Number

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Standard Form: Basics

When numbers get really big or really small, standard form helps us to make numbers shorter and easier to understand by using indices with a base of 10.

Standard form makes it easier to write really big or really small numbers.

Why do we use standard form?

The speed of light is approximately $299,792,458$ metres per second, which is close to $300,000,000$ m/s (the unit for metres per second).

As you can see, this number is quite long and difficult to read!

However, when we use standard form, the speed of light looks like this:

$3 \times 10^8$

Standard form always looks roughly the same:

$a \times 10^x$

$a$ and $x$ can change, and $a$ is a number between 0.99 and 10.

Which of the following is in correct standard form?

Let's try expressing the number $560,000$ in standard form.

The new format should be $a \times 10^x$

$a$ must be a number between $1$ and $10$ .

We need to convert $560,000$ to a number between $1$ and $10$

$560,000$ is the same as $560,000.00$ .

Let's move the decimal point to the left, so the number is between $1$ and $10$ .

How many places to the left should we move?

We've moved 5 places to the left

Therefore, we've divided the original number by $100,000$ .

Notice that $100,000$ is the same as $10^5$ .

By moving 5 places, we divide by $10^5$

Therefore, the number of places we move is the same as the index.

Now we can express in standard form

$560,000 = 5.6 \times 10^5$

What is $56,880$ in standard form to 3 significant figures?

Let's have a go at expressing $45,678$ in the form $a \times 10^x$ to 3 significant figures

$a$ must be between $1$ and $10$

Divide $45678$ by $10000$ . This leaves $4.5678$

Round $4.5678$ to 3 significant figure

$4.5678 \rightarrow 4.57 (3sf)$ as the $7$ means we need to round up.

Express $10,000$ as a power of $10$

Now express this in standard form

$45678 = 4.57 \times 10^4 (3sf)$

What is $7.23 \times 10^6$ as an ordinary number?

Small numbers can also be written in standard form, but the number of places the decimal point needs to move is written as a negative power.

0.00000375 can be written as:

$3.75 \times 10^{-6}$

Let's convert $0.0000745$ to standard form (3 significant figures)

Now we have a really small number

This time, we need to move the decimal place to the right, to get a number between $1$ and $10$ .

What is the first part of this standard form?

Find the power of 10

We moved the decimal point 5 places to the right, which means that the power of 10 is $-5$ .

Express in standard form

$0.0000745=7.45 \times 10^{-5}$

What is $0.0000004579$ in standard form to 3 significant figures?

What is $1.44 \times 10^{-7}$ as an ordinary number?