Standard Form Calculator
Convert any number to and from standard form (scientific notation). Enter an ordinary number or standard form parts to get an instant conversion with full step-by-step working. Also multiply and divide numbers in standard form.
where 1 ≤ A < 10 and n is an integer
(A × 10m) × (B × 10n) = (A × B) × 10m+n — adjust if A×B ≥ 10
First number
Second number
(A × 10m) ÷ (B × 10n) = (A ÷ B) × 10m−n — adjust if A÷B < 1
Dividend (numerator)
Divisor (denominator)
Common Standard Form Examples
What Is Standard Form?
Standard form — also called standard index form or scientific notation — is a method of writing numbers as a product of a coefficient and a power of ten. The format is:
where A is a number satisfying 1 ≤ A < 10 (the coefficient, also called the mantissa), and n is any integer (the exponent or index). Standard form is essential for GCSE Maths and A-Level Maths, and is used throughout science, engineering and computing to handle numbers that are either extremely large or extremely small.
Why Use Standard Form?
Standard form makes it far easier to:
- Write very large numbers such as the speed of light (3 × 108 m/s) concisely
- Write very small numbers such as the diameter of a hydrogen atom (1.2 × 10−10 m) clearly
- Compare numbers quickly by examining their powers of ten
- Multiply and divide large or small numbers using index laws
How to Convert to Standard Form (Step by Step)
Follow these three steps for any number:
- Identify the coefficient A: Move the decimal point until you have a number between 1 and 10 (not including 10). This is your A value.
- Count the moves: Count how many places you moved the decimal point. This gives |n|.
- Determine the sign of n: If the original number was ≥ 10, n is positive (you moved the decimal left). If the original number was < 1, n is negative (you moved the decimal right).
Example: Convert 3,400,000 to standard form.
Example: Convert 0.0052 to standard form.
Multiplying Numbers in Standard Form
To multiply (A × 10m) × (B × 10n):
- Multiply the coefficients: A × B
- Add the powers: m + n
- If A × B ≥ 10, divide it by 10 and increase n by 1
- If A × B < 1, multiply it by 10 and decrease n by 1
Example: (4 × 105) × (5 × 103) = 20 × 108 = 2 × 109
Dividing Numbers in Standard Form
To divide (A × 10m) ÷ (B × 10n):
- Divide the coefficients: A ÷ B
- Subtract the powers: m − n
- Adjust if the result is not in correct standard form
Example: (6 × 108) ÷ (2 × 103) = 3 × 105
Standard Form in Science
| Quantity | Value | Standard Form |
|---|---|---|
| Speed of light | 300,000,000 m/s | 3 × 108 m/s |
| Earth-Sun distance | 150,000,000,000 m | 1.5 × 1011 m |
| Diameter of atom | 0.0000000001 m | 1 × 10−10 m |
| Mass of electron | 0.00000000000000000000000000000091 kg | 9.1 × 10−31 kg |
| UK national debt | 2,700,000,000,000 | 2.7 × 1012 |
GCSE and A-Level Exam Tips
- Always check that A satisfies 1 ≤ A < 10 — a common error is leaving A = 0.34 or A = 34
- Remember that 100 = 1, so numbers between 1 and 10 can be written as A × 100
- When adding/subtracting in standard form, first convert both to the same power before combining
- On a calculator, standard form is often entered using the EXP or ×10x button
- The term “standard form” in UK maths always refers to A × 10n; do not confuse with the US use of “standard form” for linear equations
Frequently Asked Questions
Standard form (also called scientific notation) is a way of writing very large or very small numbers in the format A × 10n, where 1 ≤ A < 10 and n is an integer. For example, 3,400,000 written in standard form is 3.4 × 106, and 0.0052 written in standard form is 5.2 × 10−3. It is widely used in science, engineering and GCSE Maths.
To convert a number to standard form: 1) Move the decimal point until you have a number A where 1 ≤ A < 10. 2) Count how many places you moved the decimal point — this gives the power n. 3) If you moved left, n is positive; if you moved right, n is negative. For example, 56,000: move the decimal 4 places left to get 5.6, so the answer is 5.6 × 104.
To multiply two numbers in standard form (A × 10m) × (B × 10n): multiply the A values together and add the powers. So (3 × 104) × (2 × 103) = 6 × 107. If the product of the A values is ≥ 10, adjust: for example (4 × 105) × (5 × 103) = 20 × 108 = 2 × 109.
To divide two numbers in standard form (A × 10m) ÷ (B × 10n): divide the A values and subtract the powers. So (8 × 106) ÷ (2 × 102) = 4 × 104. If the quotient of the A values is less than 1, adjust: for example (3 × 105) ÷ (6 × 102) = 0.5 × 103 = 5 × 102.
Standard form and standard index form are the same thing in UK mathematics — both refer to writing a number as A × 10n where 1 ≤ A < 10. The term ‘standard index form’ emphasises that 10n is written using index notation. In the USA this is called ‘scientific notation’. All three terms describe the identical mathematical concept and format.
To add or subtract numbers in standard form, first convert them to the same power of 10, then add or subtract the A values. For example, 3.4 × 106 + 2.1 × 105 = 3.4 × 106 + 0.21 × 106 = 3.61 × 106. Alternatively, convert both numbers to ordinary form, add them, then convert the answer back to standard form.
Standard form is used in science because it makes very large or very small numbers much easier to write, compare and calculate with. For example, the distance from Earth to the Sun is approximately 1.5 × 1011 metres — far cleaner than 150,000,000,000. It also makes the order of magnitude immediately obvious and simplifies multiplication and division using the laws of indices.