since these are the same, if we know the length of just one side,
we know the square root. We can go ahead and divide this up into a grid
and
if the length of each
side was 3 then we would have 9 units of dirt.
But if instead it was 5 on each side, we would have 25 units of dirt.
But the sweet spot for us and you're going to see it here, is that if each
length
of the roots, each side of the square is 4 units long then in total we will have 16 units of dirt
sitting right here in our roots, and that's exactly what we wanted. So we know that the square root of 16
will be equal to 4.
All these numbers that I've mentioned 9, 16, 25,
they're all called perfect squares because they are the squares of whole numbers.
We could look at a non-perfect square say 20, although all numbers are perfect in their own way and
we could look, well 20 is between 16 and 25
so the square root of 20 would be somewhere between 4 and 5 and in fact, it is 4.47.
Now there's actually another solution to all of these square roots, and I hate to mention it as an afterthought
because nobody likes being an afterthought, but we could have also had for example
-4 times -4 being equal to 16 because two wrongs make a right, but
in our case having a negative side length wouldn't make a lot of sense
so we were fine to just take the positive solution.
There is actually a symbol that tells us to look underground and find the square root of a number and
that is this
heart beat kind of symbol here called the radical because it is pretty radically cool.
Now I remember when I very first
encountered the symbol on some math homework at school
but I was pretty confused because I had never seen it before I didn't know what it was called
and I didn't know how to find out.
I wanted to google it but I didn't know how to describe it in the Google search, do you type in
'tick like symbol' or 'heartbeat'?
I kind of liked the name heartbeat symbol.
Now if you like me
you might be a little bit curious as to the root of this square root symbol
and there seems to be a little bit of disagreement about where exactly it comes from
one story that I've heard as to its origin is that it comes from the Arabic word for root and
it comes from the first letter of the word which looks a little bit
like this.
The Latin word for root is actually radix and it comes from radix quadratum
which actually means the side of a square.
Some mathematicians wrote this as
Capital R x.
Now that might have made its way to becoming the modern symbol or it might have also come from a lowercase r
also for radix, but there's one more possible origin story and that is that some
mathematicians used to denote a square root as putting a
dot or a few dots above the numbers that they wanted to take the square root of
and it's thought that maybe somewhere along the way these dots
just acquired a bit of an upstroke and that became our symbol.
What we do know is that in
1637
Descartes was the first to use the square root symbol with this line over the top that has a name
It's called the vinculum and that includes everybody in the square root who was supposed to be there.
It's like a big group hug. The square root symbol typically means to just take the principal or the positive
result of the square root. So there is only one leaf that would satisfy
the square root of 16 is equal to
leaf.
Only one solution would satisfy that but if instead we were to have leaf
squared is equal to 16
there are two leafs that would satisfy this and we would have to write our answer as being
plus or minus
4.
The square root can also just as easily be written as
16 to the half, we can start using fractional powers and
for other roots say the cube root
we will just extend this to be 16 to the one-third or we could write
3 in our little square root symbol, and that would give us a different solution.
Squaring any number positive or negative will give us a positive result.
Squaring 0 will give us 0 so there is actually no way for a negative number to have any real root.
But we can make things more inclusive if we invite over our imaginary friends, if we have
we introduce our friend i squared
to be equal to minus 1
then we would have that i
is equal to the square root of -1.
Technically - i would also be equal to the square root of -1, but by convention we just take positive
i to be our square root minus ,1 our principal root.
Then if we wanted to do the square root of minus 16
we could do that and it would give us the square root of
16 times the square root of -1
leading us to an answer of 4i.
Moral of the story is don't let anyone tell you that you can't have a square root.
You can even take the square root of an imaginary number
but you can see my other video for how to do that.
So that has been our trip underground to understand square roots
if you haven't already then take a walk near some trees and try to imagine what their hidden roots look like.
I'd like to wish you happy studying and I hope that you have an
absolutely
mathematical day!