Standard form is a way of writing very large or very small numbers in a way that allows us to compare the biggest and smallest of numbers, and without loads of zeros. Instead of writing out all those zeros, we use powers of 10 to express the number in a more compact way. It’s dead useful in science, where you might be dealing with huge distances like the size of the universe. Or the tiniest of measurements, like the width of atom.

TLDR

Standard form is written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer
For large numbers (≥ 10), n is positive
For small numbers (< 1), n is negative
To convert to standard form:
Move the decimal point to get a number between 1 and 10
Count how many places you moved it
If you moved it left, n is positive; if right, n is negative
To convert from standard form:
If n is positive, move the decimal point n places right
If n is negative, move the decimal point |n| places left
For calculations, either convert to ordinary numbers first or use the laws of indices
Make sure your calculator is in scientific mode for working with standard form

What You Need to Know

You need to be able to calculate with and interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer
This includes working with and without a calculator
You should be able to interpret calculator displays involving standard form

This topic is in both Foundation and Higher tiers.

Key Facts

🔑 Key Fact: Standard form is always written as A × 10ⁿ where A is a number between 1 and 10 (including 1, but not 10), and n is an integer.

🔑 Key Fact: For large numbers (≥ 10), n is positive. For small numbers (< 1), n is negative.

🔑 Key Fact: When you move the decimal point to the left, the power of 10 is positive; when you move it to the right, the power is negative.

🔑 Key Fact: Some calculators display standard form as, for example, 2.5E6 which means 2.5 × 10⁶.

Converting to Standard Form

To convert a number to standard form:

Move the decimal point so that there is exactly one non-zero digit to the left of the decimal point
Count how many places the decimal point has moved
If the original number was ≥ 10 (you moved the decimal point left), n is positive
If the original number was < 1 (you moved the decimal point right), n is negative

Example 1

Convert 4,580,000 to standard form.

Step 1: Move the decimal point to get a number between 1 and 10.

4,580,000 → 4.58

Step 2: Count how many places the decimal point moved.

It moved 6 places to the left

Step 3: Since the decimal point moved left, the power of 10 is positive.

n = 6

Therefore, 4,580,000 = 4.58 × 10⁶

Example 2

Convert 0.00372 to standard form.

Step 1: Move the decimal point to get a number between 1 and 10.

0.00372 → 3.72

Step 2: Count how many places the decimal point moved.

It moved 3 places to the right

Step 3: Since the decimal point moved right, the power of 10 is negative.

n = -3

Therefore, 0.00372 = 3.72 × 10⁻³

Converting from Standard Form

To convert from standard form back to an ordinary number:

If n is positive, move the decimal point n places to the right
If n is negative, move the decimal point |n| places to the left
Add zeros if needed

Example 3

Convert 7.83 × 10⁵ to an ordinary number.

Step 1: Identify A and n.

A = 7.83
n = 5 (positive)

Step 2: Move the decimal point 5 places to the right.

7.83 → 783000 (adding zeros as needed)

Therefore, 7.83 × 10⁵ = 783,000

Example 4

Convert 8.2 × 10⁻⁴ to an ordinary number.

Step 1: Identify A and n.

A = 8.2
n = -4 (negative)

Step 2: Move the decimal point 4 places to the left.

8.2 → 0.00082 (adding zeros as needed)

Therefore, 8.2 × 10⁻⁴ = 0.00082

Calculations with Standard Form

Multiplication

When multiplying numbers in standard form, multiply the A values and add the powers of 10.

(A₁ × 10ⁿ¹) × (A₂ × 10ⁿ²) = (A₁ × A₂) × 10^(n₁ + n₂)

If the result gives an A value ≥ 10, adjust by moving the decimal point and increasing the power of 10.

Example 5

Calculate (3.6 × 10⁴) × (2.5 × 10⁻²).

Step 1: Multiply the A values.

3.6 × 2.5 = 9

Step 2: Add the powers of 10.

n = 4 + (-2) = 2

Step 3: Write the result in standard form.

Result = 9 × 10²

Therefore, (3.6 × 10⁴) × (2.5 × 10⁻²) = 9 × 10²

Division

When dividing numbers in standard form, divide the A values and subtract the powers of 10.

(A₁ × 10ⁿ¹) ÷ (A₂ × 10ⁿ²) = (A₁ ÷ A₂) × 10^(n₁ – n₂)

If the result gives an A value < 1, adjust by moving the decimal point and decreasing the power of 10.

Example 6

Calculate (8.4 × 10⁷) ÷ (2.1 × 10³).

Step 1: Divide the A values.

8.4 ÷ 2.1 = 4

Step 2: Subtract the powers of 10.

n = 7 – 3 = 4

Step 3: Write the result in standard form.

Result = 4 × 10⁴

Therefore, (8.4 × 10⁷) ÷ (2.1 × 10³) = 4 × 10⁴

Addition and Subtraction

For addition and subtraction, the numbers must have the same power of 10:

Convert both numbers to the same power of 10
Add or subtract the A values
Keep the same power of 10
Ensure the result is in standard form by adjusting if necessary

Example 7

Calculate (5.7 × 10⁶) + (3.2 × 10⁵).

Step 1: Convert both numbers to the same power of 10.

5.7 × 10⁶
3.2 × 10⁵ = 0.32 × 10⁶ (divide A by 10, increase n by 1)

Step 2: Add the A values with the same power.

(5.7 + 0.32) × 10⁶ = 6.02 × 10⁶

Therefore, (5.7 × 10⁶) + (3.2 × 10⁵) = 6.02 × 10⁶

Example 8

Calculate (7.5 × 10⁻³) – (8.2 × 10⁻⁴).

Step 1: Convert both numbers to the same power of 10.

7.5 × 10⁻³
8.2 × 10⁻⁴ = 0.82 × 10⁻³ (divide A by 10, increase n by 1)

Step 2: Subtract the A values with the same power.

(7.5 – 0.82) × 10⁻³ = 6.68 × 10⁻³

Therefore, (7.5 × 10⁻³) – (8.2 × 10⁻⁴) = 6.68 × 10⁻³

Calculator Use

Most scientific calculators have a special button for entering numbers in standard form. It might be labeled as "EXP", "EE", or "×10ˣ".

Example 9

To enter 4.58 × 10⁶ on a calculator:

Press 4.58
Press the EXP or ×10ˣ button
Press 6

The display might show "4.58E6" or "4.58 06".

To interpret calculator displays:

"4.58E6" means 4.58 × 10⁶
"3.72E-3" means 3.72 × 10⁻³

Common Mistakes to Avoid

Getting the sign of the power wrong:
Incorrect: Writing 0.0045 as 4.5 × 10³
Correct: 0.0045 = 4.5 × 10⁻³ (moving decimal right means negative power)
Having A outside the range 1 ≤ A < 10:
Incorrect: Writing 45,000 as 45 × 10³
Correct: 45,000 = 4.5 × 10⁴
Adding/subtracting without converting to the same power:
Incorrect: (3 × 10⁴) + (2 × 10³) = 5 × 10⁷
Correct: (3 × 10⁴) + (0.2 × 10⁴) = 3.2 × 10⁴
Forgetting to adjust after calculations:
Incorrect: 6 × 4 × 10⁵ = 24 × 10⁵
Correct: 6 × 4 × 10⁵ = 2.4 × 10⁶
Misinterpreting calculator displays:
Incorrect: Reading 4.58E6 as 4.58 × 10^-6
Correct: 4.58E6 means 4.58 × 10⁶

Questions

Try these questions to practice working with standard form:

Convert the following numbers to standard form:

a) 34,500,000

b) 0.000783
Convert the following numbers from standard form to ordinary numbers:

a) 7.32 × 10⁵

b) 9.05 × 10⁻⁴
Calculate the following, giving your answers in standard form:

a) (2.4 × 10⁶) × (5 × 10⁴)

b) (9.6 × 10⁸) ÷ (3.2 × 10³)
Calculate the following, giving your answers in standard form:

a) (8.3 × 10⁴) + (4.2 × 10³)

b) (5.7 × 10⁻²) – (3.8 × 10⁻³)
The distance from the Earth to the Sun is approximately 149,600,000 km. The speed of light is about 300,000 km/s. How long does it take light to travel from the Sun to the Earth? Give your answer in standard form and then convert to minutes.
A molecule of water has a mass of approximately 0.0000000000000000000029 grams. How many water molecules are there in 1 gram of water? Give your answer in standard form.

Solutions

Question 1

a) 34,500,000 = 3.45 × 10⁷

b) 0.000783 = 7.83 × 10⁻⁴

Question 2

a) 7.32 × 10⁵ = 732,000

b) 9.05 × 10⁻⁴ = 0.000905

Question 3

a) (2.4 × 10⁶) × (5 × 10⁴)

= (2.4 × 5) × 10^(6 + 4)

= 12 × 10¹⁰

= 1.2 × 10¹¹ (adjusting to standard form)

b) (9.6 × 10⁸) ÷ (3.2 × 10³)

= (9.6 ÷ 3.2) × 10^(8 – 3)

= 3 × 10⁵

Question 4

a) (8.3 × 10⁴) + (4.2 × 10³)

= (8.3 × 10⁴) + (0.42 × 10⁴)

= 8.72 × 10⁴

b) (5.7 × 10⁻²) – (3.8 × 10⁻³)

= (5.7 × 10⁻²) – (0.38 × 10⁻²)

= 5.32 × 10⁻²

Question 5

Step 1: Convert the distance and speed to standard form.

Distance = 149,600,000 km = 1.496 × 10⁸ km
Speed = 300,000 km/s = 3 × 10⁵ km/s

Step 2: Calculate the time (distance ÷ speed).

Time = (1.496 × 10⁸) ÷ (3 × 10⁵)
Time = (1.496 ÷ 3) × 10^(8 – 5)
Time = 0.49867 × 10³
Time = 4.9867 × 10² seconds (in standard form)

Step 3: Convert to minutes.

4.9867 × 10² seconds = 498.67 seconds
498.67 ÷ 60 = 8.31 minutes

Therefore, light takes 4.9867 × 10² seconds, or about 8.31 minutes, to travel from the Sun to the Earth.

Question 6

Step 1: Convert the mass of one water molecule to standard form.

Mass of one molecule = 0.0000000000000000000029 g
Mass of one molecule = 2.9 × 10⁻²³ g

Step 2: Calculate how many molecules in 1 gram (1 g ÷ mass of one molecule).

Number of molecules = 1 ÷ (2.9 × 10⁻²³)
Number of molecules = 1 × 10⁰ ÷ (2.9 × 10⁻²³)
Number of molecules = (1 ÷ 2.9) × 10^(0 – (-23))
Number of molecules = 0.3448… × 10²³
Number of molecules = 3.448… × 10²²

Therefore, there are approximately 3.45 × 10²² water molecules in 1 gram of water.

Summary

Standard form is a way of writing very large or very small numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer
For numbers ≥ 10, n is positive; for numbers < 1, n is negative
To convert to standard form, move the decimal point to get a number between 1 and 10, then count the number of places moved
For calculations:
Multiplication: multiply the A values and add the powers
Division: divide the A values and subtract the powers
Addition/Subtraction: convert to the same power first, then add/subtract the A values
Make sure your answers are in proper standard form, adjusting if necessary
Remember that standard form is used extensively in science to express very large and very small quantities

Standard form might seem a bit fiddly at first, but it’s one of those skills that becomes second nature with practice. Master it, and you’ll find working with extreme values in physics, chemistry, and astronomy much more manageable.

GCSE Maths: Standard Form