Video: Working with Numbers in Scientific Notation (or Standard Form)

Learn how to represent very large, or very small numbers using scientific notation (also known as standard form).

15:44

Video Transcript

In this video, we’re gonna be working with numbers in scientific notation. And some of you may know that as standard form. Basically, we’re talking about numbers in the format where we got something times 10 to the power of something else. And it’s basically a way of writing very large or very small numbers in a more compact format.

So, scientific notation is when we put numbers in this particular format, a number times 10 to the power of some other number. Now, that first number, represented by 𝑎 in this general formula, has to be between one and 10. Well, specifically, it can be equal to one, so it’s greater than or equal to one, but it can’t be quite as big as 10. It could be 9.999 recurring, but it can’t be quite as big as 10. So, it’s a number between one and 9.9 recurring.

And the other number, represented by this 𝑏, is a positive or negative integer which represents the exponent, or power, of 10. Well, the best way to understand this is to have a look at a couple of examples. So, let’s see an example.

1.5 times 10 to the power of three, or 10 to the third power.

Now, multiplying by 10 to the power of three, or 10 cubed, is like multiplying by 10, then multiplying by 10, then multiplying by 10 again. So, that’s 1.5 times 10 times 10 times 10. Now, there is a shorthand way of thinking about this which makes it very easy to work with scientific notation. Now, this isn’t strictly mathematical, but this shorthand method does make it a lot easier to work with.

Multiplying by 10 is the same as taking that decimal point and moving it one place to the right. So, 1.5 becomes 15 point, well, 15.0. Now, if we multiplied by 10 again, we’re taking that decimal point and moving it one place to the right. Well, we didn’t have a number there, but remember 15 is the same as 15.0. So, in this position here, there would’ve been a zero, and we can put a decimal point over here. So, now, we’ve multiplied by 10 twice, and we’ve got one five zero point, so 150.

We’ve just got to multiply by 10 once more. And doing that moves the decimal point yet another place to the right. So, that moves to here, and we’ve got our zero that fills in here. So, 1.5 times 10 times 10 times 10 is 1500. Now, it’s important not to leave your number in this format because that looks a real mess. This is nice and clear. 1.5 times 10 to the power of three is 1500. Remember, 1.5 times 10 was 15, 15 times 10 was 150, and 150 times 10 is 1500. Okay let’s look at another example.

9.09 times 10 to the power of five.

Well, multiplying by 10 to the power of five is like multiplying by 10 then by 10 then by 10 then by 10 then by 10 again. So, we’re multiplying by 10 five times. So, let’s look at our shorthand way of actually working that out. When we multiplied 1.5 times 10 to the power of three, we ended up with three of these little arrows here. So, multiplying by 10 to the power of five, we’re gonna end up with five little arrows over here.

That means that this decimal point is going to move one, two, three, four, five places to the right. And we need to fill in some zeros. We need a zero here, a zero here, and a zero here. We could write one after the decimal point as well if you want. So, our answer is 909000.

Now, when you’re doing your working out, you wouldn’t normally bother writing this out or any of this. You would just use this kind of like short notation here, which is rough working out, and then write your answer correctly in a nice easy uncluttered format. Okay, hopefully, that’s fairly clear. Let’s look at a couple of examples now with negative powers of 10.

For example, 7.2 times 10 to the power of negative four.

Now, in that exponent negative four, the four tells us we’re multiplying by 10 four times and the negative tells us to flip that fraction. So, instead of 10, it’s one over 10. So, 7.2 times 10 to the power of negative four means 7.2 times a tenth times a tenth times a tenth times a tenth. Now, multiplying by a tenth is the same as dividing by 10. So, that means we’re starting off with 7.2 and then we’re dividing that by 10, dividing by 10 again, dividing by 10 again, and dividing by 10 again.

Let’s just jump straight to our shorthand notation for doing this then. 7.2 divided by 10 is gonna be 0.72, so effectively that decimal point is moving left one place here. Now, we’ve got to do that four times, so one, two, three, four. So, our decimal point ends up here. We need to put in some zeros here as placeholders and a zero in front of the decimal point so we know it’s 0.00072.

Now, these zeros here were very important because they were placeholders that told us that the seven should be in the one ten thousandth column and the two should be in the one one hundred thousandths column. And again, it’s important to write out our answer clearly without all that horrible working out scribbled all over it, so 0.00072. Now, remember, all this moving the decimal point around is not necessarily mathematically correct in the way that we’re looking at it, but it’s a nice shorthand way of working out the answer to these questions. Okay let’s have a look at one more example.

3.05 times 10 to the negative six.

Well, times 10 to the negative six, remember, means that we’re dividing the negative power, means we’re dividing by 10 six times. So, we start off with 3.005 [3.05] and we divide by 10, we divide by 10, we divide by 10, we divide by 10, we divide by 10, and we divide by 10 again. So, I’m dividing by 10 lots of times. I need to leave a bit of space to the left-hand side of my number here.

Okay, divide by 10 once, the decimal point moves to here, twice, three times, four times, five times, six times, which means our decimal point’s moved here. Now, I can fill in the zeros. There’s one here. There’s one here. There’s one here, one here, one here. And I’m gonna to put a zero in front of the decimal point as well. So, that’s 0.00000305. And that’s our answer written out nice and clear.

Okay, then, so, what’s all this scientific notation about? In the examples we’ve shown you, there’s not a massive amount of difference between writing that and writing that. That’s because the examples I’ve given you were relatively easy just to get the idea across. What if I was to ask what’s the distance from Earth to Proxima Centauri, the nearest star other than the sun?

Well, it’s over 40 quadrillion metres away from the Earth. Now, that’s quite a big number to write out. And once we start talking on an astronomical scale, there are even bigger numbers than that to deal with. Now, it starts to become more efficient to write something like 4.0208 times 10 to the power of something metres. But what is that something?

Well, if we’ve got a decimal point here, in order to move it to the end of this number over here, it’s got to jump over one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16 places. So, that’s 4.0208 times 10 to the power of 16 metres. And in working that out, remember, this number here has to be bigger than or equal to one, but it must be less than 10. So, we’ve made it four point something. And in this case, that left us with 10 to the 16th power.

And at the other end of the spectrum, we’ve got very very small things. So, for example, the diameter of a proton is about a millionth of a nanometre, or 0.000000000000001 metres. Now, there must be a quicker and easier way to write that. Well, the digit on the end here is just a one, so we can write it as one times 10 to the power of negative something metres. So, what is that something?

Well, if we wrote a one with a decimal point after it, like this, we would have to move that decimal point one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, places to the left. So, we can write the number as one times 10 to the negative 15 metres. Okay, let’s have a look at some typical questions you might see in a test.

Write 3.141579265 times 10 to the power of four as an ordinary number.

Now, if you live somewhere that calls this standard form, then a number not written in standard form would be called an ordinary number. But if you live somewhere where they call this format scientific notation, then they’d call the ordinary number version standard form. How confusing’s that? Anyway, let’s not worry about that for now. You know where you live. You know what you call these things hopefully.

So, 3.141579265 times 10 to the power of four, in this case, we’re multiplying by 10 positive four times. We’re multiplying by 10 four times, so that decimal point is gonna have to move one, two, three, four places to the right. And after we’ve multiplied that number by 10 four times, we get 31415.79265.

How about this one then?

Write 6.2 times 10 to the power of seven as an ordinary number.

Well, we’re gonna write down the 6.2, and then we’re gonna multiply it by 10 seven times. So, that’s one, two, three, four, five, six, seven. So, that’s gonna be 62000000.0. Well, obviously, in this case we can leave off the point zero, and that gives us an answer of 62 million.

Now, write 9.603 times 10 to the negative three as an ordinary number.

So, we’re gonna take 9.603, and we’re gonna multiply it by 1 over 10 three times, which is dividing by 10 three times. So, this decimal point is gonna move to the left. It’s gonna make that number smaller. Divide by 10 once, twice, three times, so my decimal point’s gonna go here. I can then put in my holding zeros, and we can see that the answer is 0.009603.

Now, let’s have a look at one or two slightly more challenging questions.

Work out the value of three times 10 to the four times 1.2 times 10 to the power of seven, giving your answer in scientific notation.

Well, with multiplication, it doesn’t actually matter what order we multiply these things together in, so I can get rid of those parentheses straight away. So, that’s the same as three times 10 to the four times 1.2 times 10 to the power of seven. Now, we can multiply the 10 to the four and the 10 to the seven together, and we can multiply the three by the 1.2. Now, three times 1.2 is 3.6. And if I multiply by 10 four times and then another seven times, I’ve multiplied by 10 11 times. That’s 10 to the power of 11. So, that’s equal to 3.6 times 10 to the power of 11.

Now, remember, to be in proper scientific notation, the number must be in this format, 𝑎 times 10 to the power of 𝑏, where 𝑎 is a number between one and 10. It can be equal to one, but it can’t be equal to 10. And 𝑏 is a positive or negative power exponent of 10. Now, in our case, the 𝑎 value, this 3.6, that is in the right range. It’s bigger than or equal to one, but it’s less than 10. And we have, in this case, a positive power, or exponent, of 10. So, our answer is 3.6 times 10 to the power of 11.

Right then, just one final question before we go.

Work out the value of five times 10 to the 5 times 8.2 times 10 to the power of 11, giving your answer in scientific notation.

Well, again, multiplication is commutative, so it doesn’t matter whether we’ve got the parentheses and what order we write those in. So, I’m going to do five times 8.2 times 10 to the power of five times 10 to the power of 11, in that order. Now, five times 8.2 is 41. And if I multiplied by 10 to the power of five, that’s multiplying by 10 five times, and then by 10 to the power of 11, that’s multiplying by 10 another 11 times, that’s a total of 16 times. So, it’s times 10 to the power of 16. So, that’s equal to 41 times 10 to the power of 16.

But wait a minute! That’s not our answer! This number here has to be between one and 10. 41 isn’t between one and 10. But look, I could write it as 4.1 times 10 to the power of something. Well, in fact, in this case it would be times 10 to the power of one because I need to multiply 4.1 by 10 once to turn it into 41. So, 41 is equal to 4.1 times 10 to the power of one.

And that’s the number that has to be multiplied by 10 to the power of 16. Well, we can see that we’ve got 4.1 times 10 once times 10 another 16 times, which means that’s times 10 17 times in total. And there we have it. We’ve now got our number in proper scientific notation. That first term is 4.1. That is between one and 10. And we’ve got times 10 to the power of 17.

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