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## Arithmetic (all content)

### Course: Arithmetic (all content) > Unit 6

Lesson 14: Significant figures# Rules of significant figures

This video teaches significant figures rules, crucial for measurements and calculations. It covers identifying significant digits, including non-zero digits, zeros in between, leading zeros, and trailing zeros. Additionally, it introduces scientific notation for clearer representation of significant figures. Created by Sal Khan.

## Want to join the conversation?

- What would you use significant figures for?(10 votes)
- In engineering significant figures has to do with how accurately you are able to measure something. According to Google, Mt. Everest is 29,029 feet tall. Well, I suspect it is not EXACTLY 29,029 feet tall... to the inch? the thousandth of an inch? how accurately is it measured?

If the height were given as 29,000 feet, that would still be a pretty good measurement, but it implies the number is really only 'good' to the thousands place (plus or minus 500 feet). 20,029 implies that measurement is 'good' to the individual foot (plus or minus 6 inches)... a very different impression of accuracy.

The intent (as I understand it) is to avoid giving the impression of accuracy where it does not exist.(52 votes)

- What is the difference between 5.60, 5.600, and 5.60000? We all know that the zero is a placeholder for a non-existing value when placed after a decimal and after a non-zero digit. I understand that that would show that the measurer took the time and had an appropriate measuring device to arrive at a more and more accurate measurement, and that it is essentially superfluous to write additional zeros after the first trailing zero. What is a little ambiguous to me, is the difference between 5.0, 5.00 and 5.000... (All trailing zeros that reflect additional accuracy). I guess my real question is, what are the circumstances for counting additional trailing zeros as significant?? ...is it simply the amount of significant figures that the measuring device reads that decides this scenario?(3 votes)
- Mathematically there is no difference between the exact, precise numbers 5.0 and 5.00 and 5.000.

However for measurements in the real world (for science, engineering etc.) there is always a limit to the precision of a measure. Someone wanting to make use of a measurement needs to know how precise it is because they need to know how precise accurate their final result is.

A measurement of 5.0 (2 SF) could be anywhere between 5.04 or 4.95

A measurement of 5.00 (3 SF) could be in reality anywhere between 5.004 and 4.995

A measurement of 5.000 (4 SF) could be in reality anywhere between 5.0004 and 4.9995.

Measuring the carpet to cover a floor as 5.0 Meters (2SF) by 6.0 Meters (2SF) means the area could be as little as 4.95x5.95, 29.4525M^2. However, at that precision the area could be 5.04x6.04, 30.4416M^2. That is a difference of 1 M^2

When measuring accurately to 4 SF means a difference of 2 thousandths of a meter squared.

Knowing accuracy and knowing possible error for future calculations is the difference between 5.60, 5.600 and 5.6000.

It is the precision of the measuring device that decides the significant figures.(20 votes)

- How do we remember trailing 0s, leading, whatever? It's really confusing.(0 votes)
- Remember it like a group of three people walking on the road. The one in the front is
**leading**the others. the one in the back is**trailing**them. So, the**leading**zeroes are the ones in front (like 0.052; the first two zeroes are leading) and the ones in the back are trailing (like in 56.00, the last two are trailing). Hope this helps!(8 votes)

- Why are the zeroes significant when there is a unit?(3 votes)
- Whether or not a value has a unit does not affect whether a zero is significant or not.(2 votes)

- The 0,00700 is considered 3 significant figure for the 700 part, however, the decimal is quite further off, and we are ignoring the 0.00. Thats' conflicting. Can anyone explain?(2 votes)
- The first three zeros are just placeholders, not significant.(3 votes)

- I have several conflicting sources regarding leading zeros. I have always been told that if the number is less than one, then a single zero before the decimal is considered significant. The reason for this is because it shows a purposeful inclusion of the place in the accuracy of the measurement, rather than just leaving the place blank, and therefore ambiguous. An example would be 0.345. Can you verify this? I DO understand that leading zeros, such as the zeros in the famous ( or infamous, depending on your opinion) 007, would not be considered significant. In the Rules of significant figures video, at1:18Sal simply states that all leading zeros are not significant. Are there exceptions to this? Thank you.(3 votes)
- how many significant figures do percentages have? Do they have one? (5%) or 3? (.005)(3 votes)
- then it would just be 1 sig fig because of 0.005(1 vote)

- So like the problem shown what if the 37,000 was actually a 37,500

Would you say they had 2 significant figures or 3 significant figures by counting the 3,5, and 7? or would you have to count the zeros too?(1 vote)- If you have 37500 with no other notations (bars over zeros, decimal points, etc.) then the number as written has THREE sig figs. Not four. I have no idea how Kyle came up with four.

37500 with a bar over the first zero would be four sig figs

37500 with a bar over both zeros would be five sig figs

37500. (with a decimal at the end) would be five sig figs

Writing them in scientific notation avoids confusion because the zeros cannot be "mistaken" as only placeholder zeros. Hope that helps...(4 votes)

- How many significant figures are in 870.050(2 votes)
`870.050`

has 6 significant figures.

If it were`870.05`

, it would have 5. However, the trailing zero above is included, so there are six.(1 vote)

- How many significant figures are there in:-

a) 25,000 kg

b)5.67 x 10^3 m/s

and why?(2 votes)- To figure out the number of significant digits, turn the number into scientific notation and then cross out the 10^. The numbers you end up with are the significant digits.

For example:

25,000 in scientific notation is 2.5 x 10^4.

Get rid of the 10^ and it's just 2.5, which is a two digit number. So there are two significant digits in that number. Your second example is already in scientific notation, so 5.67 are the sig figs. This is because numbers that can be replaced by 10^() or 10^-() are not significant. For example:

20=20

2 x 10 = 20

0.05=0.05

0.5 x 10^-1= 0.05

So the 0s are not significant.

but 0.505=0.505

0.55 x 10= 5.5

This is, however, not the only rule. Zeroes used for "cosmetic" purposes (or used only to show where other numbers are; e.g. the 0s in 0.05) are not significant.(1 vote)

## 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.