GCSE Maths Number

Standard Form

# Standard Form

Here we will learn about standard form including how to convert between ordinary numbers and standard form, and how to calculate with numbers in standard form.

There are also 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. It is also known as scientific notation.

Numbers in standard form are written in this format:

$a\times10^{n}$

Where a is a number 1 ≤ x < 10 and n is an integer.

To do this we need to understand the place value of a number

E.g.

Let’s look at the number 8290000 and write the digits in a place value table,

\begin{aligned} &10^6 \quad \quad 10^5 \quad \quad 10^4 \quad \quad 10^3 \quad \quad 10^2 \quad \quad 10^1 \quad \quad 10^0 \\ &\; 8 \quad \quad \quad 2 \quad \quad \quad 9 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \end{aligned}

So 8290000 written in standard form is

$8.29\times10^{6}$

## How to calculate with standard form

### In order to write numbers in standard form

1. Identify the non-zero digits and write these as a decimal number which is greater than or equal to 1 but less than 10
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
4. Write your number in standard form

E.g.
Convert 4500 to standard form.

1. Writing 4500 as a decimal between 1 and 10 is 4.5
2. 4.5 × 1000
3. 103 = 1000
4. So 4500 written in standard form is 4.5 × 103

### 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

E.g.
Convert 7.1 × 104 to standard form.

1. 104 = 10000
2. 7.1 × 10000
3. So 7.1 × 104 as an ordinary number is 71000

Step-by step guide: Converting to and from standard form

### In order to multiply and divide in standard form

1. Multiply or divide the non-zero numbers
2. Multiply or divide the powers of ten by adding or subtracting the indices
3. Write the solution in standard form, checking that the first part of the number is
1 ≤ x < 10

E.g.
Work out (4 × 105) × (2 × 108)

1. 4 × 2 = 8
2. 105 × 108 = 1013
3. 8 × 1013 is in standard form

E.g.
Work out (2 × 103) ÷ (4 × 108)

1. 2 ÷ 4 = 0.5
2. 103 ÷ 108 = 10-5
3. 0.5 × 10-5 is not in standard form

0.5 is less than 1 so multiply 0.5 by 10 which gives the result 5.

To compensate, you need to divide the power of ten by 10 which has the effect of subtracting one from the power, which gives the result 10-6.

So the answer is 5 × 10-6 .

Step-by-step guide: Multiplying and dividing in standard form

### In order to add and subtract numbers in standard form

1. Convert the number with the lower power of ten so that both numbers have the same power of ten
2. Add the non-zero integers
3. Check your answer is in standard form

E.g.
(3 × 105) + (6 × 104)

1. 6 × 104 = 0.6 × 105
2. 3 + 0.6 = 3.6
3. 3.6 × 105 is in standard form

E.g.
(9 × 104) − ( 3 × 103)

1. 3 × 103 = 0.3 × 104
2. 9 − 0.3 = 8.7
3. 8.7 × 104 is in standard form

Step-by-step guide: Adding and subtracting in standard form

## Standard formexamples

### Example 1: writing numbers in standard form

Write this number in standard form:

52000

1. The non-zero digits need to be written as a decimal number

The number needs to lie between 1≤ x <10

So the number will begin as 5.2

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

$5.2 \times 10000 = 52000$

3Write that power of ten as an exponent

$10, 000 = 10^{4}$

4Write your number in standard form

$5.2\times10^{4}$

### Example 2: writing a number in standard form as an ordinary number

Write 9.4\times10^5 as an ordinary number.

$10^5 = 100000$

$9.4 \times 100000$

$9.4 \times 100000 = 940000$

### Example 3: writing a small number in standard form

Write 0.0068 in standard form.

So the number will begin as 6.8

$6.8 \times \frac{1}{1000} =0.0068$

$\frac{1}{1000}=10^{-3}$

$6.8\times10^{-3}$

### Example 4: multiplying numbers in standard form

Calculate (5\times10^{4}) \times (7\times10^{8}) .

$5 \times 7 = 35$

$10^{4} \times 10^{8} = 10^{12}$

$35\times10^{12}$

To convert 35 to a decimal between 1 and 10 you need to divide it by 10.
To maintain the value of the number, you need to multiply the power of ten by 10, which adds one to the exponent.

\begin{aligned} &\quad \quad \quad \quad 35\times10^{12} \\\\ &\div10 ↓ \quad \quad \quad \quad \quad \quad \times10 ↓\\\\ &\quad \quad \quad \quad 3.5\times10^{13} \end{aligned}

### Example 5: dividing numbers in standard form

Calculate (8\times10^{7})\div(2\times10^{2}) .

$8 \div 2 = 4$

$10^{7} \div 10^{2} = 10^{5}$

$4\times10^{2}$

This is in standard form as 4 is greater than 1 but less than 10.

### Example 6: adding numbers in standard form

Calculate (6\times10^{4})+(2\times10^{3}) .

$2\times10^{3}$

You need to increase the power, to do this multiply it by 10 to add one to the power.  But to maintain the value of the number you need to divide the non-zero number by 10.

\begin{aligned} &\quad \quad \quad\quad \;\; 2\times10^{3} \\\\ &\div10 ↓ \quad \quad \quad \quad \quad \quad \times10 ↓\\\\ &\quad \quad \quad \quad \;\; 0.2\times10^{4} \end{aligned}

6\times10^4 and 0.2\times10^4

You now add the two non-zero numbers

$6 + 0.2 = 6.2$

So your answer in standard form is 6.2\times10^4 .

### Example 7: subtracting numbers in standard form

Calculate (5\times10^{4})−(4\times10^{3}) .

To do this identify the number with the lowest exponent.

$4\times10^{3}$

You need to increase the power. To do this multiply it by 10 to add one to the power. But to maintain the value of the number you need to divide the non-zero number by 10.

\begin{aligned} &\quad \quad \quad\quad \;\; 4\times10^{3} \\\\ &\div10 ↓ \quad \quad \quad \quad \quad \quad \times10 ↓\\\\ &\quad \quad \quad \quad \;\; 0.4\times10^{4} \end{aligned}

5\times10^{4} and 0.4\times10^{4}

You now add the two non-zero numbers

$5 – 0.4 = 4.6$

So your answer in standard form is 4.6\times10^4 .

### Common misconceptions

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

This can happen 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.

• Identifying incorrect place value with small numbers

For example, in a number such as 0.000682, selecting the ‘2’ to determine the power rather than the ‘6’ which has a higher place value can lead to …

• Not converting solutions to standard form

After calculating with standard form, a common mistake is to not check that the first part of the number is 1 a < 10.

• Not giving the solution in the correct form

It is important to check what form the question asks for the solution in – ordinary number or standard form.

### Practice standard form questions

1. Write 86,000 in standard form.

8.6 \times 10^{3}

8.6 \times 10^{4}

86 \times 10^{3}

8.6 \times 10^{-4}

The number between 1 and 10 here is 8.6. Since 8 is in the ten thousands column,

\begin{aligned} 86000&=8.6 \times 10000\\\\ &= 8.6 \times 10^{4} \end{aligned}

2. Write 0.0097 in standard form.

9.7 \times 10^{3}

9.7 \times 10^{-3}

9.7 \times 10^{-4}

0.97 \times 10^{-2}

The number between 1 and 10 here is 9.7. Since 9 is in the thousandths position,

\begin{aligned} 0.0097 &= 9.7 \times \frac{1}{1000}\\\\ &= 9.7 \times 10^{-3} \end {aligned}

3. Write 5.9 \times 10^{3} as an ordinary number.

59000

590

5900

0.0059
10^{3}=1000

Therefore

\begin{aligned} 5.9 \times 10^{3} &= 5.9 \times 1000\\\\ &=5900 \end{aligned}

4. Work out (6\times10^{8})\times(3\times10^{4}) . Write your answer in standard form.

18 \times 10^{12}

18 \times 10^{32}

1.8 \times 10^{11}

1.8 \times 10^{13}
\begin{aligned} &6 \times 3=18\\\\ &10^{8} \times 10^{4} = 10^{12} \end{aligned}

Therefore

(6 \times 10^{8}) \times (3 \times 10^{4}) = 18 \times 10^{12}

However  18 \times 10^{12} is not in standard form since 18 is not between 1 and 10.

So we need to divide 18 by 10 and, to compensate, increase the power of ten by 1.

This gives us

1.8 \times 10^{13}

5. Work out (9\times10^{7}) \div (4\times10^{2}) . Write your answer in standard form.

2.25 \times 10^{5}

22.5 \times 10^{4}

0.225 \times 10^{6}

2.25 \times 10^{3.5}
\begin{aligned} &9 \div 4 = 2.25\\\\ &10^{7} \div 10^{2}=10^{5} \end{aligned}

Therefore

(9 \times 10^{7}) \div (4 \times 10^{2}) = 2.25 \times 10^{5}

This is already in standard form since

1 \leq 2.25 <10

6. Work out (7\times10^{5})+(2\times10^{4}) . Write your answer in standard form.

7.2 \times 10^{5}

7.2 \times 10^{4}

72 \times 10^{4}

9 \times 10^{9}
2 \times 10^{4} = 0.2 \times 10^{5}

(increase the power of ten by 1 and divide the 2 by 10 to compensate)

(7.2 \times 10^{5}) + ( 0.2 \times 10^{5}) = 7.2 \times 10^{5}

### Standard form GCSE questions

1.   The distance from Earth to Mars is 2.88\times10^8km, to three significant figures. Write this number in standard form.

(1 Mark)

288, 000, 000 km

(1)

2. (a) The speed of light is 3\times10^8m/s , rounded to one significant figure. Write this as an ordinary number.

(b) The mass of Earth is 5.97\times10^{24}kg . Work out the mass of the Earth in grams. Write your answer in standard form.

(3 Marks)

(a)

300, 000, 000 m/s

(1)

(b)

Recognising 1000g in a kilogram or multiplying by 1\times10^3

(1)

5.97\times10^{27} g

(1)

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

(2 Marks)

Converting all of the numbers to the same form for comparison or orders 3 of the four numbers correctly.

(1)

(1)

## Learning checklist

You have now learned how to:

• Write numbers in standard form
• Calculate with numbers in standard form

## Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.