hi I'm Toby and welcome to another episode of the joys of mathematics if
this is your first time watching then here's your chance to go and grab a
notepad and pen so you can work along at home and if you are a returned viewer
thanks for coming back now today we're going to work through something that's a
little more abstract in its creation and we're going to do this to hopefully see
that sometimes you can't always trust something on its first impression
sometimes you need to look a little bit closer because often things are not as
they seem so keep in mind that this piece today is
going to be more of an abstract artwork than a conventional one so I've got my
trusty titanium white color here and let's get started
the first thing that we're going to do is we're going to let a equal b. a and b
are our first two characters and they are variables they could be anything
that they ever wanted to be they have the whole world as their oyster but
today a and b have chosen to be exactly the same thing now as we go along don't
be afraid of the mathematics here I'm here with you and I'm going to help you
go through it well what we're going to do is times both sides by b that will
give us a times b is equal to b times itself b squared so we've added a couple
of happy little b's here to this line we added these to both sides so that's fine
we can add or multiply whatever we want as long as we do it on each side same
with division and subtraction there's a moral of the story here only do to
others what you want done to yourself and when you've got two sides of an
equation anything you do to one side you have to do to the other now let's add
some more happy little b's and in fact we're going to add another b squared to
each side of this equation
all right got a lot of happy little bs let's just simplify this b squared plus
b squared could just be written as 2 b squared and
this line for the side says the same now you might be sort of wondering how this
is going to end up anywhere but just trust me sometimes you need to just go
where the wind blows you and it is pretty windy out here in the garden
today I hope that it doesn't cause too many issues but let's just follow along
for the drift of the wind here and you might have to trust me a little bit okay
we've added b squared what if we do we'll got an ab what if we subtract the
same term from both sides and that term is going to be 2ab all right let's do
it to both sides this term that I subtracted from both sides could have
been literally anything since I did it to both sides and it's going to make the
result the same but up until now we haven't done anything illegal with our
algebra this is all this is all fine okay so if we have one ab and we're
taking away 2 ab's how many ab's do we have left well the answer is we'll
still have our b squared and minus one ab and on the side we've got to be
squared minus 2 ab all right what I want to do next is to factorize this
line and I'm going to continue up here to do that so there's one common term to
both sides and that is b squared minus ab on this side we could take a 1 out of
the bracket if we really wanted to somewhat unnecessary but we have 1 times
B squared minus a B and that is equal to well we can take a 2 outside the
brackets now 2 times b-squared minus a B getting a
little close to the edge there but I think we're alright okay so now we've
got this line still nothing illegal has happened you want to check my algebra
you can go through there but now we have something all right we have B squared
minus a B on both sides of the equation what we may want to do is to
recognize that if we divide both sides by the same thing that's alright like we
can divide both sides by B squared minus a B and that would be the same as
cancelling out this term if we'd cancelled out this term in brackets then
what we have left with is that one is equal to two now here's where it appears
to be pretty abstract and in fact something doesn't quite add up here I'm
going to put a little box of shame around this because this means we've
done something weird or something illegal a long way we can't just
suddenly have one being equal to two because that would fundamentally change
so many things about mathematics we rely on two being exactly double one and not
being equal to each other so have a little look at the algebra and see if
you can identify where we went wrong and why don't worry if you can't get it I'm
going to show you in just a second but if you've formed your opinion and then
let's have a look I stand by that all of this algebra is correct
the issue comes in at this line here now we divided both sides by B squared minus
a B and that would be fine as long as B squared minus a B the stuff in brackets
was not equal to zero if this stuff was equal to zero this would amount to
dividing both sides by zero and as you might know dividing by zero is kind of
forbidden and mathematics the answer to that is undefined when you start to
viding by zero things in the universe stop making much sense
you could already sort of argue that the world doesn't make a lot of sense to us
you know good things happen to bad people all the time and there's a lot of
things that are unexplained but right here this would give us some weird
things dividing by zero leads to all sorts of things such as that one is
equal to two and frankly this is nonsense so we need to sort this out so
let's have a look B squared since a and B are equal to each other is actually
the same as a times B so this in brackets is in fact equal to zero and
what we have on this line up here is that one times zero is equal to two
times zero now I mean sure anything times zero is going to be equal to zero
so these 1 and two could have been replaced by literally anything all this
really shows is that zero is equal to zero so if we've proved anything that's
what we've proved here not that one is equal to two so there is a bit of a
moral here and it was to not trust your first impressions all the time you know
when we did this step it might have looked like it was legal and this might
have looked like a legal result but we used our sort of critical thinking we
realized it probably doesn't make sense and there's probably a reason why so the
rule for mathematics is that whenever you're going to divide by something
check that that something is not zero but also in life you know sometimes
things are too good or too weird to be true so always look a little closer and
don't just take things at face value now I'm going to rub this off here and make
room for a little message a duster can get pretty chalky so you just beat the
devil out of it this episode of the joys of mathematics is brough to you Skillshare
Skillshare are an online learning community with classes in things like
business design and technology I'm currently working through some courses
on writing short stories and editing videos in final cut if you want to
learn new skills then as a special offer for my viewers
first 500 people to sign up for Skillshare at the link in the
description will get two months of unlimited access for free that's access
to over 20,000 courses so I'm sure you can find one there that you like this is
also your personal invitation to subscribe to my channel for more videos
like this don't forget to take a closer look at the world around you and from
the bottom of my heart I hope that you have an absolutely mathematical day
here's some bonus after the credits footage but I just wanted to show you
guys something in the garden it's over here and it's a moth or a butterfly
perched on a little branch there now that's pretty nice but don't look too
close past him well you might see what scared me earlier it seems to be a
spider I'll zoom in because I don't want to get too close it's either the spider
or like the skin of the spider but to be honest I didn't know that spiders head
skin so I'm getting out of here