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Place value Arithmetic Laws of indices Negative numbers

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GCSE Maths Number

Standard Form

Converting to and from Standard Form

Here we will learn how to convert to and from standard form, including how to adjust numbers to write them in standard form notation.

There are also converting to and from standard form worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is converting to and from standard form?

Converting to and from standard form is where we convert an ordinary number to a number written in standard form or scientific notation.

Standard form is a way of writing very large or very small numbers by comparing the powers of ten.
Numbers in standard form are written in this format:

\[ a\times10^{n}\]

Where $a$ is a number $1\leq{a}\lt10$ and $n$ is an integer.

Converting between ordinary numbers and numbers in standard form can help us to compare numbers and interpret answers given in standard form on a calculator.
To do this we need to understand the place value of a number.

E.g.
Let’s look at the number $350000$ and place the digits in a place value table:

\[\begin{aligned} &10^5 \quad \quad 10^4 \quad \quad 10^3\quad \quad 10^2 \quad \quad 10^1 \quad \quad 10^0 \\ &\; 3 \quad \quad \quad 5 \quad \quad \quad 0 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \end{aligned}\]

So 350000 written in standard form is:

\[3.5\times10^{5}\]

What is converting to and from standard form?

How to convert ordinary numbers to standard form

In order to convert ordinary numbers to standard form:

Identify the non-zero digits and write these as a decimal number which is $\pmb{ 1\leq{x}\lt10}$ .
In order to maintain the place value of the number, this decimal number needs to be multiplied by a power of ten.
Write the power of ten as an exponent.

Explain how to convert ordinary numbers to standard form

Standard form calculator worksheet

Get your free standard form calculator worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Standard form calculator worksheet

Get your free standard form calculator worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Converting ordinary numbers to standard form examples

Example 1: writing numbers in standard form with positive powers

Write this number in standard form:

480000

The non-zero digits need to be written in decimal notation.

The number needs to lie between $1\leq{x}\lt10$

So the number will begin as $4.8\dots$ .

2You now need to maintain the value of the number by multiplying that decimal by a power of ten.

\[4.8 \times 100000 = 480000\]

3Write that power of ten as an exponent.

\[100000 = 10^{5}\]

4Write your number in standard form.

\[4.8\times10^{5}\]

Example 2: writing numbers in standard form with positive powers

Write this number in standard form:

5420000

The non-zero digits need to be written in decimal notation.

The number needs to lie between $1\leq{x}\lt10$

So the number will begin as $5.42\dots$ .

You now need to maintain the value of the number by multiplying that decimal by a power of ten.

\[5.42 \times 1000000 = 5420000\]

Write that power of ten as an exponent.

\[1000000 = 10^{6}\]

Write your number in standard form.

\[5.42\times10^{6}\]

Example 3: converting a small number to standard form

Write $0.00081$ in standard form.

Identify the non-zero digits and write these as a decimal number which is $\pmb{1\leq{x}\lt10}.$

This number will begin with $8.1...$

Identify what power of ten the decimal needs to be multiplied by in order to preserve place value.

$8.1 \div 10000 = 0.00081$

Write the power of ten as an exponent.

\[\frac{1}{10000} = 10^{-4}\]

Write your number in standard form.

\[8.1\times10^{-4}\]

Example 4: converting a small number to standard form

Write $0.00718$ in standard form.

Identify the non-zero digits and write these as a decimal number which is $\pmb{{1\leq{x}\lt10}.}$

The number will begin with 7.18…

Identify what power of ten the decimal needs to be multiplied by in order to preserve place value.

$7.8 \div 1000 = 0.0078$

Therefore,

\[7.18 \times \frac{1}{1000}=0.00718\]

Write the power of ten as an exponent.

\[\frac{1}{1000} = 10^{-3}\]

Write your number in standard form.

\[7.18\times10^{-3}\]

How to convert standard form to ordinary numbers

In order to convert from standard form to ordinary numbers:

Convert the power of ten to an ordinary number
Multiply the decimal number by this power of ten
Write your number as an ordinary number

Explain how to convert standard form to ordinary numbers

Converting standard form to ordinary numbers examples

Example 5: converting standard form to an ordinary number

Write $6.2\times10^4$ as an ordinary number.

Write the exponent as a power of ten.

\[10^{4} = 10000\]

Multiply the decimal number by that power of ten.

\[6.2 \times 10000\]

Write your answer as an ordinary number.

\[6.2 \times 10000 = 62000\]

Example 6: converting standard form to an ordinary number

Write $1.9\times10^{-3}$ as an ordinary number.

Write the exponent as a power of ten.

\[10^{-3} = 0.001\]

Multiply the decimal number by that power of ten.

\[1.9 \times 0.001\]

Write your answer as an ordinary number.

\[1.9 \times 0.001 = 0.0019\]

How to adjust numbers to standard form

Sometimes we might have a number that looks like it is in standard form however the decimal number is not between $1$ and $10$ ,
E.g.
$36103$ or $0.2104$ .
In this case we need to adjust the number.

In order to adjust numbers to standard form:

Identify what power of ten the decimal number needs to be multiplied by so that the value is $\pmb{1\leq{x}\lt10}.$
Apply the inverse of this to the power of ten.

Adjusting numbers to standard form examples

Example 7: adjusting number in standard form

Write $48\times10^5$ in standard form.

Identify what the first number needs to be multiplied or divided by so that it lies between $\pmb{1\leq{x}\lt10}.$

$48$ needs to be divided by $10$ so $48$ becomes $4.8$ .

Apply the inverse operation to the power of ten.

$10^5$ needs to be multiplied by $10$ which adds one to its power, so it becomes $10^6$ .

Write your number in standard form.

\[ 4.8\times10^{6}\]

Example 8: adjusting numbers to standard form

Write $0.68\times10^{4}$ in standard form.

Identify what the first number needs to be multiplied or divided by so that it lies between $\pmb{1\leq{x}\lt10}.$

$0.68$ needs to be multiplied by $10$ so it becomes $6.8$ .

Apply the inverse operation to the power of ten.

$10^4$ needs to be divided by $10$ which subtracts one from its power, so it becomes $10^3$ .

Write your number in standard form.

\[ 6.8\times10^{3}\]

Example 9: adjusting numbers to standard form

Write $290\times10^{-4}$ in standard form.

Identify what the first number needs to be multiplied or divided by so that it lies between $\pmb{1\leq{x}\lt10}.$

$290$ needs to be divided by $100$ so it becomes $2.9$ .

Apply the inverse operation to the power of ten.

$10^{-4}$ needs to be multiplied by $100$ which adds two to its power, so it becomes $10^{-2}$ .

Write your number in standard form.

\[ 2.9\times10^{-2}\]

Common misconceptions

Writing a number with the incorrect power for a large or small number

This error is often made by counting the zeros following the first non zero digit for large numbers or zeros after the decimal point for small numbers, then writing this as the power, rather than considering the place value of the given number.

Identifying incorrect place value with small numbers

In a number such as $0.000682$ , selecting the $‘2’$ to determine the exponent rather than the $‘6’$ which has a higher place value.
In standard form, this number would be $6.82$ × $10^{-4}$ .

Errors with negative numbers

When checking the standard form of a number, incorrectly adjusting the negative powers due to not applying negative numbers rules correctly.
E.g.
With small numbers, adding one to the power of $10^{-5}$ will result in $10^{-4}$ not $10^{-6}$ .

Converting to and from standard form is part of our series of lessons to support revision on standard form. You may find it helpful to start with the main standard form lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice standard form calculator questions

27 \times 10^{4}

2.7 \times 10^{4}

2.7 \times 10^{5}

0.27 \times 10^{6}

The number between $1$ and $10$ here is $2.7.$

\begin{aligned} 27000&=2.7 \times 100000\\\\ &=2.7 \times 10^{5} \end{aligned}

7.9 \times 10^{-3}

7.9 \times 10^{-4}

7.9 \times 10^{-5}

0.79 \times 10^{-4}

The number between $1$ and $10$ here is $7.9.$

\begin{aligned} 0.00079&=7.9 \times \frac{1}{10000}\\\\ &=7.9 \times 10^{-4} \end{aligned}

61000

6100

0.61 \times 10^{5}

610000

10^{4}=10000

Therefore,

\begin{aligned} 6.1 \times 10^{4} &= 6.1 \times 10000\\\\ &=61000 \end{aligned}

380000

0.00038

0.0000038

0.000038

10^{-5}=\frac{1}{100000}

Therefore,

\begin{aligned} 3.8 \times 10^{-5} &= 3.8 \times \frac{1}{100000}\\\\ &=3.8 \div 100000\\\\ &=0.000038 \end{aligned}

840

8.4 \times 10^{2}

8.4 \times 10^{3}

8.4 \times 10

This number is not in standard form as $84$ is not between $1$ and $10.$

We need to divide $84$ by $10$ and, to compensate, multiply $10^2$ by $10$ , increasing the power by $1.$

This gives us

8.4 \times 10^{3}

9.2 \times 10^{-4}

9.2 \times 10^{-5}

9.2 \times 10^{-6}

0.92 \times 10^{-4}

This number is not in standard form because $0.92$ is not between $1$ and $10.$

We need to multiply $0.92$ by $10$ and, to compensate, divide $10^{-5}$ by $10$ , decreasing the power by $1.$

This gives us

9.2 \times 10^{-6}

Standard form calculator GCSE questions

(a) Write $8.23\times10^{-6}$ , as an ordinary number.

(b) Write the number $0.00702$ in standard form.

(2 marks)

Show answer

(a) $0.00000823$

(1)

(b) $7.02\times10^{-3}$

(1)

(a) The population of the USA is $3.3\times10^{8}$ , rounded to two significant figures.

Write this distance as an ordinary number.

(b) The population of Washington DC is $690000$ rounded to two significant figures.

Write this number in standard form.

(2 marks)

Show answer

(a) $3 30000000 km$

(1)

(b) $6.9\times10^{5}km$

(1)

3. Put the below numbers in order. Start with the smallest number.

$0.092 \quad \quad 2.9\times10^{-3} \quad \quad 0.00029 \quad \quad 209\times10^{-4}$

(2 marks)

Show answer

Converting all of the numbers to the same form or standard notation for comparison or $3$ of the four numbers ordered correctly.

(1)

$0.00029, \quad \quad 2.9\times10^{-3} , \quad \quad 209\times10^{-4}, \quad \quad 0.092$

(1)

Learning checklist

You have now learned how to:

Convert ordinary numbers standard form

Convert standard form to ordinary numbers

Adjust numbers to standard form notation

The next lessons are

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