WEBVTT
00:00:00.930 --> 00:00:05.460
In this video, our topic is representing small values of physical quantities.
00:00:05.900 --> 00:00:09.040
We’ll specifically be looking at numerical ways to do this.
00:00:09.290 --> 00:00:22.380
And in the process, we’ll learn a series of unit prefixes as well as how to convert small values written in scientific notation into decimal form and then back in the other direction as well, from decimal to scientific notation.
00:00:23.010 --> 00:00:29.200
To get started, the first thing we can notice is that in physics, it’s not unusual to work with small physical values.
00:00:29.640 --> 00:00:37.650
For example, the charge of a single electron is approximately equal to negative 1.6 times 10 to the negative 19th coulombs.
00:00:38.070 --> 00:00:41.680
Or consider another value, the universal gravitational constant.
00:00:42.170 --> 00:00:48.010
This is approximately 6.7 times 10 to the negative 11th cubic meters per kilogram second squared.
00:00:48.520 --> 00:01:00.220
Along with values like these, we might perform a calculation using, say, the mass of a proton, or the average time needed for an electron to decay spontaneously to a lower energy state.
00:01:00.680 --> 00:01:04.920
Or even more simply, we might just want to measure the mass of a few grains of sand.
00:01:05.390 --> 00:01:09.950
So we see that in physics, small values of physical quantities often are involved.
00:01:10.320 --> 00:01:18.200
And so, for the sake of accuracy and simplicity in our calculations, we would like to find convenient ways to represent these small numbers.
00:01:18.570 --> 00:01:28.110
With these two values here, the charge of an electron and the universal gravitational constant, we’ve already taken a step in the right direction by writing these values in scientific notation.
00:01:28.500 --> 00:01:36.640
It’s easier to write and to understand numbers written like this, rather than expressing them in a form that may be more familiar, that is, decimal form.
00:01:37.180 --> 00:01:40.580
So, for small physical values, scientific notation is good.
00:01:40.970 --> 00:01:42.900
But it turns out that we can do even better.
00:01:43.540 --> 00:01:53.110
To see how this works, let’s consider for a second the visible portion of the electromagnetic spectrum; that is, these are the specific frequencies of light that our eyes are sensitive to.
00:01:53.660 --> 00:02:04.100
When we consider either end of the visible spectrum, red light over here and violet light over here, we can write down the approximate wavelengths of these colors in scientific notation.
00:02:04.580 --> 00:02:13.530
Red light has a wavelength of about seven times 10 to the negative seventh meters, while the wavelength of violent radiation is about four times 10 to the negative seventh meters.
00:02:14.030 --> 00:02:18.360
Even just in naming those two wavelengths, though, we can see that they’re a bit of a mouthful.
00:02:18.760 --> 00:02:29.560
If we were performing an experiment, say, collecting lots of data points of visible wavelength radiation, we might find ourselves doing a lot of extra work to express the numbers this way.
00:02:30.050 --> 00:02:38.960
To help streamline cases like this where we’re working with relatively small values, a system of what are called unit prefixes was developed.
00:02:39.400 --> 00:02:43.230
We see from this name that a unit prefix will involve some kind of unit.
00:02:43.560 --> 00:02:52.390
In the case of our visible light radiation, that unit would likely be meters, and then that unit is prefaced or preceded by something called the prefix.
00:02:52.850 --> 00:02:58.010
Unit prefixes are something we’ve seen before, even if we didn’t recognize them as such in the movement.
00:02:58.610 --> 00:03:03.680
For example, say that we measure out 7.5 milligrams of some substance.
00:03:04.030 --> 00:03:07.540
In this quantity, we have a unit, grams, that has a prefix.
00:03:08.010 --> 00:03:13.020
That prefix is milli-, and we see its represented symbolically by a lowercase m.
00:03:13.340 --> 00:03:19.880
And one milligram indicates 10 to the negative third, or one one thousandth of a gram.
00:03:20.400 --> 00:03:24.690
The next smallest unit prefix commonly used is the prefix micro-.
00:03:25.110 --> 00:03:33.780
It’s represented using the Greek letter 𝜇 and it corresponds to one one millionth or 10 to the negative sixth times whatever unit is involved.
00:03:34.230 --> 00:03:38.500
Smaller still is the unit prefix nano-, represented by the letter n.
00:03:39.030 --> 00:03:44.120
This shows us one one billionth or 10 to the negative ninth of the unit we’re considering.
00:03:44.550 --> 00:03:49.710
It’s this prefix, by the way, that’s often used to represent these wavelengths of light in the visible spectrum.
00:03:50.070 --> 00:03:57.370
Instead of writing or saying seven times 10 to the negative seventh meters, for example, we might instead say 700 nanometers.
00:03:57.790 --> 00:04:03.410
And likewise for four times 10 to the negative seventh meters, where instead we would say and write 400 nanometers.
00:04:04.000 --> 00:04:09.180
We can see that this is a bit of an easier way to talk about these numbers and also to compare them to one another.
00:04:09.770 --> 00:04:20.640
Continuing on then down our prefixes list, we have the prefix pico-, represented by the letter p corresponding to 10 to the negative 12th or one trillionth of some unit.
00:04:21.330 --> 00:04:26.890
In the study of very short pulses of laser light, this prefix pico- often comes into use.
00:04:27.270 --> 00:04:32.640
We might say, for example, that a certain laser pulse lasted, say, 75 picoseconds.
00:04:33.040 --> 00:04:35.980
Even smaller still is the unit prefix femto-.
00:04:36.390 --> 00:04:45.560
A femto something, whether a femtosecond or a femtometer, is equal to one quadrillionth, 10 to the negative 15th, of the unit being considered.
00:04:46.070 --> 00:04:52.530
One example of a practical use for this particular prefix is in describing the size of subatomic particles.
00:04:52.790 --> 00:04:58.600
It turns out, for example, that one femtometer is approximately equal to the diameter of a proton.
00:04:59.110 --> 00:05:04.400
So, let’s consider how this idea of unit prefixes could apply to these values that we wrote up here.
00:05:04.780 --> 00:05:16.530
Instead of expressing the charge of an electron as negative 1.6 times 10 to the negative 19th coulombs, alternatively, we could write it as negative 0.00016 femtocoulombs.
00:05:16.910 --> 00:05:19.720
Or what about the universal gravitational constant 𝑔?
00:05:20.180 --> 00:05:25.750
We could write this value as 67 pico cubic meters per kilogram-second squared.
00:05:26.320 --> 00:05:32.920
Here, the complexity of this unit partially but doesn’t completely obscure the advantage of using a unit prefix.
00:05:33.270 --> 00:05:49.290
In both of these instances, for 𝑔 as well as the charge of an electron, notice that using unit prefixes allows us to write these values in ways that are a bit more clear and intuitive compared, say, to writing them in scientific notation or even in their full decimal format.
00:05:49.930 --> 00:06:01.670
Before we go on to an example exercise, let’s consider just how it is that we can switch between two of these different representations — that is, representing a number in scientific notation or written as a decimal.
00:06:02.210 --> 00:06:06.540
For a given physical value, we’d like to be able to switch back and forth between these two.
00:06:06.930 --> 00:06:10.150
So, let’s clear a bit of space and consider how to do that.
00:06:10.520 --> 00:06:19.070
Now, whenever we’re considering a small value, we can say that when we write that value in scientific notation, it’s going to involve taking some number.
00:06:19.250 --> 00:06:23.780
We can call it 𝑎, where 𝑎 is less than 10 and greater than or equal to one.
00:06:24.120 --> 00:06:29.510
And multiplying this value by 10 raised to some negative integer value; here, 𝑛 is an integer.
00:06:30.000 --> 00:06:34.560
Given this way of writing a number, we’d like to know how to express it instead as a decimal.
00:06:35.070 --> 00:06:37.510
To do this, we can start with the value 𝑎.
00:06:37.930 --> 00:06:46.510
Now, 𝑎 may be a whole number, like three or seven, or it might itself be written to a number of decimal places like 1.275.
00:06:47.040 --> 00:06:51.380
Either way, we want to identify the place where the decimal point is located in 𝑎.
00:06:51.800 --> 00:07:01.260
It’s either located someplace explicitly, or if 𝑎 is a whole number like we mentioned, such as seven, then the decimal point implicitly follows that digit.
00:07:01.680 --> 00:07:06.720
Wherever the decimal point is then in this value 𝑎, we figure that out and we write it in place.
00:07:07.160 --> 00:07:12.330
Our next step will involve moving this decimal place a certain number of spots to the left.
00:07:12.800 --> 00:07:19.040
The reason we do this is because in scientific notation, we’re multiplying 𝑎 by 10 to some negative integer.
00:07:19.470 --> 00:07:24.230
Because this is 10 to a negative power, that’s why the decimal place moves to the left and not to the right.
00:07:24.850 --> 00:07:28.560
Just for the sake of illustration, let’s pick a particular value for 𝑛.
00:07:28.740 --> 00:07:30.520
Let’s say that 𝑛 is equal to five.
00:07:31.030 --> 00:07:38.850
That means then that we’ll move our decimal point over here one, two, three, four, five spots to the left.
00:07:39.200 --> 00:07:43.760
And then regarding the spots that are currently empty, we fill those, we could say, with zeros.
00:07:44.220 --> 00:07:47.230
As a final step, we put a zero in front of our decimal point.
00:07:47.620 --> 00:07:56.250
And we now have this small value, originally written in scientific notation where we let 𝑛 equal five, expressed in its equivalent decimal form.
00:07:56.830 --> 00:08:03.390
Now, if we were to consider the case where 𝑛 is any positive integer, then we could write that in decimal form this way.
00:08:03.780 --> 00:08:08.910
We could say that between the decimal point and our value 𝑎, there are 𝑛 minus one zeros.
00:08:09.380 --> 00:08:16.080
Seeing how to convert from scientific notation into decimal form also gives us a sense for how to go in the other direction.
00:08:16.360 --> 00:08:31.180
If we have some small value written in decimal form like this, then we can count the number of zeros we find in between the decimal point and the first nonzero digit, add one to that number, and then that’s our exponent 𝑛 where we write this value in scientific notation.
00:08:31.630 --> 00:08:37.860
And we then take our nonzero digits, the number 𝑎 where 𝑎 is greater than or equal to one and less than 10.
00:08:38.200 --> 00:08:42.110
And we put that in front of this factor of 10 to the negative 𝑛th.
00:08:42.870 --> 00:08:45.340
The best way to really learn all this is through practice.
00:08:45.340 --> 00:08:47.310
So, let’s try out an example exercise.
00:08:47.800 --> 00:08:51.800
A bullet comes to rest in five times 10 to the negative fourth seconds.
00:08:52.200 --> 00:08:56.210
What is the time taken for the bullet to come to rest, expressed in decimal form?
00:08:56.780 --> 00:09:01.360
Okay, so here we have this value in seconds that’s expressed in scientific notation.
00:09:01.730 --> 00:09:06.840
We know that because this number starts with a value that’s greater than or equal to one and less than 10.
00:09:07.180 --> 00:09:11.250
And then, this is multiplied by 10 raised to an integer value, negative four.
00:09:11.670 --> 00:09:17.070
Our question asks us, “what is this time expressed not in scientific notation but in decimal form?
00:09:17.530 --> 00:09:39.580
Now, in general, to convert between these two ways of writing a number, we can say that if we have a value expressed in scientific notation 𝑎 times 10 to the negative 𝑛, where 𝑎 is greater than or equal to one and less than 10 and 𝑛 is some positive integer, then we can write that as zero with a decimal place following with that followed by a number of zeros equal to 𝑛 minus one.
00:09:39.920 --> 00:09:42.310
And then at the end of all this comes the value 𝑎.
00:09:42.740 --> 00:09:48.370
We can apply this conversion approach to our particular value of the time in which the bullet comes to rest.
00:09:48.620 --> 00:09:54.620
In this time value in scientific notation, the number five corresponds to the value 𝑎 over here.
00:09:54.960 --> 00:09:56.210
So we’ll write that down.
00:09:56.570 --> 00:10:01.700
And then in our exponent, we can see that four corresponds to 𝑛 in our general expression.
00:10:02.180 --> 00:10:07.470
This general rule tells us that we have 𝑛 minus one zeros to the left of our value 𝑎.
00:10:07.750 --> 00:10:10.420
When 𝑛 is equal to four, 𝑛 minus one is three.
00:10:10.780 --> 00:10:14.350
So, we put in one, two, three zeros to the left of five.
00:10:14.550 --> 00:10:17.830
And then to the left of that comes a decimal point and a final zero.
00:10:18.160 --> 00:10:23.900
What we’ve done here is we followed our general rule for converting a number from scientific notation to decimal form.
00:10:24.210 --> 00:10:27.580
The last thing we’ll do is include the unit seconds on this number.
00:10:27.870 --> 00:10:32.160
And in doing that, we’ve written the time it takes for this bullet to come to rest in decimal form.
00:10:32.590 --> 00:10:35.280
It’s 0.0005 seconds.
00:10:35.950 --> 00:10:38.300
Let’s look now at a second example exercise.
00:10:38.700 --> 00:10:43.230
Which of the following is equal to one nanowatt when multiplied by one watt?
00:10:43.780 --> 00:10:53.710
(A) 10 to the ninth, (B) 10 to the negative sixth, (C) 10 to the negative eighth, (D) 10 to the negative ninth, (E) 10 to the sixth.
00:10:54.320 --> 00:11:02.960
Okay, so this question is asking which number from among these five would be equal to one nanowatt if we multiplied it by one watt.
00:11:03.640 --> 00:11:12.160
So basically we’re saying, “What number, if we call that number capital 𝑁, could we multiply by one watt in order to yield one nanowatt?”
00:11:12.720 --> 00:11:18.640
To answer this question, to solve for 𝑁, we’ll need to know how one nanowatt relates with one watt.
00:11:19.210 --> 00:11:23.130
This symbol here, lowercase 𝑛, refers to this prefix of nano-.
00:11:23.460 --> 00:11:29.660
And we can recall that this prefix nano- corresponds to one one billionth of whatever unit it’s attached to.
00:11:30.040 --> 00:11:33.470
So, in this case, one nanowatt is a billionth of a watt.
00:11:34.020 --> 00:11:38.760
To represent one billionth numerically, we can use this value here, 10 to the negative ninth.
00:11:39.160 --> 00:11:46.830
This means that if we replace this number, capital 𝑁, with 10 to the negative ninth, then that replacement makes this equation true.
00:11:47.380 --> 00:11:55.270
It is the case that if we take one watt and we multiply it by 10 to the negative ninth, then we’ll get a billionth of a watt or one nanowatt.
00:11:55.880 --> 00:12:00.800
So, then, we’ll look for this value among our answer options, and we see it at option (D).
00:12:01.290 --> 00:12:05.520
10 to the negative ninth multiplied by one watt is equal to one nanowatt.
00:12:06.560 --> 00:12:10.980
Let’s now summarize what we’ve learned about representing small values of physical quantities.
00:12:11.490 --> 00:12:19.180
In this lesson, we saw that for the sake of clarity and ease of comparison, unit prefixes for small values have been developed.
00:12:19.570 --> 00:12:36.080
These prefixes include milli-, representing 10 to the negative third of some unit; micro-, representing a millionth; nano-, representing a billionth; pico-, representing a trillionth; and femto-, corresponding to 10 to the negative 15th or one quadrillionth of some unit.
00:12:36.560 --> 00:12:42.310
And finally, we saw that a value can be converted from scientific notation to decimal form and back.
00:12:42.920 --> 00:12:54.930
We can do this by recognizing that a small number written as 𝑎 times 10 to the negative 𝑛 is equal to zero followed by a decimal point followed by 𝑛 minus one zeros followed by 𝑎.
00:12:55.700 --> 00:12:59.650
This is a summary of representing small values of physical quantities.