Standard form

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In order to access this I need to be confident with:

Place value Arithmetic Laws of indices Negative numbers

This topic is relevant for:

GCSE Maths Number

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 ≤ a < 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

\[ 8.29\times10^{6}\]

What is standard form?

Standard form notation

Standard form notation is a representation of place value which compliments the decimal number system, as shown in the table below.


Words	Decimal form	Fraction form	Fraction form with powers of ten	Powers of ten	Standard form
Thousand	1000	1000	$10^3$	$10^3$	$1\times10^{3}$
Hundred	100	100	$10^2$	$10^2$	$1\times10^{2}$
Ten	10	10	$10^1$	$10^1$	$1\times10^{1}$
Unit	1	1	$10^0$	$10^0$	$1\times10^{0}$
Tenth	0.1	$\frac{1}{10}$	$\frac{1}{10^1}$	$10^{-1}$	$1\times10^{-1}$
Hundredth	0.01	$\frac{1}{100}$	$\frac{1}{10^2}$	$10^{-2}$	$1\times10^{-2}$
Thousand-th	0.001	$\frac{1}{1000}$	$\frac{1}{10^3}$	$10^{-3}$	$1\times10^{-3}$

Any integer or terminating decimal can be written using the standard form notation $a\times10^{n}$ . The table below shows how the value of a can remain the same while the power of ten ‘n’ changes the place value of those digits.


Decimal form	Standard form
4500	$4.5\times10^{3}$
450	$4.5\times10^{2}$
45	$4.5\times10^{1}$
4.5	$4.5\times10^{0}$
0.45	$4.5\times10^{-1}$
0.045	$4.5\times10^{-2}$
0.0045	$4.5\times10^{-3}$

Using standard form notation enables us to write very large or very small numbers.

E.g.

67 500 000 000 000 000 000 000 = 6.75\times10^{22}

0.000 000 000 000 037 = 3.7\times10^{-14}

Using standard form notation also enables us to compare the size of very large or very small numbers easily.

E.g.

Which is larger, 8560000000000 or 45320000000000?

At a glance it is difficult to tell which is larger, but written in standard form you can compare these numbers very quickly as shown.

Which is larger, $8.56\times10^{12}$ or $4.532\times10^{13}$ ?

Instantly we can see that $4.532\times10^{13}$ is the larger number as it has the higher power of ten.

Standard form calculation

When numbers are written in standard form it can make some calculations neater and quicker to compute. This is particularly the case for calculations involving multiplication, division and powers. However, this is reliant on an understanding of the rules of indices.

Multiplication rule $a^{b} \times a^{c} = a^{b+c}$

Division rule $a^{b} \div a^{c} = a^{b-c}$

Zero power rule $a^{0} = 1$

Power of power rule $(a^{b})^{c} = a^{b \times c}$

To review this topic see step-by-step guide: Laws of indices

E.g.

$340000 \times 0.002$

In standard form this changes to $(3.4 \times 10^{5}) \times (2 \times 10^{-3})$

Brackets are often used to show each separate number in standard form but they are not actually necessary $3.4 \times 10^{5} \times 2 \times 10^{-3}$

As multiplication is commutative (the order does not matter) we can reorder this calculation to $3.4 \times 2 \times 10^{5} \times 10^{-3}$

$3.4 \times 2 = 6.8$

$10^{5} \times 10^{-3} = 10^{5+-3} = 10^{2}$ (applying the multiplication rule of indices $a^{b} \times a^{c} = a^{b+c})$

$(3.4 \times 10^{5}) \times (2 \times 10^{-3}) = 6.8 \times 10^{2}$

$340000 \times 0.002 = 680$

Let’s look at how to calculate with standard form in more detail.

How to calculate with standard form

In order to write numbers in standard form

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

E.g.
Convert 4500 to standard form.

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

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

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

10⁴ = 10000
7.1 × 10000
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

Multiply or divide the non-zero numbers
Multiply or divide the powers of ten by adding or subtracting the indices
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)

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

E.g.
Work out (2 × 10³) ÷ (4 × 10⁸)

2 ÷ 4 = 0.5
10³ ÷ 10⁸ = 10^-5
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

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

E.g.
(3 × 10⁵) + (6 × 10⁴)

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

E.g.
(9 × 10⁴) − ( 3 × 10³)

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

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

Explain how to calculate with standard form

Standard form worksheet

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

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Standard form worksheet

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

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Standard form examples

Example 1: writing numbers in standard form

Write this number in standard form:

$52000$

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.

Write the exponent as a power of ten

\[10^5 = 100000\]

Multiply the decimal number by that power of ten

\[9.4 \times 100000\]

Write your answer as an ordinary number

\[9.4 \times 100000 = 940000\]

Example 3: writing a small number in standard form

Write $0.0068$ in standard form.

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 $6.8$

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

\[6.8 \times \frac{1}{1000} =0.0068\]

Write that power of ten as an exponent

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

Write your number in standard form

\[6.8\times10^{-3}\]

Example 4: multiplying numbers in standard form

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

Write your answer in standard form.

Multiply the non-zero digits

\[5 \times 7 = 35\]

Multiply the powers of ten by adding the powers

\[10^{4} \times 10^{8} = 10^{12}\]

Put these two parts together

\[35\times10^{12}\]

However, this number is not in standard form as $35$ is not a decimal number.

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

Write your answer in standard form.

Divide the non-zero digits

\[8 \div 2 = 4\]

Divide the powers of ten by subtracting the powers

\[10^{7} \div 10^{2} = 10^{5}\]

Put these two parts together

\[4\times10^{5}\]

Check your number is in standard form

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

Example 6: adding numbers in standard form

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

Write your answer in standard form.

Convert the number with the smaller power so that both numbers have the same power

\[2\times10^{3}\]

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 $10$ .

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

You will now have two numbers with the same exponent

$6\times10^4$ and $0.2\times10^4$

You now add the two non-zero numbers

\[6 + 0.2 = 6.2\]

The exponents are the same after what you did in step $1$

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

Example 7: subtracting numbers in standard form

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

Write your answer in standard form.

Convert the number with the smaller power so that both numbers have the same power

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 $10$ .

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

You will now have two numbers with the same exponent

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

You now add the two non-zero numbers

\[5 – 0.4 = 4.6\]

The exponents are the same after what you did in step $1$

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

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}

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}

59000

590

5900

0.0059

10^{3}=1000

Therefore

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

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}

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

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)

Show answer

$288, 000, 000 km$

(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)

Show answer

(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.

$62\times10^{-2}, \quad \quad 0.0068, \quad \quad 6.9\times10^{-3}, \quad \quad 0.0607$

(2 Marks)

Show answer

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

(1)

$0.0068, \quad \quad 6.9\times10^{-3}, \quad \quad 0.0607, \quad \quad 62\times10^{-2}$

(1)

Learning checklist

You have now learned how to:

Write numbers in standard form

Calculate with numbers in standard form

The next lessons are

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