Free Standard Form Calculator with Steps
Convert between standard form and ordinary numbers with step-by-step working. Multiply, divide, add and subtract. Perfect for GCSE and A-Level maths.
Enter any number to convert to standard form (A × 10ⁿ)
How helpful was this?
Help other students find great tools
What is Standard Form?
Standard form (also called scientific notation in the US) is a way of writing very large or very small numbers using powers of 10. It is the preferred notation in UK school exams.
In GCSE maths, standard form questions appear in the "Number" unit and are worth 2–4 marks. You need to know how to convert between standard form and ordinary numbers and perform calculations.
Standard Form Format
A × 10ⁿ
Large vs Small Numbers
Large Numbers (≥ 10)
Move decimal LEFT → POSITIVE exponent
45,000 = 4.5 × 10⁴
3,200,000 = 3.2 × 10⁶
670 = 6.7 × 10²
Small Numbers (< 1)
Move decimal RIGHT → NEGATIVE exponent
0.0032 = 3.2 × 10⁻³
0.00007 = 7 × 10⁻⁵
0.45 = 4.5 × 10⁻¹
Between 1 and 10
No movement needed → exponent = 0
5.6 = 5.6 × 10⁰
1 = 1 × 10⁰
Converting to Standard Form
Follow these steps to convert any ordinary number to standard form:
- 1Identify the number — is it large (≥10) or small (<1)?
- 2Move the decimal until you get a number between 1 and 10.
- 3Count the places moved. Left = positive exponent, Right = negative.
- 4Write A × 10ⁿ. Verify by multiplying back.
Converting to Ordinary Numbers
Reverse the process: use the exponent to determine how many places to move the decimal.
- 1Identify the exponent sign: positive → move right, negative → move left.
- 2Move the decimal the absolute value of the exponent number of places.
- 3Fill empty spaces with zeros.
Multiplying & Dividing in Standard Form
Multiply: ADD exponents
(a × 10ᵐ)(b × 10ⁿ) = ab × 10^(m+n)
(3 × 10⁴)(2 × 10³) = 6 × 10⁷
(4 × 10⁵)(5 × 10⁴) = 20 × 10⁹ → 2 × 10¹⁰
Divide: SUBTRACT exponents
(a × 10ᵐ) ÷ (b × 10ⁿ) = (a÷b) × 10^(m−n)
(8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴
(9 × 10⁷) ÷ (3 × 10⁴) = 3 × 10³
Adjustment Rule
If the result coefficient falls outside 1–10, adjust: shift decimal and update exponent.
15 × 10⁶ → 1.5 × 10⁷
Adding & Subtracting in Standard Form
Unlike multiplication, you cannot simply combine exponents. The safest method is to convert both numbers to ordinary form first.
Method
- 1. Convert both to ordinary numbers
- 2. Add or subtract normally
- 3. Convert result back to standard form
(3×10⁴) + (5×10³)
= 30000 + 5000 = 35000
= 3.5 × 10⁴
(7×10⁵) − (2×10⁴)
= 700000 − 20000 = 680000
= 6.8 × 10⁵
(1.2×10³) + (8×10²)
= 1200 + 800 = 2000
= 2 × 10³
Standard Form Adjustment Rule
After performing operations, the intermediate coefficient might not be between 1 and 10. Use the adjustment rule to fix this.
Coefficient ≥ 10
Move decimal LEFT, add 1 to exponent (per move)
12 × 10⁵ → 1.2 × 10⁶
350 × 10² → 3.5 × 10⁴
Coefficient < 1
Move decimal RIGHT, subtract 1 from exponent (per move)
0.8 × 10⁴ → 8 × 10³
0.03 × 10⁶ → 3 × 10⁴
Standard Form in Real Life
🌍 Astronomy
⚗️ Chemistry
⚡ Physics
🔬 Biology
🛰️ Engineering
💻 Computing
Frequently Asked Questions
What is standard form in maths?
Standard form is A × 10ⁿ where A is between 1 and 10, and n is an integer. It's used to write very large or small numbers compactly.
How do I know if the exponent is positive or negative?
Large numbers (≥10) have positive exponents. Small numbers (<1) have negative exponents. Remember: moved LEFT = positive, moved RIGHT = negative.
What's the difference between standard form and scientific notation?
They are the same thing! "Standard form" is the UK term used in GCSE exams. "Scientific notation" is the American term.
How do I multiply in standard form?
Multiply the coefficients (A values) and ADD the exponents. Then adjust if the new coefficient isn't between 1 and 10.
How do I add numbers in standard form?
Convert both to ordinary numbers first, add them together, then convert the answer back to standard form.
Is this calculator aligned with GCSE?
Yes! It covers the entire GCSE standard form topic for Edexcel, AQA, and OCR, showing the exact working expected in exam answers.
Can standard form numbers be negative?
Yes, the coefficient A can be negative (e.g. −3.2 × 10⁴ = −32,000). The constraint is |A| ≥ 1, so the absolute value is between 1 and 10.
Why must the coefficient be between 1 and 10?
The constraint 1 ≤ A < 10 makes standard form unique — every number has exactly one representation. Without it, 45,000 could be 4.5×10⁴, 45×10³, or 450×10², creating ambiguity.
How do you compare two standard form numbers?
First compare the exponents — larger exponent means larger number. If exponents are equal, compare the coefficients. Example: 3.2×10⁵ > 9.9×10⁴.
What real-world problems use standard form?
Astronomy (distances to stars), chemistry (Avogadro's number), physics (speed of light), biology (cell sizes), and computing (memory capacities) all use standard form.
What is the adjustment rule in standard form?
After an operation, if the coefficient is ≥10 or <1, adjust: shift the decimal left (add to exponent) or right (subtract from exponent) until 1 ≤ A < 10.
How does the Learn Mode work?
Learn Mode breaks each calculation into guided steps with multiple-choice questions. Each step teaches a specific concept — direction of movement, counting places, applying index laws — without revealing the answer.
Explore More Free Tools
All our tools are 100% free with step-by-step learning