## What you need to know

**Standard form** is a shorthand of expressing very large or very small numbers. It is sometimes called “scientific notation” because often it is used across all kinds of sciences.

There are strict rules that determine when a number is in standard form. We say a number is in standard when it is written like

A\times10^n

Where n must be a whole number, and 1\leq A<10, or in other words, A must be greater than or equal to 1 but strictly less than 10. For example,

4.2\times10^5

is a number in standard form. But what does it actually mean? Well, it’s just a normal number written in an unusual way, so the question really is: what number is it?

Note: **decimal notation** is the technical term for numbers written in the usual, familiar way.

**Example: **Write 4.2\times10^5 in decimal notation.

Firstly, recall that 10^5=10\times10\times10\times10\times10. Then, we get

4.2\times10^5=4.2\times10\times10\times10\times10\times10

To multiply a number by 10 once, we move the decimal point one space to the **right**. Here, we’re multiplying by 10 five times, so we must the decimal point 5 spaces to the right (if it helps, think of 4.2 like 4.20000). Doing this, we get

4.2\times10\times10\times10\times10\times10=420,000

Therefore, we have concluded that 4.2\times10^{5}=420,000.

**Example: **Write 2.8\times10^{-4} in decimal notation.

This is different because the power is negative, but it’s actually no harder. Firstly, recall that

10^{-4}=\dfrac{1}{10^4}

Now, we know 10^4=10\times10\times10\times10. Having this on the bottom of the fraction means that rather than multiplying by 10 four times, we must *divide *by 10 four times. Dividing by 10 means moving the decimal point to the **left**, so moving it 4 spaces left, we get

2.8\times\dfrac{1}{10\times10\times10\times10}=0.00028

Therefore, we have concluded that 2.8\times10^{-4}=0.00028.

So, if the power is a positive number, it’s a big number, and if the power is negative, it’s a small number. Now we’re going to have a look at how to go from decimal notation to standard form.

**Example: **Write 56,700,000 in standard form.

It says above that the in standard form, your number should be between 1 and 10. So, what we need to do to write this in standard form is: move the decimal point left until this very big number becomes a number between 1 and 10, and then the number of spaces you had to move the decimal point will become the power of 10.

So, if we move the decimal point from the end of 56,700,000 **nine **spaces to the left, it becomes 5.67, which is between 1 and 10. Therefore, this number written in standard form is

5.67\times10^{9}

Remember, it’s a big number, so the power should be **positive**.

**Example: **Write 0.0000099 in standard form.

Just like in the last example, we need to move the decimal point until we get a number that falls between 1 and 10, and then count the number of times we moved it. The difference is, the power will be the **negative** of that number, since it is a very small value we’re dealing.

So, if we move the decimal point in 0.0000099 **six** spaces to the right, it becomes 9.9, which is between 1 and 10. Therefore, this number written in standard form is

9.9\times10^{-6}

Now we’ve seen what standard form is and how to get to/from it, we’re going to take a look at how to multiply/divide. two numbers together without taking them out of standard form. We’re going to be applying the **laws of indices**; rules of indices revision for more information on them.

**Example: **Find the standard form value of (3\times10^8)\times(7\times10^4).

We could change both these numbers out of standard form and then do the multiplication then put the answer back into standard form, but that’s a lot of work with no calculator. Instead, we’re just going to change the order around of the things being multiplied. We get

(3\times10^8)\times(7\times10^4)=3\times7\times10^8\times10^4

There’s no reason why we can’t do this. We can multiply 2\times5\times6 and get the same answer as 5\times6\times2 – when multiplying numbers, order doesn’t matter. We know that 3\times7=21, and using the **multiplication law** of indices, we also know that 10^8\times10^4=10^{12}. So, we get that

3\times7\times10^8\times10^4=21\times10^{12}

Great! Well, almost. This answer is not in standard form (21 is **not **between 1 and 10), and we need it to be. Fortunately, if we recognise that 21=2.1\times10, then we get that

21\times10^{12}=2.1\times10\times10^{12}=2.1\times10^{13}

2.1 is between 1 and 10, so we have successfully completed the multiplication in standard form.

**Example: **Find the standard form value of (8\times10^{-5})\div(2\times10^6).

This is the exact same idea as the last example. We’re going to break it up and divide the first numbers, 8 and 2, and the powers of 10 separately. Using the **division law** of indices this time, we get 10^{-5}\div10^6=10^{-5-6}=10^{-11}, and so we have

\dfrac{8\times10^{-5}}{2\times10^6}=\dfrac{8}{2}\times\dfrac{10^{-5}}{10^6}=4\times10^{-11}

4 is between 1 and 10, so this answer *is* in standard form, and so we are done.

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### Example Questions

1) Write 1.15\times10^{-6} in decimal notation.

Because the power is negative, this is going to be a very small number. As the power of ten is -6, we want to divide the number 1.15 by 10 six times, and so we will move the decimal point to the left six spaces. Doing so, we get

1.15\times10^{-6}=0.00000115.

2) Write 5,980,000 in standard form.

Because this is a big number, the power of 10 is going to be positive. In standard form, the number must be between 1 and 10, so we will move the decimal point to the left until we have a number between 1 and 10, and we will count the number of spaces we moved. That number will be the power of 10.

So, if we move the decimal point in 5,980,000 to the left six spaces it becomes 5.98. Therefore, we get that

5,980,000=5.98\times10^{6}

3) Find the standard form value of (2.5\times10^{4})\times(6\times10^{-9}).

We will split up this multiplication, multiplying the initial numbers together and the powers of 10 together separately. Firstly,

2.5\times6=15.

Secondly, using the multiplication law of indices,

10^4\times10^{-9}=10^{4+(-9)}=10^-5

So, we get

\begin{aligned}(2.5\times10^{4})\times(6\times10^{-9})&=2.5\times6\times10^4\times10^{-9}\\&=15\times10^{-5}\end{aligned}

We’re almost done, but this isn’t in standard form since 15 isn’t between 1 and 10. However, recognising that 15=1.5\times10, and again applying the multiplication law of indices, we get

15\times10^{-5}=1.5\times10\times10^{-5}=1.5\times10^{-4}

This is in standard form, and so we are done.

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