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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? •   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.
• 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? • 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.
• How do we remember trailing 0s, leading, whatever? It's really confusing. • 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!
• Why are the zeroes significant when there is a unit? • 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? • 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, at Sal simply states that all leading zeros are not significant. Are there exceptions to this? Thank you. • how many significant figures do percentages have? Do they have one? (5%) or 3? (.005) • 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...
• How many significant figures are in 870.050 • How many significant figures are there in:-
a) 25,000 kg
b)5.67 x 10^3 m/s

and why? • 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)