Standard Form

Topic Overview

Standard form, or standard index form as it is often known, is a very useful mathematical process when you need to write very large or very small numbers.

In standard form, a number is always written in the format:

\[A*{10^n}\]

In this format, 'A' is always between 1 and 10 and 'n' tells you how many places you need to move the decimal point.

For example, a very large number such as 500,000 would be written in the form 5 x 105. In this example, 'n' is 5 because we need to move the decimal point 5 places to the left so that 'A' is more than 1 but less than 10.

Similarly, a very small number such as 0.09292 would be written in the form 9.2 x 10-2. In this example, 'n' is -2 because we need to move the decimal point 2 places to the right so that 'A' is more than 1 but less than 10.

Key Concepts

In the new linear GCSE Maths paper, you will be required to work out several problems involving standard form. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

  • Approximate to specified or appropriate degrees of accuracy including a given power of 10, number of decimal places and significant figures
  • Use a calculator effectively and efficiently

Listed below are a series of summaries and worked examples to help you solidify your knowledge about standard form.

1 - How to write numbers in standard form
As mentioned in the topic overview, standard form is used to write very large or very small numbers. During your exam, you will be asked to write a series of very large or very small numbers in standard form.

Example
(a) - Write 37, 000, 000 in standard form

Solution
(a) - Using the format A x 10n , you can write 37,000 000 as:

\[3.7*10,000,000\]

This can be rewritten as:

\[3.7*10*10*10*10*10*10*10\]
\[ = 3.7*{10^7}\]

Therefore 37,000,000 written in standard form:

\[3.7*{10^7}\]

You may also be asked to convert numbers which have been written in standard form back to ordinary numbers.

Example
(b) - Convert

\[5.9*{10^4}\]

back from standard form.

Solution
(b)

\[5.9*{10^4} = 5.9*10,000 = 59,000\]

Therefore, when converted back from standard form,

\[5.9*{10^4} = 59,000\]

2 - Adding and subtracting numbers in standard form
There is a simple process which you can follow when adding and subtracting numbers in standard form:

  • Convert the values in question into ordinary numbers
  • Do the calculation
  • Change the answer back into standard form using the format A x 10^n

Example
(a) Calculate

\[3.6*{10^4} + 9.48*{10^5}\]

Solution

  • Convert the values into ordinary numbers
\[36,000 + 948,000 = 984,000\]

• Change the answer back into standard form using the format A*10^n

\[984,000 = 9.84*{10^5}\]

Therefore,

\[3.6*{10^4} + 9.48*{10^5} = 9.84*{10^5}\]

3 - Multiplying and dividing numbers in standard form When multiplying and dividing numbers which are in standard form, you can use the fundamental rules for multiplying and dividing powers:

  • When multiplying powers you add the values, e.g.
\[{10^4}*{10^3} = {10^7}\]

When dividing powers you subtract the values, e.g.

\[{10^6}/{10^4} = {10^{6-4}}=10^2\]

Example
(a) - Simplify

\[(4*{10^5})*(2*{10^4})\]

Solution
(a) - Multiply 4 by 2 and add the powers of 10:

\[(4*{10^5})*(2*{10^4}) = (4*2)*{10^{5 + 4}} = 8*{10^9}\]

Therefore,

\[(4*{10^5})*(2*{10^4})\]

can be simplified to,

\[8*{10^9}\]

Example
(b) - Simplify

\[(45*{10^6})/(9*{10^2})\]

Solution
(b) - Divide 45 by 9 and subtract the powers of 10:

\[(45*{10^6})/(9*{10^2}) = (45/9)*{10^{6 - 2}} = 5*{10^4}\]

Therefore,

\[(45*{10^6})/(9*{10^2})\]

Can be simplified to,

\[5*{10^4}\]

4 - Calculator standard form
It is important that you know how to enter standard form into your calculator.

The 'EXP' button on your calculator is used for entering exponents into your calculator.

An exponent is the term used to describe numbers which have been multiplied by a certain power of ten.

During the calculator paper of your maths exam, you may be required to enter numbers into your calculator in standard form. To do this, follow the process below:

  • Enter the number 4 × 10^15 on your calculator by typing in:

4 [EXP] 15 [=]

This will enter the value into your calculator in correct standard form. This process can be highly useful if you are asked to make multiple calculations of very large or very small numbers.

Exam Tips

  1. Memorise the format A x 10^n . Once you understand this format, you can easily apply it to any standard form exam question
  2. Always double check your answers and count the number of decimal places
  3. Remember that when multiplying you add the powers and when dividing you subtract the powers
  4. Before sitting the calculator paper, check that your calculator is functioning properly and contains new batteries
  5. Write down ALL of your working out in order to score maximum method marks

Topic Overview

Once you understand the basic principles of standard form, you will see that it is actually a simple and highly useful process. As long as you correctly place the decimal point and double check your working, standard form can be a great topic through which you can earn plenty of marks!

Related Topics

  • Arithmetic with Fractions
  • Indices
  • Problem Solving with Decimals and Percentages
  • Accuracy of Measurement Problems
  • Arithmetic with Positive and Negative Integers
  • Rational and Irrational Numbers
  • Combinations