1
00:00:01,200 --> 00:00:07,995
Let's see if we can learn a thing or two about significant figures, sometimes called significant digits.

2
00:00:07,995 --> 00:00:14,610
And the idea behind significant figures is just to make sure that when you do a big computation and you have a bunch of digits there,

3
00:00:14,610 --> 00:00:17,754
that you're not over representing the amount of precision you have

4
00:00:17,754 --> 00:00:24,733
that your result isn't more precise than the things that you actually measured - that you usually use to get that result.

5
00:00:24,733 --> 00:00:27,533
But before we go into the depths of it and how you use it with computation

6
00:00:27,533 --> 00:00:32,636
let's just do a bunch of examples of identifying significant figures, then we'll try to come up with some rules of thumb.

7
00:00:32,636 --> 00:00:40,102
But the general way to think about it is - "Which digits are really giving me information about how precise my measurement is?"

8
00:00:40,102 --> 00:00:46,333
So on this first thing right over here, the significant figures are this seven-zero-zero.

9
00:00:46,333 --> 00:00:51,933
So over here you have three significant figures.

10
00:00:51,933 --> 00:00:58,344
And it might make you a little uncomfortable that we're not including these zeros that are after the decimal point and before this seven.

11
00:00:58,344 --> 00:01:03,000
That we're not including those - because that does help define the number.

12
00:01:03,000 --> 00:01:06,967
And that is true but it's not telling us how precise our measurement is.

13
00:01:06,967 --> 00:01:08,863
And to try to understand this a little bit better,

14
00:01:08,863 --> 00:01:12,564
imagine if this right over here was a measurement of kilometers.

15
00:01:12,564 --> 00:01:18,277
So, if we measured zero point zero zero seven zero zero kilometers.

16
00:01:18,277 --> 00:01:26,559
That same measurement we could have - this would have been the exact same thing as seven point zero zero meters.

17
00:01:26,600 --> 00:01:32,071
Maybe in fact we just used a meter stick. And we said it's exactly seven point zero zero meters.

18
00:01:32,071 --> 00:01:34,533
So we measured to the nearest centimeter.

19
00:01:34,533 --> 00:01:37,067
And we just felt like writing it in kilometers.

20
00:01:37,067 --> 00:01:42,200
These two numbers are the exact same thing - they're just different units. But I think when you look over here

21
00:01:42,200 --> 00:01:45,800
it makes a lot more sense why you only have three significant figures.

22
00:01:45,800 --> 00:01:53,738
These zeroes are just kind of telling you - are just shifting it based on the units of measurement that you're using.

23
00:01:53,738 --> 00:01:58,600
But he numbers that are really giving you the precision are the seven, the zero and the zero.

24
00:01:58,600 --> 00:02:04,292
And the reason why we're counting these trailing zeros is that whoever wrote this number didn't have to write them down.

25
00:02:04,333 --> 00:02:07,267
They wrote them down to explicitly say "Look, I measured this far."

26
00:02:07,267 --> 00:02:14,677
If they didn't measure this far they would have just left these zeros off, and they would have just told you seven meters - not seven point zero zero.

27
00:02:14,677 --> 00:02:21,744
Let's do the next one - so based on the same idea we have the five and the two - the non zero digits are going to be significant figures.

28
00:02:21,744 --> 00:02:28,933
You don't include this leading zero by the same logic that if this was point zero five two kilometers

29
00:02:28,933 --> 00:02:37,667
this would be the same thing as fifty two meters, which clearly has two significant figures.

30
00:02:37,667 --> 00:02:52,729
So you don't want to count leading zeros before the first non zero digit, I guess we could say.

31
00:02:52,729 --> 00:02:58,333
You don't on't want to include those. You just want to include all the non zero digits and everything in between.

32
00:02:58,333 --> 00:03:04,200
and - and trailing zeros - trailing zeros if a decimal point is involved.

33
00:03:04,200 --> 00:03:09,067
I'll make those ideas a little bit more formal. So over here, the person did three hundred seventy,

34
00:03:09,067 --> 00:03:10,867
and then they wrote the decimal point.

35
00:03:10,867 --> 00:03:14,738
If they didn't write the decimal point it would be a little unclear on how precise this was.

36
00:03:14,738 --> 00:03:18,938
But becuase they wrote the decimal point it means they measured it to be exactly three hundred seventy.

37
00:03:18,938 --> 00:03:26,425
They didn't get three hundred seventy two and then round down or they didn't have a kind of roughness only to the nearest 10s place.

38
00:03:26,425 --> 00:03:29,400
This decimal tells you that all three of these are significant.

39
00:03:29,400 --> 00:03:34,200
So this is three significant figures over here.

40
00:03:34,200 --> 00:03:40,600
Then on this next one, once again, this decimal tells us that not only did we get to the nearest one,

41
00:03:40,600 --> 00:03:44,200
but then we put another trailing zero here which means we got to the nearest tenth.

42
00:03:44,200 --> 00:03:48,867
So in this situation once again we have three significant figures.

43
00:03:48,867 --> 00:03:55,877
Over here - the seven is in the hundreds but we got all the way down - the measurement went all the way down to the thousandths place

44
00:03:55,877 --> 00:03:59,600
And even though there are zeros in between, those zeros are part of our measurement,

45
00:03:59,600 --> 00:04:03,067
because they are in between non zero digits.

46
00:04:03,067 --> 00:04:12,692
So in this situation every digit - the way it's written - is a significant digit. So you have six significant digits.

47
00:04:12,692 --> 00:04:19,405
Now this last one is ambiguous. The thirty seven thousand - it's not clear whether you measured exactly thirty seven thousand.

48
00:04:19,452 --> 00:04:23,800
Maybe you measured to the nearest one, and you got an exact number - you got

49
00:04:23,800 --> 00:04:29,538
exactly thirty seven thousand. Or, maybe you only measured to the nearest thousand.

50
00:04:29,538 --> 00:04:36,933
So it depends on what - there's a little bit of ambiguity here - if you've just seen something written exactly like this,

51
00:04:36,933 --> 00:04:48,021
you'd probably say if you had to guess - or not guess - but if there wasn't any more information, you would say that there's just two significant figures or significant digits.

52
00:04:48,021 --> 00:04:52,641
For this person to be less ambiguous they would want to put a decimal point right over there.

53
00:04:52,641 --> 00:04:58,667
And that let's you know that there was actually five digits of precision - that we actually go to five significant figures.

54
00:04:58,667 --> 99:59:59,999
So if you don't see the decimal point, I would go with two.