Video Transcript
In this video, we will learn about a mathematical notation that’s very useful in science, scientific notation. We’ll learn why this notation is useful, how to write and interpret this notation, and how to convert regular numbers into and out of scientific notation.
In science, we often encounter numbers that describe very large quantities of things. For example, the Earth is about 150 million kilometers away from the Sun on average, and it’s about 4,600,000,000 years old. We’ll also frequently encounter numbers that describe very small quantities of things. For example, a single water molecule is about 0.275 nanometers across which in meters is 0.000000000275. This water molecule’s weight in grams would be 0.00000000000. Well, I won’t bother reading all those zeros, but that’s zero point 22 zeros followed by the numbers two nine nine one five grams.
Dealing with numbers whose magnitudes are this large and this small is difficult even when you have a calculator. Another problem with these numbers is that when we make measurements, we can only be sure that a small number of digits are actually precise. For instance, in the age of the Earth, we’re sure about the first two numbers, the four and the six. But the following zeros could really be any value. The Earth could actually be 4,600,003,000 years old. It could be 4,600,000,042 years. Or maybe it’s 4,600,500,010 years old. We’re just not able to make a measurement of the age of the Earth that’s more precise than 4,600,000,000. All of these zeros are really only here to tell us that the four is in the billions place and the six is in the hundred millions place. So, all of the zeros in this large number are really just here to tell us the place of the four and the six.
But even in small numbers, like the mass of a water molecule, where the zeros are not just placeholders, they still don’t convey a lot of information. And they hide the fact that the only numbers that we can be sure are precise are the four and the six. So, wouldn’t it be nice if we had a way to simplify how we express these numbers? This is exactly what scientific notation, which is sometimes called standard form, is used for.
Numbers that are written in scientific notation have two parts. The first part of a number that’s written in scientific notation will be the digits that were actually important or significant, which in math is called the significand. So, for a number with a large magnitude like the age of the Earth, those would be the numbers that are out in front, in this case, the four and the six. And for a number with a small magnitude like the size of a water molecule, those would be the digits that are at the end, in this case, the two, seven, and five.
The second part will be an indication of the place of those numbers, which is called their order of magnitude. So, in the age of the Earth, for example, that four is in the billions place. Now, there’s a convenient way in math to express the second part, which are powers of 10. We can break apart numbers into different powers of 10. For instance, 100 is 10 times 10 or 10 to the two. 1000 is equal to 10 times 10 times 10 or 10 to the three and so on.
So, let’s see how that fits together by converting a regular number into scientific notation. To stress this out, let’s try converting the number five million into scientific notation. The five here is that significant digit out in front, the significand in math lingo. So, in order to convert this number into scientific notation, we need to figure out how many times we need to multiply five by 10 to get five million.
In order to convert five million into scientific notation, we essentially want to know how many times we want to multiply five by 10. That’s because when we’re expressing a number in scientific notation, we want the number out in front, the significand, to be greater than or equal to one and less than 10. So, five million expressed in scientific notation will end up being five times a power of 10. If we ended up with something like 0.5 or 50, that number wouldn’t be in scientific notation since 0.5 and 50 aren’t between one and 10.
So, let’s figure out what five million is in scientific notation. Well, five million is equal to 500,000 times 10 which is 10 to the one, which is equal to 50,000 times 10 times 10 or 10 to the power of two. And that’s equal to 5000 times 10 times 10 times 10 or 10 to the three. So, you might notice the pattern here. Each time we decrease the number of zeros in this number, we increase the number in the power of 10.
To be more specific, each time we divide the number by 10, we’re also multiplying by 10 by increasing the number in the exponent. So, overall, the number stays the same. So, that’s equal to 500 times 10 to the four or 50 times 10 to the five or five times 10 to the six. So, this is five million expressed in scientific notation, five times 10 to the six. We know it’s in scientific notation because our significand is between one and 10. And then, we have our power of 10 expressing the order of magnitude.
Of course, that doesn’t just work for numbers with large magnitudes. We can also convert numbers with small magnitudes into scientific notation as well. That’s because numbers like 0.1 is equal to one divided by 10 or 10 to the negative one. And similarly, 0.01 is equal to one divided by 10 times 10 or 10 to the negative two. And 0.001 is equal to one divided by 10 times 10 times 10 or 10 to the negative three and so on.
So, let’s see if we can convert the number 0.007 into scientific notation. Well, 0.007 would be equal to 0.07 times 10 to the negative one. So, here, when we decrease the number of zeros, the number in the exponent also decreases. That’s because we’re multiplying by 10 to go from 0.007 to 0.07. So, we also want to divide by 10 so the number is the same as before, which we can do by decreasing the number in the exponent. And this number is also equal to 0.7 times 10 to the negative two. Here, we again decrease the power in the exponent by one.
But we’re not yet in scientific notation because 0.7 isn’t between one and 10. So, we need to go once more, giving us seven times 10 to the negative three. So, now, we have our number expressed in scientific notation. The significand, the seven, is between one and 10. And then, we have our power of 10 to express the magnitude of that number.
Now, we might not want to go through this whole process every time. Luckily, there’s a shortcut that we can use to convert numbers. To do this trick, we want to put the decimal place where we want it to end up when our number is in scientific notation. So, in the age of the Earth, we want to put it between the four and the six so that we end up with a number that’s between one and 10. Then starting where the decimal is currently, we count how many places we need to move until we end up at where we want the decimal place to be. So that’s one, two, three, four, five, six, seven, eight, nine places. Now, if we use this trick, if we move to the left, the power of 10 increases. So that would give us 4.6 times 10 to the nine years for the age of the Earth expressed in scientific notation.
Now let’s try it with this number on the bottom, the size of a water molecule. We want the decimal place to end up here between the two and the seven so that we get a number between one and 10. Then, we want to move one, two, three, four, five, six, seven, eight, nine, and 10 places. Now, also, I’m sure you can guess every time we move to the right, the power of 10 decreases, which would give us 2.75 times 10 to the minus 10 meters for the size of a water molecule when we express it in scientific notation.
Now, here, we are not actually moving the decimal place. We’re just seeing what order of magnitude we need to express the number in scientific notation. But it’s still a handy trick to quickly convert a number into scientific notation.
Now we can, of course, go the other way. We can convert a number that’s in scientific notation back into a regular number. Let’s try this with this number that’s in scientific notation, 6.7 times 10 to the six. This 10 to the six means that we need to multiply the significand, 6.7, by 10 six times. So, 6.7 times 10 is 67, and 67 times 10 is 670. 670 times 10 is 6700. Now, we’ve multiplied by 10 one, two, three times, so three times more. Multiplying by 10 again gives us 67,000. Multiplying by 10 again gives us 670,000. And then, multiplying by 10 one last time gives us 6,700,000. So, the number in scientific notation 6.7 times 10 to the six expressed as a regular number is 6,700,000.
We can also use our trick here to figure out what this number would be as a regular number. Because this exponent is a positive number, we know that we’ll be making the significand larger. So, we’ll be moving the decimal place to the right, and we’ll want to move it to the right six times because the power in the exponent is six. So, one, two, three, four, five, six. The decimal point will end up there, and we’ll fill in the rest of the places with zeros. This, of course, ends us up with the same number that we got before, 6,700,000. Remember, we’re not actually moving the decimal point, but this is a really helpful trick to help us convert back and forth between scientific notation easily.
Let’s try another example, 3.2 times 10 to the negative three. Since our power of 10 here is a negative three, that means that we’d be dividing our significand, 3.2, by 10 three times. Dividing 3.2 by 10 one time gives us 0.32. If we divide by 10 again, we’d get 0.032. And dividing by 10 one last time would give us 0.0032. So, 3.2 times 10 to the negative three expressed as a regular number is 0.0032. And of course, we can use our trick moving the decimal point here again. Since the number in the exponent is negative three, we need to move the decimal place one, two, three times. So, the decimal place ends up there, and we’ll fill in the places that we moved with zeros, which gave us the same number that we got using the other method, 0.0032.
So, now, we know what scientific notation is, why it’s useful, and how we can convert regular numbers into and out of scientific notation. So, let’s try a problem, and then we’ll conclude this video.
Which of the following numbers is the greatest number of atoms? (A) Four times 10 to the four, (B) one times 10 to the seven, (C) six times 10 to the three, (D) nine times 10 to the five, (E) three times 10 to the six.
All of these numbers are expressed in scientific notation. Numbers expressed in scientific notation have two parts. The first is called the significand, which is a number that will be between one and 10. The second part is the power of 10 that tells us the order of magnitude of the significand. The second part essentially tells us how many times we need to multiply the significand by 10 in order to get the right number.
So, for example, our first answer choice four times 10 to the four means that we need to multiply four by 10 four times, which will be equal to 40,000. So, if we wanted to, we could convert all of these numbers out of scientific notation using the same method. One times 10 to the seven is one multiplied by 10 seven times, which will give us 10 million. Six times 10 to the three is six times 10 three times or 6000. Nine times 10 to the five is nine times 10 five times or 900,000. And three times 10 to the six is three times 10 six times, which is three million.
So, looking through the numbers in our answer choices, we can see that (B) one times 10 to the seven is the greatest number, which is what we’re looking for for this problem. But before we wrap this problem up, we might have noticed a pattern here. We could have skipped all this work by simply looking through our answer choices and seeing which one had the highest power of 10, since the number with the largest power of 10 will be multiplied by 10 more times, resulting in a larger number. So instead of doing all this work, we simply could have looked through our answer choices, noticed that one times 10 to the seven is the largest power of 10 so that one would be the greatest number.
Of course, if we had two answer choices that had the same power of 10, for example, one times 10 to the seven and four times 10 to the seven, we would then have to compare the significand and see which one was larger. So, if four times 10 to the seven was one of our answer choices, that one would have represented the greatest number of atoms since four times 10 to the seven corresponds to the number 40 million and one times 10 to the seven is only 10 million. But that wasn’t an answer choice. So, of the numbers we were given, the one that represented the greatest number of atoms was one times 10 to the seven.
In this video, we learned how scientific notation is useful for expressing numbers that have large or small magnitudes. And we learned that numbers in scientific notation are always of the form 𝑎 times 10 to the 𝑏. 𝑎 will always be a number between one and 10, and the power of 10 tells us the order of magnitude of 𝑎.
We also learned a handy trick for converting numbers into and out of scientific notation, where we think about where we want the decimal point to end up when the number is in scientific notation and how many places there are between where the decimal point is now and where we want the decimal point to end up.
