Video: Representing Small Values of Physical Quantities

In this video, we will learn how to use scientific notation and unit prefixes to multiply and divide values of physical quantities by various powers of ten.

13:00

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.

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