Identification & Indeterminate
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NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologerMathologer PayPal: paypal.me/mathologer (see the Patreon page for details) You've all been indoctrinated into accepting that you cannot divide by zero. Find out about the beautiful mathematics that results when you do it anyway in calculus. Featuring some of the most notorious "forbidden" expressions like 0/0 and 1∞ as well as Apple's Siri and Sir Isaac Newton. In his book “Yearning for the impossible” one my favourite authors John Stillwell says “…mathematics is a story of close encounters with the impossible and all its great discoveries are close encounters with the impossible.” What we talk about in this video and quite a few other Mathologer videos are great examples of these sort of close encounters. For those of you desperate to get hold of the t-shirt check out this link: http://shirt.woot.com/offers/how-natu...Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles. Enjoy :)
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00:04
Welcome to another Mathologer video. Today
00:07
i'll talk about these crazy expressions
00:10
here. Now in school they tell you that if you
00:12
don't stay away from these eternal
00:14
damnation awaits. And that's actually a
00:16
pretty good rule to pass on to the
00:18
masses to prevent them from suiciding but
00:21
if you actually stop there a lot of
00:23
modern mathematics is not possible. So
00:26
let me explain this a little bit.
00:28
Alright, so why do they tell tell you that
00:30
you can't divide 3 by 0 or that 0 / 0 is
00:34
undefined?
00:35
Well let's have a look at something that
00:37
nobody has a problem with --- 3/8. So in
00:41
maths 3/8 actually just stands for the
00:44
one and only solution of this simple
00:47
equation there 8 times x is equal to 3.
00:50
Ok, so let's change 8 to a 0 and see what happens.
00:54
Well we are immediately in trouble here because
00:57
no number satisfies this equation.
00:59
The equation will always be 0 is equal to
01:02
3 which is wrong. So it sort of makes
01:05
sense to stay away from something like
01:07
this, right? And so what about 0/0? Well a
01:11
different problem comes up here. Every
01:14
single number actually satisfies this
01:16
equation so there is also trouble.
01:20
Let's stay away from it. So for most
01:22
people that's all they really need to
01:23
know but now if we had stopped there in
01:26
mathematics there would actually be no
01:28
calculus and nobody would know Isaac
01:31
Newton. That would be really sad right?
01:33
Ok, so what's calculus about? Calculus is all
01:36
about derivatives and integrals. Here I
01:40
have drawn a nice function and we want
01:43
to find the derivative of this function
01:45
at a certain value. Now you all know that
01:48
this stuff is incredibly important but
01:51
it has a really simple geometrical
01:53
interpretation. The interpretation is the
01:55
derivative here is just the slope of
01:57
this touching line. Now just reading off
02:02
from the function it's not clear what
02:03
that slope should be. Wow what's easy to
02:06
calculate is actually the slope of a
02:08
cutting line like this. And then
02:12
the idea is that if I move these two points of
02:14
intersection together
02:16
then the more they come together the
02:19
closer the cutting line will be to the
02:21
tangent line and the closer the slopes of
02:23
the cutting lines will be to the slope
02:25
that I'm really interested in.
02:27
Ok now how do I actually calculate the
02:30
slope of the cutting lines?
02:32
Well you can just immediately read off
02:34
what the height here is and what the width is
02:37
and then we get the slope, just height divided by
02:39
width, pretty obvious. But now see
02:42
what happens when i move the two points
02:44
together. Both the height and width
02:46
approach zero, so overall the slope
02:49
approaches this forbidden 0/0. But now
02:53
nothing really terrible happens, right, you
02:56
would expect something terrible to happen
02:57
but nothing terrible happens.We are just
02:58
getting closer and closer to this slope
03:01
of the touching line. That seems a bit
03:03
strange but just remember that equation
03:06
that corresponds to 0/0 has all
03:09
numbers as solutions so what really
03:12
happens here is that we've established
03:14
a context, a very narrow context in which
03:17
0/0 kind of makes sense as one
03:21
particular number and you can see that
03:24
as the context changes we get different
03:27
numbers. So let's have a look at a
03:29
specific example. Let's calculate the
03:32
derivative of x squared at 1/2. Alright we
03:38
have to see here what's the width here and what's the
03:41
height. Width call it w so that gives
03:43
us a second value. Now we evaluate the
03:45
function at these two values. That gives
03:47
us 1/2 squared and 1/2 plus w
03:51
squared. Now the height is of course the
03:53
difference between the two so that one here.
03:55
Expand the top, notice that these two
03:58
things cancel out and you've got the
04:00
height and with the height
04:02
we've got the slope and you can see the
04:04
slope really what does it do at the top
04:06
it goes to 0 at the bottom it goes to 0
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but as long as we staying off 0 we can
04:13
actually cancel the w and what this
04:15
gives us this nice expression here and
04:18
just by looking at that it's completely
04:19
obvious what the limit of this
04:22
expression is as w approaches 0. That's
04:26
the slope we are after.
04:28
This means that the derivative of x
04:30
squared at 1/2 is equal to 1,
04:32
isn't this neat. Now 1/2 was actually not
04:35
very special here. We could have used any
04:37
initial value here and if we did that we
04:39
would find that the derivative of x
04:42
squared is 2x. So once we've got that
04:44
we've got this whole thing under control
04:46
as far as calculus is concerned.
04:48
Alright so we write this like that in
04:51
calculus books and actually if you have
04:53
a really close look you also find that
04:54
the 2 was nothing special. We
04:56
could have done the same thing for 3
04:58
and you should really try this, or 4
05:00
or for 5 or just in general for any
05:02
positive integer, not a big deal. And now
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we just run with the scheme and make up
05:07
a huge table of derivatives for all
05:10
sorts of functions that we are interested
05:12
in. And so you have it in a nutshell
05:13
calculus courtesy of 0/0, pretty neat isn't it.
05:20
So really that apple that hit Newton there at
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some point in time and made him invented
05:24
calculus was definitely a 0/0.
05:26
Before we go on and maybe
05:29
don't believe anything I said so far
05:31
let's just ask our smartphone what 0/0 is.
05:33
So Siri ...
05:49
That's fun but now I also promised you
05:51
all of these guys. It is very important
05:54
to make sense of these and actually it's
05:55
done in a calculus book in the chapter on
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"indeterminate forms" and if you look
05:59
there you find that these slope quotients
06:02
that we had a look at so far are actually
06:04
just a special incarnation of this sort
06:06
of setup. So you have a quotient of two
06:08
functions
06:09
g and h depending on a variable, in this
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case t. And as the variable approaches a
06:17
critical value both g and approach 0.
06:21
Whenever you get something like this
06:23
happening then you say that the quotient
06:27
takes on the indeterminate form 0/0 at
06:30
the critical value and actually just here
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I better say the
06:35
critical value can also be infinity so
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t can also go off to infinity.
06:40
What makes this whole thing
06:41
indeterminate, well, by just looking at the
06:45
information we've got so far it's not a all
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clear what the quotient does as we
06:48
approach the critical value. It could go
06:50
to a specific value, it could go to
06:53
infinity or it could do nothing
06:55
reasonable. On the other hand, you know if
06:58
you were to consider the product you
07:00
don't have to know anything about g and h
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except that they both go to 0 to
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conclude that the product will go to 0
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and even Siri knows about indeterminate.
07:10
So if you ask about any of those strange
07:12
expressions that i showed you Siri will
07:15
tell you they're all indeterminate. So
07:17
let's have a look at an example, 1 to the
07:19
power of infinity. Infinity and
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I really have to say this, in
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elementary calculus never stands for
07:26
anything like a number. It always stands
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for some sort of function or process
07:30
that goes to infinity. The same sort of
07:32
thing here. Both 1 and infinity actually
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stand for functions, the first one
07:37
approaches 1, the second one approaches
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infinity as t goes to a critical value.
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And this is also indeterminate meaning
07:44
that if you don't know exactly what
07:45
you're talking, about which g and which
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h you're talking about, the g to the
07:50
power of h could approach any sort
07:53
of finite value or it could go to
07:55
infinity or it could go anywhere you
07:56
want.
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Ok so here's a really famous example of
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something like this: (1+1/t)t
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So here the critical
08:06
value is actually infinity. So t gets
08:08
bigger and bigger and as t gets bigger and
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bigger this 1/t will approach
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0 and so the whole orange bit will
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approach 1 and of course the exponent
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will approach infinity and it's really
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really important to say that you can't
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do first one limit and then the other
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one. Tt's really the speeds at which one
08:30
expression goes to 1 and the other
08:32
expression goes to infinity that
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determine the behavior of the whole
08:36
expression. In this case it approaches
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this value here, very famous, it's the
08:43
base of the natural logarithm, e, and
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there is a whole video that I've done about
08:47
this indeterminate form. Maybe watch it
08:50
again after you finish with this video.
08:52
And so we have a look at this and you
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think that well to make sense of all of
08:55
these strange expression you have
08:56
to invent a lot of different sorts of
08:58
mathematics. But it actually turns out that
09:00
all of these expressions and that comes
09:02
really comes as a surprise to many many people
09:04
are just 0/0 in disguise. So all of these
09:08
expressions can be reduced to just the
09:10
consideration of 0/0. I just want to show
09:12
you how that works for 1 to the power of
09:14
infinity. So again that just corresponds
09:18
to this. Now this whole thing is equal to
09:21
e ^ .....
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And now we're dealing with a
09:25
logarithm so we can pull the exponent
09:28
in front of the logarithm, like that.
09:29
g goes to 1. Now ln(1) is 0. This means
09:34
that this whole thing here goes to 0.
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h goes to infinity but we can also write
09:40
the whole exponent here in this form
09:42
and then the 1/h actually goes
09:44
to 0. So what we've done now is basically
09:47
reduce 1 to the power of infinity to 0/0 :)
09:52
Pretty neat, right, and you can do this
09:55
for all the other ones. So far, apart from
09:57
this very, very simple x squared example
09:59
we haven't actually figured out any
10:02
other expressions. How do you actually do
10:03
this in practice. What I've tried to push here is
10:06
that the apple that hit Newton was really a
10:08
0/0 apple. The 0/0 apple
10:10
makes calculus but you can also go the
10:13
other way around. Once you've got
10:15
calculus you can actually hit 0/0
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with it.
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How does that work? Well that's a bit of
10:22
magic and its called l'Hopital's Rule. So
10:25
here is an indeterminate form 0/0 at 1.
10:29
So, if we let t go to 1, both
10:31
the top and the bottom go to 0 and
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now how would you actually figure out
10:37
where the whole expression goes to. You
10:39
could try and evaluate this at ts that
10:43
are very close to 1 t,hat's a perfectly
10:45
fine strategy but there's actually a
10:47
really nice shortcut. So what you do is
10:49
you take the top and find the derivative
10:52
and you take the bottom and find the
10:54
derivative. So the derivative of the top
10:56
is 1/t and the
10:58
derivative of the bottom is just 1 and we
11:01
will see what happens to this expression
11:03
as t approaches the critical value.
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Of course nothing terrible happens here at
11:08
all, this just becomes 1, right? And now if
11:13
the functions that we're dealing with
11:14
here are nice, differentiable, check out
11:17
the details in a calculus textbook, then we
11:19
can actually conclude at this stage that
11:21
what we're really interested in also
11:23
goes to 1. A really really nice trick.
11:25
So, overall, what we've seen is that to
11:28
make sense of 0/0 in calculus just means
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setting a special kind of context and
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sneaking up on the zeros. The same
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sort of thing if you want to make sense
11:39
of 3/0. You also do this by sneaking up
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on 0 and of course you know that
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things explode magnitudewise and, you
11:47
know, you can make sense of this in this
11:48
way. But that's not the end of it all
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So in higher levels of calculus it
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actually makes sense to treat infinity
11:57
like a number and to actually write
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equations like 3/0 is equal to infinity and
12:05
you really mean it.
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So these things don't stand for any kind
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of functions they really stand for, you
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know, 3 as a number divided by 0 as a
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number is equal to infinity (sort of) as a
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number. In other branches of mathematics
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you sometimes find that it actually does make
12:20
sense to set 0/0 equal to 1. But that's
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a topic for another video, so let's
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just leave it at that.