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# Rules of significant figures

Learn the "rules of thumb" for significant figures. Created by Sal Khan.

Video transcript

Based on the examples
in the last video, let's see if we can come
up with some rules of thumb for figuring out how
many significant figures or how many significant
digits there are in a number
or a measurement. So the first thing
that is pretty obvious is that any non-zero
digit and any of the zero digits in
between are significant. Clearly, the 7 and the
5 here are significant. And the 0 in between them, it's
also going to be significant. So let's write this over here. So any non-zero digits
and zeros in between are going to be significant. That's pretty straightforward. Now, the zeros that are not in
between nonzero digits, these become a little
bit more confusing. So let's just make sure we
can rule out some of them. So you can always rule out
when you're thinking about significant figures,
the leading 0's. And when I'm talking
about leading zeros, I'm talking about
the zeros that come before your non-zero digit. So these are
leading zeroes here. These are leading zeroes. There is no leading zeroes here. No leading zeroes in this
one, this one, and this one. But in any situation,
the leading zeros are not significant. So leading zeros
not significant, I'll write it over here. Leading zeros not significant. And so the last question,
all you have left, I mean you only have non-zero
digits and zeros in between. You could have some leading
zeros, which you've already said are not significant. And so the only thing left
that you have to figure out is what do you do with the
trailing zeroes, the zeroes behind the last
non-zero, or to the right of the last non-zero digit. So these trailing zeroes here. There's actually two
trailing zeroes over here. And then there's three
trailing zeroes over here. So let me make a little--
so trailing zeroes. Trailing zeroes, what
do we do with them? So the easy way
to think about is if you have a decimal, if
there's a decimal anywhere in your number, count them. If you have a
decimal, count them. Count them as significant. They are significant,
count them as significant. If there's no decimal
anywhere in the number, then it's kind of ambiguous. You're kind of not sure
and this is a situation. So clearly over here, there's
a decimal in the number, so you count the
trailing zeroes. These are adding
to the precision. Over here there's a decimal,
so you count the trailing zero. There's a decimal here, so
you count the trailing zeroes. There are no
trailing zeroes here. And over here-- well, the way
I later put a decimal here. Here you would count it. So if you have
the decimal there, you would count all five. If you didn't have the
decimal, if you just had 37,000 like
that, it's ambiguous. And if someone doesn't
give you more information, your best assumption is
probably that they just measured to the
nearest thousand. That they didn't measure
exactly the one and just happened to get
exactly on 37,000. So if there's no
decimal, let me write it this way-- it's
ambiguous, which means that you're not
sure what it means, it's not clear what it means. And you're probably safer
assuming to not count it. If someone really does measure,
if you were to really measure something to the
exact one, then you should put a decimal
at the end like that. And there is a notation
for specifying. Let's say you do measure-- and
let me do a different number. Let's say you do measure 56,000. And there is a notation for
specifying that 6 definitely is the last significant digit. And sometimes you'll see
a bar put over the 6, sometimes you'll see
a bar put under the 6. And that could be
useful because maybe your last significant digit
is this zero over here. Maybe you were able to
measure to the hundreds with a reasonable
level of precision. And so then you would write
something like 56,000, but then you would put
the bar above that zero, or the bar below
that zero to say that that was the last
significant digit. So if you saw
something like this, you would say three
significant digits. This isn't used so frequently. A better way to show
that you've measured to three significant
digits would be to write it in
scientific notation. There's a whole video on that. But to write this in
scientific notation, you could write this as 5.60
times 10 to the fourth power. Because if you multiply
this times to the fourth, you would move this
decimal over four spaces and get us to 56,000. So 5.60 times 10 to the fourth. And if this confuses
you, watch the video on scientific notation. It will hopefully clarify
things a little bit. But when you write a number
in scientific notation, it makes it very clear
about your precision and how many significant
digits you're dealing with. So instead of doing
this notation that's a little bit outdated-- I
haven't seen it used much with these bars below or above
the high significant digit, instead you could
represent it with a decimal in scientific notation. And there it's
very clear that you have three significant digits. So hopefully that helps you out. In the next couple of videos,
we'll explore a little bit more why significant digits are
important, especially when you do calculations with
multiple measurements.