ROB EASTAWAY: Can I show you the symbol, first of all,
because that's kind of at the center of it.
And I'm going to draw the quick and dirty zequals, which
looks like this.
And it's a bit like an equals sign, but it's zigzaggy.
And the reason for that is it's sort of an approximation
symbol, but it's more precise than that.
And the zigzagginess and the zeds are supposed to make you
kind of think of zeroes because this technique
involves lots of zeroes, especially for big
calculations.
Really, what we're doing is reducing every number to one
significant figure.
And let's start with a really simple calculation, one of
your basic times tables like 3 times 7.
Well, hopefully, everyone knows that that's 21.
But actually, if we say what it zequals, then we round it
to the nearest round number with one digit
followed by a zero.
So 3 times 7 zequals 20.
OK, let's take another one.
So 5 times 8 equals--
bad example.
[LAUGHTER]
BRADY HARAN: Let's do one below it.
ROB EASTAWAY: Let's do one below it.
BRADY HARAN: Even I know what that zequals.
ROB EASTAWAY: Yeah, that's right.
Let's do 7 times 8, because no one can ever remember that.
But I'll tell you what it equals.
It equals 56, but it zequals 60 because you round it
ruthlessly to one significant figure.
BRADY HARAN: It's almost more work at the moment to do the
zequaling than the equaling.
ROB EASTAWAY: Yeah, I can see why you'd think that, but if
we get onto a more complicated calculation, I think you'll
begin to see why it's actually quite handy to just simplify
everything at the first opportunity.
Should we do a big one?
436 times 68.
Well, if we're zequaling this, the first thing we do is
simplify these numbers.
So that zequals 400 times 70.
So we've simplified both those to make this calculation,
because then it's just going to be 4 times 7, and then add
on all the zeroes at the end.
So that equals--
well, seven 4 are 28, followed by three zeroes,
so it equals 28,000.
But actually, we're being really ruthless because we
just want a rough answer.
It zequals 30,000.
So 436 times 68 zequals 30,000.
And you know what?
That's pretty close to the right answer.
Just check what the actual answer is 29,648.
Now, in this case, that is really close.
So zequals has been amazingly accurate.
It's not always quite as accurate as that.
But in fact, in most estimation challenges, you're
really looking for something that's an order of magnitude,
and zequals works in most situations.
BRADY HARAN: Hi there, everyone.
I'm going to interrupt my own video at this point because I
can hear some of the numberphiles screaming at the
screen, upset by this ruthless rounding, this
dismissing of digits.
And I can kind of understand that.
So I spoke to my graphing expert, Dave Wiley, told him
about zequals, and he actually put this together.
This is a graph showing where zequals is at its strongest
and also where it's furthest from the true answer.
I thought it might be worth a look, and I've put more
details about it on my blog.
I'll talk about this more at the end of this video, but
let's get back to hearing more about what Rob has to say
about zequals.
ROB EASTAWAY: I think the most valuable thing here is it's
giving you a feel for numbers.
Because there's a lot of obsession these days with
people need to learn how to do long multiplication.
Well, we all use a calculator for anything difficult these
days, and we don't need, most of us, most of the time, to do
a really accurate calculation.
A calculator can do that for us.
But we do need to have a feel for numbers.
And I suspect that many people who use calculators have no
idea when the answer comes out whether that number
is vaguely or not.
And the great power of zequals is it just allows you to very
quickly work out roughly what the answer is so you can say,
well, look, is the number I've come up with, does it even
make sense?
Or have I typed something wrong into this calculator or
into my laptop or whatever?
So what this is trying to generate, more than anything
else, is a feel for numbers and the power to be able to
say, you know what?
I don't know what exactly the answer is, but I do know
roughly what the answer is.
BRADY HARAN: Is this something people do intuitively anyway?
ROB EASTAWAY: I think they do.
I'm sure--
I have been estimating for years.
And I think the difference here, really, between this and
any old estimation is just how ruthlessly you're simplifying
things to one significant figure.
So when we did the tables, seven 8's are 56, seven 8's
zequals 60.
Most people don't think to make the simplification at
that point.
So I'm just taking it one stage further to become a bit
of a rounding extremist.
You could be even more extreme than zequals, incidentally.
You could round 60, say, we'll call it 100.
It's close enough.
I'm just a little bit short of that extreme.
I would say I'm not a fifth decimal place kind of person.
I really am a zequals-y kind of person.
As a degree, I actually did engineering, and there it's
all about having a feel for numbers.
There's a joke about how you can tell the difference
between an engineer and a mathematician.
So you ask him what pi is.
And a mathematician will say, it's 3.14 and it goes on
forever, a transcendental number.
Whereas an engineer will say, well, it's about three.
But let's call it 10 just to be on the safe side.
BRADY HARAN: Is this the thin end of a wedge?
I sometimes think that the emergence of text messaging
and things like that has really eroded people's
spelling ability.
Would the emergence of zequals in a kind of lazy, rounded
estimation take away some of the real skill and craft that
we want to engender in our future mathematicians?
ROB EASTAWAY: I would say completely the opposite.
If only your typical 14-year-old frankly even could
just say off the top of their head what seven 8's is, that
it's 56 in the first place, let alone be able to then
round it to 60.
So I think this skill is missing in so much of the
population.
Let's get them to this stage.
And actually, strangely enough, I think the ability to
estimate, even though it's all rough and ready, sometimes you
actually need a lot of math understanding to be able to do
in the first place.
It probably has its most common application in
financial stuff, working something out.
You're in a shop.
You're working out a deal.
There's a question I've been asking teenagers, quite
interesting to ask them.
You say, right, I've got 70 people, each want a Mars bar,
and let's say the Mars bar costs 80p,
and there's 70 people.
So how much is that going to cost?
How much are we going to have to spend on that?
And they kind of know it's seven 8's.
Maybe they know seven 8's 56.
But they'll probably say 5 pounds 60, because that sounds
like about what you'd spend on Mars bars.
So being able to say, well, hang on a sec.
70 times 80, it's like kind of 50 times 100, and what's that?
Oh, 5,000, oh, 50, OK.
So it's sort of, does this answer make sense?
I can give you an example of when zequals would perhaps not
be a good idea.
For example, if you're taking your child's temperature and
you like to use old-fashioned Fahrenheit for that and say,
well, it's 98.4.
Let's zequals that.
Let's call it 100.
Well, suddenly you're thinking, well, maybe I should
take him to the doctor then.
Knowing when it's appropriate to round numbers and when not
is kind of--
I've rather hidden that away in all of the zequals stuff,
that that is a skill, a life skill, an experience skill.
I hope they'll see the point.
I hope they'll see that I'm not some ignorant person who's
imposing some impurity on this great subject, but actually,
this is a handy tool.
I'm going to guess quite a few of your viewers actually kind
of do this with variations themselves.
So I'm just offering one technique where
it's a shared language.
We all play the same game.
We round to one figure.
But I'm not saying it's the best.
I'm just saying it's one that's got a
name now and a symbol.
BRADY HARAN: Now, Rob's got a book out called "More Maths
for Mums and Dads, The Teenage Years," and it's in that book
he talks about zequals.
Zequals is just a very small part of the book.
It's full of all sorts of cool stuff, and I've put a link
under this video if you want to check that out.
Also, that graph I talked about that Dave put together
where we've had a look at zequals, you can
find that on my blog.
Again, links underneath.
And if you want to see even more from this video, where I
asked Rob even more questions, stuff that I've left out of
this video for time reasons, I've put some of those extra
questions and answers as an unlisted Numberphile video,
and there will be links on the screen and underneath as well.
So if you're really into this and you want to hear Rob talk
about it more, because he's got a lot to say about it, go
check that out.