Standard Form | GCSE Maths | Third Space Learning

Hi, this is Paul from First Base Learning and today we're going to look at standard form. Standard form is a way of writing very large or very small numbers. And we can write it in the form a multiplied by 10 to the power of n, where A is a number between 1 and 10, not including 10 and n is an integer, a whole number. In this video we'll look at converting ordinary numbers to standard form and standard form to ordinary numbers. As usual, I'll use our online classroom to go through the topic and look at some questions.

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Of the video for access to tons of free resources and information on our one to one personalised tuition. Okay, let's have a look at converting standard form. Grab a pen and let's dive in. Let's start by converting an ordinary number to standard form. So let's start with a big number, 4,500. Well, we need to take those digits and write them between 1 and 10. So for 4,500, that would just be 4.5. So 4,500 equals 4.5 multiplied by 10 to the power of something. So what's that something? Well, 4.5 multiplied by 10 to the power of 1 is the same as multiplying by 10, so that's 45. 4.5 multiplied by 10 to THE POWER OF 2 is the same as multiplying by 100. Well, that will give me 450. And 4.5 multiplied by 10 to The POWER OF 3 is the same as multiplying by a thousand. So that will give me 4,500, which is what we want. So 4,500 written in standard form is 4.5 multiplied by 10 to the power of 3. Let's try them together.

So the question says write 48,000 in standard form. Now we know standard form means writing.

The number between 1 and 10, not.

Including 10 and multiplying it by a power of 10. So let's start by writing the non 0 digits of 48,000 between 1 and 10. So for this number, that will be 4.8. Now we need to multiply that by a power of 10. So if we multiplied it by 10.

To the power of 1, we would.

Get 48 multiplied by 10 to the power of 2, we get 480. Multiply it by 10 to the power of 3, we get 4800 and multiply it by 10 to the power of 4 and we get 48,000. So the answer in standard form is 4.8 multiplied by 10 to the power of 4. Now another way we could think about this would be to write our 4.8 and then move the numbers to the left in our heads. So decimal point, it doesn't actually move, but it would appear here. So it would be 48. If we multiply it by 10, multiplied it by 10 again we get 480, and again we get 4800, and again we get 48,000. Either way we get 4.8 multiplied by 10 to the power of 4, and that gets us one mark. Okay, your turn. Pause the video and have a go. Okay, let's see how you got on. So it says write 4020 in standard form. So let's start by writing the number between 1 and 10. So that's going to be 4.02. And we need to multiply that by a power of 10 to get 4020. Well, multiplying it by 10 and we get 40.2 because all the digits move one place to the left by 10 again and we get 402 and by 10 again and we get 4002, 20. So we multiply 4.02 by 10 to the power of 1, 2, 3, and again we could think about it like this if we wanted to. So we could do 4.02, and we can imagine those numbers moving to the left. So if they move to the left once, the decimal point will now be here. Twice it'll be here. And if we move the numbers three times to the left, the decimal point is now here. So we get 4.02 multiplied by 10 to the power of 3. And that's worth a mark.

Next, let's convert a really small number to standard form. So let's take 0.025. So let's start by writing the non 0 digits between 1 and 10, not including 10. So if 0.025, that will be 2.5. So 0.025 is equal to 2.5 multiplied by 10 to the power of something. So what's that something? Well, because 0.025 is less than 2.5, let's start by multiplying by 1 over a power of 10. So 2.5 multiplied by 1 over 10 to the power of 1, that's the same as dividing by 10. And do that we'll get 0.25. So we're nearly there, but we're not quite. So let's do 2.5 multiplied by 1 over 10 to the power of 2 because that's the same as dividing by 100. If we do that, we get 0.025, which is what we want.

So. So that's great.

Now, rather than writing 1 over 10 to the power of 2, we can write that as 10 to the power of negative 2. So 0.025, written in standard form is 2.5 multiplied by 10 to the power of negative 2. Let's have a look at a few examples of this, and I'll show you a really easy way to do it.

The question says write 0.00047 in standard form. So I start by writing the digits between 1 and 10, not including 10. So for this number, it's going to be 4.7. Now we need to multiply that by a power of 10 so that it will give me 0.00047. But the question is, what is that power of 10? So because 0.00047 is smaller than 4.7, let's start by dividing it. So let's do 4.7 divided by 10. So all those digits will move one place to the right. So we'll get 0.47. If we do that a second time, we'll get that. If we do that a third time, we get that. And if we do that a fourth time, we get that. So we have divided 4.7 by 10 four times. Now, that is the same as multiplying by 10 to the power of negative four. So we write 4.7 multiplied by 10 to the power of negative 4. Now, we can visualize this in our head if we want to, and we could say, okay, well, here's 4.7, and I want to get 0.00047.

So.

So all those digits are going to keep moving to the right. So if we imagine they move to the right one place, the decimal point is now here. If they move again, the decimal point is here, and I put a zero in again. Decimal point is now here, and we put a zero in. And one more time, the decimal point is now here, and we put a zero in. So we get 0.000. And we've moved the digits four times. So whichever way you do it, we get 4.7 multiplied by 10 to the power of negative 4. Okay, your turn. Pause the video and have a go. Best of luck. Right, let's go through it. So it says write 0.0142 in standard form. So let's write 0.0142 as a number between 1 and 10. So that will be 1.42. So it's 1.42, and we're going to multiply that by a power of 10. Now, we don't know what the power of 10 is just yet, so let's work that out, because 1.42 is bigger than 0.0142. Let's start by dividing it by powers of 2, 10. So if we move the digits one place to the right and divide it by 10, we'll get 0.142. And if we do that again, move all the digits one more place to the right, we get 0.0142, which is what we want. So we have done that twice. This means the answer is 1.42 multiplied by 10 to the power of negative. And again, if we want to, we can visualize this in our head and we can do 1.42. And we're going to visualize those numbers moving to the right. So if we do that, the decimal point will go here. If the numbers move to the right one place. If we do that again, the decimal point will now be here. And we put a zero in there and a zero in here. So it's 0.0142. So we've moved the digits twice and have gone to the right. Whichever way you do it, the answer is 1.42 multiplied by 10 to the power of negative 2. Now, let's look at a number that's.

Already in standard form and convert it back into an ordinary number. So let's start with a really big number, 1.6 multiplied by 10 to the power of 3. That number is in standard form, and we're going to write it as an ordinary number. Well, 1.6 multiplied by 10 to the power of 3 means 1.6 multiplied by 10 multiplied by 10 multiplied by 10.

So let's do that.

1.6 multiplied by 10 is equal to 16. Multiply by 10 again and we get 160. Multiply by 10 a third time and we get 10. So 1.6 multiplied by 10 to THE POWER OF 3 is equal to 1600.

Let's try a few examples together. So the exam question says write 1.2 multiplied by 10 to the power of 3 as an ordinary number. So let's start then by multiplying 1.2 by 10. So if we do that, all the numbers move to the left by one place. If we multiply it by 10 again, they move one place to the left again. And if we Multiply it by 10 by a third time, the numbers move to the left a third time. So our answer written as an ordinary number is 1200. Now, if we Wanted to. We could do this in our heads and we could say, well, here's 1.2, and if I'm going to multiply this by 10 three times, all the numbers are going to move to the left three times. So let's visualize that if they move once to the left, the decimal point will be here. Twice would be here, and we put a zero in, and three times the decimal point will be there, and we'll put a zero in. Whichever way you do it, the answer will be the same. 1200. Okay, your turn. Pause the video and have a go. Best of luck. Right, let's go through it. So it says write 5.23 times 10 to the power 4 as an ordinary number. So let's start by writing 5.23, and we're going to multiply that by 104 times. So if we multiply it by 10 once, all the numbers move one place to the left, multiply by 10 again, and the numbers move one more place to the left. Multiply by 10 a third time, and the numbers move another place to the left, and then multiply it by 10 a fourth time, and the numbers move one more place to the left. So there we are, 5.23 times 10 to the power of 4 is 52,300 written as an ordinary number. And again, let's see what this would.

Look like if we were going to.

Just do it in our heads. So if we had 5.23 and we multiply it by 10 four times, that means all the digits will move to the left four times. So if we move them once, the decimal point will be here twice the decimal point will be here three times. The decimal point will go here and put a zero in and four times, and the decimal point goes at the end of the number and we get 52,300. And that's worth one mark.

Now let's have a look at converting a really small number that's already in standard form back to an ordinary number. Let's have a look at 2.3 multiplied by 10 to the power of negative 2. Now, 2.3 multiplied by 10 to THE power of negative 2 is the same as 2.3 multiplied by 1 over 10 multiplied by 1 over 10 10.

So let's do that.

2.3 multiplied by 1 over ten is the same as dividing by ten. So we get 0.23 multiplied by 1 over ten again, and we get 0.023. So 2.3 multiplied by ten to the power of negative 2 is equal to 0.023.

Let's have a look at a few examples together. The question says, Write 3.4 multiplied by.

10 to the power of negative 3.

As an ordinary number. So what this means is we're going to take 3.4 and we're going to multiply it by 1 10, 3 times. Now, that's the same as dividing by 103 times. So let's do that. If we divide 3.4 by 10, we get 0.34 by 10 again, and we get 0.034 and by 10 again and we get 0.0034 as those digits keep on moving to the right. So that means that 3.4 multiplied by 10 to the power of negative 3 is 0.0034. Now, if we want to, we can think about it like this, and we can write 3.4, and we're going to visualize those numbers moving to the right three times. So if we do that and then move to the right once, the decimal point is now here twice. It's now here, and we put a zero in and three times and it's now there. So we get 0.0034. Whichever way you do it, you still get 0.0034 as our ordinary number. And that's worth a mark. Your turn. Pause the video and have a go.

1.25 times 10 to the power of.

Negative 4 as an ordinary number. Best of luck. We'll go through it in a sec. Right, let's do it. So let's start by writing 1.25, and we're going to divide this by 10 four times. So if we divide it by 10 once, we get 0.15 as all those digits move to the right once, let's divide by 10 again. So 0.0125 by 10 again, 0.00125, and by 10 a fourth time, 0.000125. So we've divided by 10 1, 2, 3, 4 times. So the answer is 0, 0.000125. And if you want to, we can think about this in our heads and we can do 1.25, and we can visualize those numbers moving to the right four times. So if we move the numbers to the right once, the decimal place will now be here twice. It's here and put a zero in three times. It's here and put a zero in four times. It's here and put A zero in. So the answer is 0.000125. Whichever way you do it you'll get four marks.

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