Video: Representing Large Values of Physical Quantities

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

17:30

Video Transcript

In this video, we’re talking about representing large values of physical quantities. As we study the physical world, some of the numbers we encounter are really, really big. For example, the number of atoms in our galaxy. In this lesson, we’re going to learn a way to write out such large values using a shorthand notation. Along with this, we’ll learn some prefixes which will help us describe very large as well as very small numbers.

To get started, let’s consider another large number. We’ll think of the number of stars that are in the observable universe. In that space, in the part of the universe we can observe, we count about 10 billion galaxies. And each galaxy, we estimate, has about 100 billion stars. If we multiply these two numbers together, 10 billion galaxies and 100 billion stars per galaxy. We get a result which is a one followed by one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 zeros.

Now, let’s just make some observations about this number. Maybe the most impressive thing about this number is how many zeroes it has. In order to make sure the number was written correctly, we had to go through and count each one of the zeros. That took some time. And it would be easy to make a mistake in counting and accidentally write down the wrong number. So the second thing we notice is that this number is easy to write incorrectly. It takes some careful looking to make sure we have the right number of zeros. Lastly, we could notice that this number takes up a lot of space on screen. We had to make space for all of these zeroes. So if we wanted to write this number several times on our screen, say, as part of a calculation, we would need to be careful to give it plenty of space so we can do that.

Now, it’s worth pointing out that this number, as we’ve written it, is written out in what’s called decimal notation. This way of counting is based on the decimal system. In this system, as we’ve seen before and are familiar with, the values that we can write in any given place in the number, say this place right here, can range from zero all the way up to nine. To get a bit more clear on that, let’s consider this number right here, 1234.56. In this number, we know that each one of the digits occupies a certain place within the number. For example, the four is in the ones place in our number. The three is in the tens place. The two is in the hundreds place, and the one is in the thousands place.

Realizing this, there’s another way that we could right these individual numbers. We could write the digit in our ones place, the four, as four times 10 raised to the zeroth power. Recall that any number raised to the zeroth power is equal to one. So this is equal to four times one. And four times one we know is equal to four. The digit we have in our ones place. If we consider then the digit in our tens place, the three, we can represent that number as three times 10 not to the zero but to the one.

Now, why is this? Why are we using three times 10 to the one instead of three times 10 to the zero? The reason is because our three is not in the ones place like the four. But it’s in the tens place. And so, we multiply it by a factor of 10. 10 to the one is equal to 10. Having a three in the tens place represents a value of three times 10 or 30. One way to think about this is if we were to add this number, 30, to the number in our ones place, four, we would get a result of 34. And indeed, if we only look at the digits in the tens and the ones place in our overall number, we see that that’s the value represented, 34.

We can continue doing this with our number in the hundreds place, the two. This value is represented by two, the digit in that place, multiplied by 10 raised to the two. And we can recall that 10 raised to the two is equal to 10 times 10 or 100. Two multiplied by that is then equal to 200. And then, moving on to our thousands place where our digit is a one. This value can be represented by one times 10 to the third. 10 to the third is equal to 10 times 10 times 10, which is equal to 1000. So this value overall is equal to one times 1000 or 1000.

Notice the overall trend we’re seeing here. That for the digit in our ones place, in our case this was a four, we multiply that by 10 to the zero. Then for the value in the tens place, we multiply that by 10 to the one. Then in our hundreds place value, we multiply that by 10 to the two. And then 10 to the three in our thousands value. And by the way, if we had done the same thing for our values five and six that are after the decimal point, we would’ve seen a similar trend. In this position, the five can be written as five times 10 to the negative one. And we can express the six as six times 10 to the negative two. When we do this, we get a result of 0.06. And five times 10 to the negative one is equal to 0.5.

Now, with all this writing on screen, let’s focus for a moment on the numbers that are multiplied by 10 raised to some power. So we’ll look at this expression here, this expression here, this one, this one over here, this one over here, and this one here. Considering these six expressions altogether, what can we notice about them? One thing we notice is that all six of these numbers start with some number that’s between one and 10. And when we say the number that they start with, we’re referring to the one up here, the two here, the three here, the four here and so on. That starting number is always greater than or equal to one and less than 10.

Another thing that stands out is that each one of these numbers is multiplied by 10 raised to some integer power. 10 to the one, 10 to the two, 10 to the negative two, 10 to the positive three. We’ve seen all those in this example. Now, here’s where it gets interesting. Because even though we’ve called these observations, what if we thought of them as rules for writing down numbers? What if we said that any number we wanted to write down had to start with a number greater than or equal to one and less than 10. And then it had to be multiplied by 10 raised to some integer or whole number power. We can see examples of how that would work from our original number here, 1234.56.

Say, for instance, that we wanted to write down the number 1000 using this recipe, using these two rules. We’ve seen how that would work. 1000 written this way is one times 10 to the third power. We’ve started with a number greater than or equal to one and less than 10. And then multiplied that number by 10 raised to an integer power. If we look at this instance a bit more carefully, we can see that we’re starting out with a one. That one is right here. And we could represent that one as a one followed by a decimal point. By doing this, we follow the first rule of our recipe. And then, we apply rule number two.

We multiply this one by 10 raised to the third power. Doing that results in a shift in our decimal point. It starts to move to the right. It moves one, two, three spaces, one space for each power of 10 by which we multiply one. And this is how we end up with the number 1000. We take our original starting number. We put a decimal point after it. And then, we move that decimal point in a way that corresponds to the number of powers of 10 by which we’re multiplying our starting number.

And by the way, if our integer power is negative, like it is in the example of six times 10 to the negative two, this process still applies. We can still start out by writing our beginning or leading number, in this case a six, and following it with a decimal point. But now, because our integer is negative, instead of moving the decimal point to the right, when we multiply six by 10 raised to this power, we move it to the left. And in particular, we move it one, two spaces. This leads to a number 0.06.

All right, so far, we’ve used this two-step method to write out numbers as large as 1000. But what if we could use this same process to write out a shorthand notation for much larger numbers. Let’s consider our number representing the number of stars in the observable universe to see how this might work. We can see that this number, as big as it is, starts with a value greater than or equal to one and less than 10. In particular, it starts with one. And then, that one, as we saw before, is followed up by 21, we counted them, zeroes. So if we were to put a decimal place in this number, that decimal place would go right here. Then if we write our starting number, one, as a one followed by a decimal point. We can see that in order for this number here to represent the actual number we want it to represent, this decimal point we’ll need to move 21 spots over to the right.

Now, of course, if we wrote that out, all we would have is this number written down once more. But if we consider our example from earlier, one times 10 to the third being equal to 1000, we can build on that example to write this number in a shorthand notation. From our starting number of one earlier, we wanted to add one, two, three zeros. And so, we multiplied our starting number by 10 to the third. Which means that with our very large number representing the number of stars in the universe. If we want 21 zeros, then we’ll want to multiply our starting number, one, by 10 raised to the 21st power. Doing this would calculate out to one followed by one, two, three, four, five, six, seven, eight, nine, and so forth up to 21 zeros.

So then, using this two-step process, we’ve expressed a very large number in a shorthand way. And notice, unlike the original number written down, this one doesn’t have many zeros that we’ll need to handle. And it doesn’t take up too much space to write it out. It’s a very large number expressed in a fairly short, compact way. Writing a number this way is known as expressing it in what’s called scientific notation. And working with such a large number, we can easily see the difference between writing it in decimal notation or scientific notation.

Writing a number in scientific notation involves expressing it this way, using these two rules. One great thing about this notation is we can apply it to any number. We don’t need a number that starts with one and then is followed by a bunch of zeros, like we have in this case. To see that we can indeed write any number in scientific notation, let’s do it for this original value we had, 1234.56.

Okay, looking at rule number one of our process, we see we need to start off with a number that’s greater than or equal to one and less than 10. The way it’s written right now, our number doesn’t meet this condition. So we need to change it so that it does. How could we do that? Well, we could do it by shifting the decimal point in this number. If we move the decimal point one, two, three spots to the left, then after that shift, our number would be 1.23456. This number is greater than or equal to one, and it’s less than 10. So it meets our first rule. But then, if we compare our original number with the number we have now, we see that they’re not the same. But we can make them the same by applying rule number two in our process.

By multiplying our starting number by 10 raised to some integer power, we can refer to that integer as 𝑏, we want to make it so this overall number is equal to our original number. To do that, we can see that this value of 𝑏 we’ll need to make it so that our decimal point moves one, two, three spots back to the right. In other words, it effectively reverses the change we made in moving the decimal point to the left to get our starting number. The value of 𝑏, the integer, that will do that is three. And now, we can say that these two numbers are equal to one another. And we can see that by noticing what would happen if we multiplied our starting number by 10 to the third. Doing this would shift our decimal point three spots to the right, which would give us our original value of 1234.56.

So then, any number to any level of precision can be written in scientific notation. This is a common way within the scientific community of writing very large or very small values. Considering such numbers, ones that are very large or very small, one way to describe them is using what are called prefixes. Now, even if the name is not familiar, prefixes are something we’ve seen and worked with before. Say that we’re measuring the mass of some object. We could write the object’s mass in units of grams. But another completely legitimate way to do it is to write the object’s mass in kilograms. Here kilo is a prefix to gram. It tells us that we’re not talking about individual grams but rather collections of thousands of grams.

Likewise, milli can be a prefix for gram, which tells us we’re working with thousandths of grams rather than grams themselves. Kilo and milli are examples of prefixes we use to describe large and small numbers. And, of course, there are others. Let’s consider this larger list. Here in this table, we have these two columns. In the first column, we have the prefixes along with their abbreviations. And in the second column, we have the power of 10 that that prefix is equivalent to. So if we use tera as a prefix, symbolized using a capital T, then that implies multiplying some value by 10 raised to the 12th power. That’s a one followed by 12 zeros.

Then going on down these columns, we also get into small values indicated by a prefix like micro. Using this prefix to describe a quantity means we’re taking that original quantity and effectively multiplying it by 10 to the negative sixth. In other words, dividing it by one million. Now, one important thing to remember about prefixes is that they need to apply to some kind of unit. They can’t stand on their own. If we just said a giga or a centi or a nano and didn’t specify what those prefixes apply to, it wouldn’t make any sense. Let’s say, though, that we did have a quantity and that that quantity was volts. In this case, it would make sense to talk about a gigavolt or centivolt or a nanovolt. The prefix can change, and it changes based on what size of number we want to describe.

To see how these prefixes work, let’s imagine that we have some number of nanovolts. And by the way, we can abbreviate that lower case n capital V. Let’s say that we have, oh, 17 nanovolts. The question may come up, how many volts is this? To figure that out, we can see that the prefix nano implies taking whatever our original unit is, in this case volts, and multiplying it by 10 to the negative ninth. This is equivalent to saying that one nanovolt or one nano anything, the unit doesn’t matter, is equal to 10 to the negative ninth times that original unit. So then, to figure out how many volts 17 nanovolts is, we could multiply both sides of this equation by 17. When we do, we see that 17 nanovolts is equal to 17 times 10 to the negative ninth volts. This shows us how using prefixes — such as, in this case, nano — helps us write out numbers in a bit of a shorter form.

Speaking of writing numbers in a certain form, here’s a question. Is this number in scientific notation? We can see it involves multiplying a number by 10 raised to an integer power. In that way, it satisfies one of the two conditions for writing a number in scientific notation. But, notice that our starting number, 17, is not between one and 10. That was the other condition that we had. So written as is, this number in volts is not in scientific notation, but we could write it that way. We could do it by noticing that right now our decimal point is effectively there. And we would like to move it one spot to the left. To compensate for that movement though, we’ll need to adjust our exponent. We’ll need to change this number.

One question is, do we make it larger or do we make it smaller? Does it become negative 10 or negative eight? Well, whatever we do to our exponent, we want it to undo this change of moving our decimal point to the left. The burden of our exponent, we could say, is to move the decimal point back to where it was before. To do this, to move the decimal to the right, our exponent will need to get bigger.

Now, in this case, understanding what bigger is is a little bit tricky because our exponent is negative. So we’ll need to recall that 10 raised to the negative eight is actually a larger number than 10 raised to the negative nine. And if that fact is surprising, it comes down to the fact that our exponent is negative. So we do shift our decimal place so that now our leading number is between one and 10. And to compensate for that, we adjust our exponent to negative eight. And having done that, we now have an expression in scientific notation for the number of volts equivalent to 17 nanovolts.

Let’s take a moment now to summarize what we’ve learned about representing large values of physical quantities. Starting off, we saw that when large numbers are written in decimal notation, they take up lots of space. And they’re easy to write inaccurately. In scientific notation, on the other hand, large numbers are more compact, and they’re easier to write correctly. There’s much less of a need for accounting up a large number of zeros and making sure we’ve written down each one.

To write out a number in scientific notation, we follow two guidelines. First, the number must start with a value greater than or equal to one and less than 10. And then, that value is multiplied by 10 raised to some integer power. In the end, we get a number that looks something like this, where 𝑎 is our starting number and 𝑏 is our integer power. And lastly, in this lesson, we saw that prefixes help us quickly refer to large and small numbers in speech and in writing.

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