# Significant Digits

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

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#### Transcript Provided by YouTube:

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

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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.

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This post was previously published on YouTube.

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Photo credit: Screenshot from video.