Significant Figures

Let's see if we can
learn a thing or two

about significant
figures, sometimes

called significant digits.

And the idea behind
significant figures

is just to make sure that
when you do a big computation

and you have a bunch
of digits there,

that you're not
over-representing

the amount of
precision that you had,

that the result isn't more
precise than the things

that you actually measured, that
you used to get that result.

Before we go into
the depths of it

and how you use it
with computation,

let's just do a
bunch of examples

of identifying
significant figures.

Then we'll try to come up
with some rules of thumb.

But the general way to think
about it is, which digits

are really giving me
information about how precise

my measurement is?

So on this first
thing right over here,

the significant figures
are this 7, 0, 0.

So over here, you have
three significant figures.

And it might make you a little
uncomfortable that we're not

including these 0's that
are after the decimal point

and before this 7, that
we're not including those.

Because you're just like, that
does help define the number.

And that is true, but
it's not telling us

how precise our measurement is.

And to try to understand
this a little bit better,

imagine if this right over
here was a measurement

of kilometers, so if we
measured 0.00700 kilometers.

This would be the exact
same thing as 7.00 meters.

Maybe, in fact, we just
used a meter stick.

And we said it's
exactly 7.00 meters.

So we measured to the
nearest centimeter.

And we just felt like
writing it in kilometers.

These two numbers are
the exact same thing.

They're just different units.

But I think when
you look over here,

it makes a lot more
sense why you only

have three significant figures.

These 0's are just
shifting it based

on the units of measurement
that you're using.

But the numbers that are
really giving you the precision

are the 7, the 0, and the 0.

And the reason why we're
counting these trailing 0's is

that whoever wrote this number
didn't have to write them down.

They wrote them down
to explicitly say,

look, I measured this far.

If they didn't
measure this far, they

would have just
left these 0's off.

And they would have just
told you 7 meters, not 7.00.

Let's do the next one.

So based on the same idea,
we have the 5 and the 2.

The non-zero digits are going
to be significant figures.

You don't include
this leading 0,

by the same logic that if
this was 0.052 kilometers,

this would be the same thing as
52 meters, which clearly only

has two significant figures.

So you don't want
to count leading

0's before the first non-zero
digit, I guess we could say.

You don't want to include those.

You just want to include all the
non-zero digits and everything

in between, and trailing 0's
if a decimal point is involved.

I'll make those ideas a
little bit more formal.

So over here, the
person did 370.

And then they wrote
the decimal point.

If they didn't write
the decimal point,

it would be a little unclear
on how precise this was.

But because they wrote
the decimal point,

it means that they measured
it exactly to be 370.

They didn't get 372
and then round down.

Or they didn't have
kind of a roughness only

to the nearest tens place.

This decimal tells you that all
three of these are significant.

So this is three significant
figures over here.

Then on this next one,
once again, this decimal

tells us that not only did
we get to the nearest one,

but then we put another
trailing 0 here,

which means we got
to the nearest tenth.

So in this situation,
once again,

we have three
significant figures.

Over here, the 7
is in the hundreds.

But the measurement
went all the way down

to the thousandths place.

And even though there
are 0's in between,

those 0's are part
of our measurement,

because they are in
between non-zero digits.

So in this situation,
every digit here,

the way it's written,
is a significant digit.

So you have six
significant digits.

Now, this last one is ambiguous.

The 37,000-- it's
not clear whether you

measured exactly 37,000.

Maybe you measured
to the nearest one,

and you got an exact number.

You got exactly 37,000.

Or maybe you only measured
to the nearest thousand.

So there's a little
bit of ambiguity here.

If you just see something
written exactly like this,

you would probably say, if you
had to guess-- or not guess.

If there wasn't any
more information,

you would say that there's
just two significant figures

or significant digits.

For this person to
be less ambiguous,

they would want to put a
decimal point right over there.

And that lets you know
that this is actually

five digits of precision,
that we actually

go to five significant figures.

So if you don't see that decimal
point, I would go with two.