# Significant Digits

## Related Post Mr. Andersen explains significant digits and shows you how to use them in calculations.

00:04
Hi. This is Mr. Andersen and today I’m going to give you a podcast on significant
00:09
digits, also known as significant figures or sometimes we call them just sig figs. And
00:15
so if I do my job right, you should be able to take a problem like this, 10.6 meters divided
00:19
by 13.960 seconds and come up with an answer that not only has the right number of units
00:24
or the right units, but also has the correct number of significant digits. So let’s get
00:29
started. We’ve got some snipers here. And what snipers try to be is they try to be both
00:35
accurate and precise. What does that mean? Well accuracy refers to truth. In other words
00:42
how close you are to the right accepted answer. Precision however reports to the repeatability.
00:50
And so let’s look at the bull’s eyes down here. This bull’s eye down here, this sniper
00:54
has been fairly accurate. In other words all the shots are pretty close to the bull’s eye
00:58
which is going to be right in the middle. So we would call this accurate shooting. But
01:03
not precise. If we look over to here, this time all the shots are way off to the side.
01:08
And so it’s not true anymore. In other word’s it’s not accurate, but it’s really precise.
01:13
In other words they have a really tight grouping right here. And so what do we hope to be as
01:17
a sniper? We hope to be both accurate and precise. And what do we hope to be as a scientists?
01:22
We hope to be accurate and precise as well. So let’s say you have a wasp that you want
01:28
to measure. And so if we measure this wasp from its head down to the need of its body,
01:34
we find that it is 1, 2 and somewhere between 2 and 3. And so I might say that the wasp
01:41
has a length of 2 point, let me approximate, 5 centimeters in length. Why can’t I get more
01:49
precise than that? Well, my ruler is no better than that. And so if I get a better ruler,
01:55
now I see we’ve got a 1 here. We’ve got a 2 here. We’ve got a 3 here. But I also have
02:00
these delineations as well. And so this is a 2.5. And this right here is a 2.6. And so
02:08
I can be more precise in my measurement. And so what is the length of the wasp right now?
02:15
Well it is 2.55 centimeters. And so this right here is a more precise measurement because
02:22
I have a more precise measuring device. Or a more precise ruler. These number, 1, 2,
02:29
3, are called significant digits or significant figures. And so this measurement would have
02:34
3. And this measurement would only have 2. So let’s play around with some of these things.
02:40
What kind of digits are significant? And there are 4 types of digits that are going to be
02:44
significant. And so if you are working through a problem and you see a non-zero number, so
02:49
let’s say you see 32.6, how many significant digits are there in that number? Well the
02:56
3 is. The 2 is. The 6 is. And so there would be 3 significant digits. Or let’s say we had
03:03
this measurement. 12.48. That would have 4 significant digits. Because there are no zeroes
03:10
in it. So that’s pretty easy. Let’s go to the next one. Final zeroes after the decimal
03:15
place are always going to be significant as well. So what does that mean? Let’s say we’ve
03:19
got 2.0. How many significant digits are there? Well this 2 is. And this 0 is also significant
03:28
because it’s a final 0, in other words at the end. And it’s also after the decimal place.
03:33
And so this would have 2. Or if we did something like this. 28.40 Well, 1, 2, 3, and now this
03:43
one, according to that second rule is also going to be significant. So we would have
03:47
4 significant digits right there. What else is significant? I like to refer to these next
03:52
ones as “sandwiched” zeroes. And so let’s say that we have 209. Well this is significant,
04:00
that is significant, because they’re not zeroes. But this one is sandwiched between the two,
04:05
and so it’s also significant. And so you could have for example 12.090. Let’s apply all of
04:12
our rules. How many do we have now? Well these guys are all significant. This 0 is sandwiched
04:19
between the 2 and the 9. So it’s significant. And this one is a final 0 after the decimal
04:24
place. And so this one right here would have 5 significant digits. So it seems like everything
04:30
is significant. Let’s go to the next one. All numbers in scientific notation are significant
04:33
as well. What does that mean? Let’s say I have a number like this. 3800000. In science
04:42
we use what is called scientific notation to write this out. And so if the decimal place
04:46
is here, remember I can count back 1, 2, 3, 4, 5, 6. And so we would write this as 3.8
04:55
times 10 to the 6th. That’s significant. That’s significant. And so this would have 2 significant
05:02
digits. Alright. So then let’s go to the next page. What actually is not a significant?
05:07
So what numbers aren’t going to be significant? Well there is only one group of numbers that
05:10
aren’t. And those are place holding zeroes. And so an example of that. Let’s say you had
05:16
230. Well this is significant. So is this. But this 0 right here is just spacing the
05:25
numbers 2 and 3 from the decimal place. So it’s a place holder. And so we would now say
05:30
that’s not significant. This only has 2. Or if we take a number like this. 0.00069. How
05:38
many significant digits are there? Well all of these zeroes are simply place holders.
05:44
So they’re not significant. And so we’d only have two significant digits there. Okay. So
05:50
what do we do? Well in calculations you have to make sure that your answer is no more precise
05:55
than the measurements that you actually make. And so we’re going to try some calculations
05:58
or try some practice. And if this doesn’t make sense, slow it down, go back again and
06:02
take a look. So let’s start with the law of multiplication and division. Law of multiplication
06:06
and division says, the number of significant digits in the answer should equal the least
06:15
number of significant digits in any of the numbers being multiplied or divided. What
06:20
does that mean? Let’s try one. So for example let’s say we take, I have one down here, 26.4
06:28
and we multiply that times 120. Okay. If we multiply those numbers in a calculator we
06:37
get a really large number. It is 3 1 6 8 point 0 0 0. So it keeps going like that. So what do we get
06:49
for an answer? We’ll this has 1, 2, 3 significant digits. This one has 1, 2, that is not significant
06:58
because it’s just a place holder. So that has 2 significant digits. And so since this
07:03
one has three and this one has two, my answer can only have 2 significant digits. So what
07:10
does that mean? I’m going to have to round. And so there’s one significant digit. The
07:15
next one, the 1 is the second significant digit. And since this number right to the
07:20
right of it is larger than 5, or equal to 5, I’m going to round this up. And so what
07:25
is the right answer? The right answer is 3200. How many significant digits does this have?
07:32
Well these two zeroes here are just place holders. And so this is going to have two
07:36
significant digits. Which is equal to the least number is my two calculations. And why
07:41
do we do that? Well we want to make sure that the measurements we make are no more precise
07:47
than the answer that we get at the end. Or the answer we get is no more precise than
07:50
those measurements. Let’s try another one of those. So let’s say we’re doing division
07:54
for a second. We’ll make an easier one. Let’s say we take the 19 and we divide that by the
08:00
number 3. What do we get for an answer? Well in our calculator we get 6.333333. It just
08:08
keeps repeating like that. But you would never turn in an answer like this in science class
08:13
or in math class because it’s not, it’s way more precise than the measurements we actually
08:17
made. And so let’s go through and use our rules. How many significant digits does this
08:21
have? Two. How many significant digits does this measurement have? One. And so how many
08:28
significant digits can my answer have? Well it can only have one significant digit. And
08:33
so what is my answer? Well this is a 6. This is a point 3. And so my answer would be 6.
08:40
In other words I’m going to use this number to round so I can get to one significant digit.
08:46
And so the answer wouldn’t be 6.333333. The answer would simply be 6. And so significant
08:53
digits actually make your job a little bit easier. Now addition and subtraction are a
08:57
little bit different. In addition and subtraction it’s the number of decimal places in the answer
09:04
that should be equal to the least number of decimal points, or decimal places in any of
09:10
the numbers being added or subtracted. What does that mean? Let’s say we have a measurement
09:13
like this. 13 plus 1.6 equals blank. Okay. Now in this one we have to look at the number
09:25
of decimal places. In other words this one is measured to the ones place. And this one
09:32
is measured to the tenth places. And so even though the answer if we add these up, you
09:36
can see is going to be 14.6, my answer can’t go and give me another decimal place right
09:43
here. And so the right answer would be 15. In other words, I have to round that 4 up
09:50
to a 5. Because I can’t get an answer that has more decimal places than my least decimal
09:56
place answer to the right. And so addition and subtraction work that way. Sometimes when
10:01
I’m solving these ones what I’ll do is I’ll line them up. So all the decimal places are
10:05
on top of each other. And then I can see which one has the least number of decimal places.
10:09
Okay. So if I go to the end I said after you watch this you should be able to answer a
10:13
question like this. So let’s take a stab at it. So this 10.6 meters. How many significant
10:19
digits would that have? It’s going to have 3. Now we’ve got 13.960. How many significant
10:27
digits does that have? 5. And so my answer can only have 3 significant digits. So even
10:35
though my calculator might say the answer is .759312321. I don’t want to turn this answer in. I want
10:49
to get an answer that has 3 significant digits. And so the right answer would be .759. That’s
10:58
it. Because this is 3, I’m not going to round this nine. And so the right answer would be
11:04
.759. So that’s how you use significant digits. The best way to get better at doing significant
11:10
digit problems is to just practice them until you eventually get it right. And so I hope
11:14
that’s helpful. And always come ask for help if you get lost.

This post was previously published on YouTube.

Photo credit: Screenshot from video.