if the zeroes of 0.000005 is not significant then might as well throw em all in the trash and put it as 5
So, 0.000005 is = 5 which is obviously wrong
​
Can someone explain to me what perspective I need to understand them? And also why my mindset is wrong
In: Mathematics
[removed]
Sure. The number of zeros ***is*** significant. You can’t just throw them away. Zeros are only insignificant if they are leading zeros *to the left* of the decimal point.
This idea usually better expressed in scientific notation https://www.mathsisfun.com/numbers/scientific-notation.html
Because the “magnitude” of the value is also put in the exponent of 10^, what’s left is how many digits you want to write in the “coefficient”.
**********
edits:
Don’t get stuck with the name “significant”
The idea is, sometime we get the number 1.23456789 or 98765.4321
But we don’t need all the precision,
sometimes we only want 1.23 or 98800
how would you call this?
People call this “rounding to 3 significant digit” what do you think people should call it instead?
Significant figures are primarily about how *precise* your numbers are – and rounding rules for significant figures is about showing how precise the result is based on the numbers you put in. Zeroes before the significant digits simply show magnitude, and as a result are not significant.
Typically when doing measurements, the last significant digit of your measurement is considered approximate, since it’s the limit of whatever measurement technique you’re using.
So to take your example of 0.000005, let’s say it’s a measurement. If it’s by itself, the magnitude (the zeroes) are important, because it shows how large the value in question is. However, if you now have another measurement, let’s say 0.003, and you need to add the two together, significant digits would become important. Saying the result of adding the two is 0.003005 would not be correct, because it suggests a precision that you don’t have in your second measurement (since the last digit is approximate, a measured value of .003 could be .00299 or .00305 in reality) – you have to round to .003.
Another way to look at it is that if you can replace zeroes with a change in unit, then those zeroes aren’t significant. So if your measurement of 0.000005 is in grams, you can get rid of the zeroes by instead stating it as 5 micrograms.
Significant figures become more obvious when written in scientific notation
0.000005 is 5×10^-6 so its pretty clear that you have a single significant digit (5) but the leading zeros are still important for scaling. Consider a real measurement like kg, 0.000005 would be 5 milligrams which is clearly only 1 sig fig.
Now if you had something like 0.000005000 that’d be 5000 micrograms and if you were measuring it with something that actually had microgram levels of precision then you have 4 sig figs and would write is as 5.000×10^-6 to denote that its actually 5.000 and not like 4.9 or even 5.014.
You need to be careful about including leading or trailing zeros that aren’t actually part of your measurement precision, if we say a ship weighs 1000 tons the precision there is really in terms of +- 0.5 tons (or maybe more) so it’d be misleading to say it weighs 1,000,000.000 kg because you don’t have precision down to the kg let alone fractional kilograms
To understand significant figures, you have to understand the problem it’s solving.
We often talk about real-world measurements in science, like “I added 450 milliliters of distilled water”.
The problem is that you might have been off in that measurement. You probably used a measuring beaker that had lines that you visually judge, you might have gotten a little bit above or below the line. The measuring line itself is almost a millimeter thick, even if you hit the line exactly you still might be in the bottom or top part of the line instead of dead center. Plus a few drops stuck to the bottom and sides when you poured it out. Plus there might have been slight differences in the size of the beaker and the placement of the marks in the manufacturing process that created it.
All this stuff is the *error* in the measurement, how much you think the measurement would be off by.
In real scientific calculations you often talk about this error using a separate term, like this: 450 mL ± 3 mL. That means you think your measurement might have been off by up to 3 mL in either direction.
Significant figures solve the following problem: *If you have a number that doesn’t explicitly tell you the error, how much error should you assume it has?*
If I tell you it’s 2000 miles from New York to Los Angeles, it’s pretty clear that’s not very accurate. Based on that statement, the actual distance could be anywhere from 1500 to 2500 miles.
On the other hand, if I tell you a track is 2.00 meters long, how long do you think it could actually be? I didn’t need to include those extra zeros after the decimal point, the only reason I included them is to tell you that I’m confident it’s between 1.99 and 2.01 meters.
If I’m doing some experiment and I tell you the laser spot moved 0.005 meters, you’ll be wildly wrong if you convert that measurement to 5 meters. What I meant is that it’s between 0.0045 and 0.0055 meters long, but I can’t measure anything smaller than a millimeter. If I was using some more accurate measuring device / technique that gave me tenths of millimeters, I’d say it was 0.0050 meters long.
*Significant figures of a number shouldn’t change based on your measurement units* so 0.005 meters has one significant figure regardless of whether we choose to talk about it as 0.005m or 5 mm or 5000 micrometers.
When you’re dealing with significant digits, the number of zeros *after* the last non-zero digit indicate how precise your number is.
In other words, 0.5 grams might mean that you actually have anywhere from 0.45 to 0.54 grams. The scale you used to measure it isn’t precise enough for you to know for certain. If you have 0.50 grams, however, then it means that you have somewhere between 0.495 grams and 0.504 grams. You have a better scale in this case, so you have a more precise measurement.
When you start adding/multiplying numbers with significant digits, you have to be careful, because your result can only be as precise as your inputs. In other words, even if you have an extremely precise watch, you can’t know very precisely how fast something is moving if you can’t measure the distance it’s travelling precisely. Likewise, if you can’t measure the weight of something to more than the nearest kg, then you can’t say that it’s gone from 5kg to 5.0005 kg because you added a 0.5 g weight to it. You didn’t know that it was 5.0000kg in the first place.
significant figures/digits just tell you how many digits are significant in say gauging the error in measurement. Essentially you cannot tell what digit follows after the significant digit.
that does not mean 0.000005 = 5. It just means both 0.000005 and 5 have 1 significant digit, i.e., 5. The 0.000005 can be 0.0000051 or it can be 0.0000056 (your instrument/measurement procedure is not good enough to tell you that, but it is sure up to the 5).
Take another example. Your precise body weight is say 154.5197403 lbs. But when you get on a scale, it shows 154.5, i.e., 4 significant digits, because it is not sensitive enough to go beyond that.
Latest Answers