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