Video Transcript
In this video, our topic is
representing small values of physical quantities. We’ll specifically be looking at
numerical ways to do this. 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.
To get started, the first thing we
can notice is that in physics, it’s not unusual to work with small physical
values. For example, the charge of a single
electron is approximately equal to negative 1.6 times 10 to the negative 19th
coulombs. Or consider another value, the
universal gravitational constant. This is approximately 6.7 times 10
to the negative 11th cubic meters per kilogram second squared. 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. Or even more simply, we might just
want to measure the mass of a few grains of sand.
So we see that in physics, small
values of physical quantities often are involved. And so, for the sake of accuracy
and simplicity in our calculations, we would like to find convenient ways to
represent these small numbers. 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. 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.
So, for small physical values,
scientific notation is good. But it turns out that we can do
even better. 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. 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. 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.
Even just in naming those two
wavelengths, though, we can see that they’re a bit of a mouthful. 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. To help streamline cases like this
where we’re working with relatively small values, a system of what are called unit
prefixes was developed. We see from this name that a unit
prefix will involve some kind of unit. 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.
Unit prefixes are something we’ve
seen before, even if we didn’t recognize them as such in the movement. For example, say that we measure
out 7.5 milligrams of some substance. In this quantity, we have a unit,
grams, that has a prefix. That prefix is milli-, and we see
its represented symbolically by a lowercase m. And one milligram indicates 10 to
the negative third, or one one thousandth of a gram. The next smallest unit prefix
commonly used is the prefix micro-. 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.
Smaller still is the unit prefix
nano-, represented by the letter n. This shows us one one billionth or
10 to the negative ninth of the unit we’re considering. It’s this prefix, by the way,
that’s often used to represent these wavelengths of light in the visible
spectrum. Instead of writing or saying seven
times 10 to the negative seventh meters, for example, we might instead say 700
nanometers. And likewise for four times 10 to
the negative seventh meters, where instead we would say and write 400
nanometers. 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.
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. In the study of very short pulses
of laser light, this prefix pico- often comes into use. We might say, for example, that a
certain laser pulse lasted, say, 75 picoseconds.
Even smaller still is the unit
prefix femto-. A femto something, whether a
femtosecond or a femtometer, is equal to one quadrillionth, 10 to the negative 15th,
of the unit being considered. One example of a practical use for
this particular prefix is in describing the size of subatomic particles. It turns out, for example, that one
femtometer is approximately equal to the diameter of a proton. So, let’s consider how this idea of
unit prefixes could apply to these values that we wrote up here.
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. Or what about the universal
gravitational constant 𝑔? We could write this value as 67
pico cubic meters per kilogram-second squared. Here, the complexity of this unit
partially but doesn’t completely obscure the advantage of using a unit prefix. 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.
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. For a given physical value, we’d
like to be able to switch back and forth between these two. So, let’s clear a bit of space and
consider how to do that.
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. We can call it 𝑎, where 𝑎 is less
than 10 and greater than or equal to one. And multiplying this value by 10
raised to some negative integer value; here, 𝑛 is an integer. Given this way of writing a number,
we’d like to know how to express it instead as a decimal. To do this, we can start with the
value 𝑎. 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.
Either way, we want to identify the
place where the decimal point is located in 𝑎. 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. Wherever the decimal point is then
in this value 𝑎, we figure that out and we write it in place. Our next step will involve moving
this decimal place a certain number of spots to the left. The reason we do this is because in
scientific notation, we’re multiplying 𝑎 by 10 to some negative integer. Because this is 10 to a negative
power, that’s why the decimal place moves to the left and not to the right.
Just for the sake of illustration,
let’s pick a particular value for 𝑛. Let’s say that 𝑛 is equal to
five. That means then that we’ll move our
decimal point over here one, two, three, four, five spots to the left. And then regarding the spots that
are currently empty, we fill those, we could say, with zeros. As a final step, we put a zero in
front of our decimal point. And we now have this small value,
originally written in scientific notation where we let 𝑛 equal five, expressed in
its equivalent decimal form. Now, if we were to consider the
case where 𝑛 is any positive integer, then we could write that in decimal form this
way. We could say that between the
decimal point and our value 𝑎, there are 𝑛 minus one zeros.
Seeing how to convert from
scientific notation into decimal form also gives us a sense for how to go in the
other direction. 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. And we then take our nonzero
digits, the number 𝑎 where 𝑎 is greater than or equal to one and less than 10. And we put that in front of this
factor of 10 to the negative 𝑛th. The best way to really learn all
this is through practice. So, let’s try out an example
exercise.
A bullet comes to rest in five
times 10 to the negative fourth seconds. What is the time taken for the
bullet to come to rest, expressed in decimal form?
Okay, so here we have this value in
seconds that’s expressed in scientific notation. We know that because this number
starts with a value that’s greater than or equal to one and less than 10. And then, this is multiplied by 10
raised to an integer value, negative four. Our question asks us, “what is this
time expressed not in scientific notation but in decimal form?
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. And then at the end of all this
comes the value 𝑎.
We can apply this conversion
approach to our particular value of the time in which the bullet comes to rest. In this time value in scientific
notation, the number five corresponds to the value 𝑎 over here. So we’ll write that down. And then in our exponent, we can
see that four corresponds to 𝑛 in our general expression. This general rule tells us that we
have 𝑛 minus one zeros to the left of our value 𝑎. When 𝑛 is equal to four, 𝑛 minus
one is three. So, we put in one, two, three zeros
to the left of five. And then to the left of that comes
a decimal point and a final zero.
What we’ve done here is we followed
our general rule for converting a number from scientific notation to decimal
form. The last thing we’ll do is include
the unit seconds on this number. And in doing that, we’ve written
the time it takes for this bullet to come to rest in decimal form. It’s 0.0005 seconds.
Let’s look now at a second example
exercise.
Which of the following is equal to
one nanowatt when multiplied by one watt? (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.
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. 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?” To answer this question, to solve
for 𝑁, we’ll need to know how one nanowatt relates with one watt. This symbol here, lowercase 𝑛,
refers to this prefix of nano-. And we can recall that this prefix
nano- corresponds to one one billionth of whatever unit it’s attached to. So, in this case, one nanowatt is a
billionth of a watt.
To represent one billionth
numerically, we can use this value here, 10 to the negative ninth. This means that if we replace this
number, capital 𝑁, with 10 to the negative ninth, then that replacement makes this
equation true. 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. So, then, we’ll look for this value
among our answer options, and we see it at option (D). 10 to the negative ninth multiplied
by one watt is equal to one nanowatt.
Let’s now summarize what we’ve
learned about representing small values of physical quantities. In this lesson, we saw that for the
sake of clarity and ease of comparison, unit prefixes for small values have been
developed. 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.
And finally, we saw that a value
can be converted from scientific notation to decimal form and back. 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. This is a summary of representing
small values of physical quantities.
