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 standard form?

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:

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

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

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:

So 350000 written in standard form is:

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:

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

Explain how to convert ordinary numbers to standard form

Converting ordinary numbers to standard form examples

Here are some examples of standard form in math:

Example 1: writing numbers in standard form with positive powers

Write this number in standard form:

1. 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….

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

3Write that power of ten as an exponent.

4Write your number in standard form.

Example 2: writing numbers in standard form with positive powers

Write this number in standard form:

Example 3: converting a small number to standard form

Write 0.00081 in standard form.

Example 4: converting a small number to standard form

Write 0.00718 in standard form.

How to convert standard form to ordinary numbers

In order to convert from standard form to ordinary numbers:

1. Convert the power of ten to an ordinary number
2. Multiply the decimal number by this power of ten
3. 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.

Example 6: converting standard form to an ordinary number

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

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:

1. Identify what power of ten the decimal number needs to be multiplied by so that the value is \pmb{1\leq{x}\lt10}.
2. 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.

Example 8: adjusting numbers to standard form

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

Example 9: adjusting numbers to standard form

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

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}.

Related lessons

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:

• Standard form
• Multiplying and dividing in standard form
• Adding and subtracting in standard form

The next lessons are

• Linear equations
• Quadratic equations
• Surds

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