📖Definition
Scientific notation is a way of writing very large or very small numbers compactly, as a number between 1 and 10 multiplied by a power of 10. It is written as a × 10ⁿ, where 1 ≤ a < 10.
📐Formula
Move the decimal point so only one non-zero digit remains before it. The number of places you moved becomes the exponent n: positive for large numbers, negative for small numbers.
📝Step-by-Step Guide
Move the decimal point
Shift the decimal point until exactly one non-zero digit remains to its left.
Count the moves
Count how many places you moved the decimal point. This count is your exponent.
Choose the sign
Use a positive exponent for numbers 10 or greater, and a negative exponent for numbers between 0 and 1.
Multiply or divide by adjusting exponents
When multiplying two numbers in scientific notation, multiply the decimal parts and add the exponents.
🧭Standard form vs. scientific notation
In UK curricula, this topic is usually called "standard form" rather than scientific notation. The rules and the a × 10ⁿ format are identical.
Both names describe the same convention used across science and engineering to handle very large numbers, like the distance to a star, or very small ones, like the size of an atom.
🧭Renormalizing after a calculation
When two decimal parts multiply to 10 or more, the result is no longer between 1 and 10, so it needs an extra adjustment.
For example, (4 × 10⁵) × (5 × 10⁻²) gives 20 × 10³ before renormalizing. Since 20 = 2 × 10¹, the exponents combine to give 2 × 10⁴.
⚠️Common Mistakes to Avoid
- Leaving more than one non-zero digit before the decimal point
- Using the wrong sign on the exponent for small numbers
- Forgetting to adjust the exponent after multiplying the decimal parts
- Miscounting the number of places the decimal point moved
- Not converting back to standard form when a question asks for it
✏️Practice Problems
Write 4500 in scientific notation
💡 Hint: Move the decimal point until one digit remains before it.
Answer: 4.5 × 10³
Write 0.00032 in scientific notation
💡 Hint: Small numbers get a negative exponent.
Answer: 3.2 × 10⁻⁴
Write 7.1 × 10⁻² in standard form
💡 Hint: A negative exponent means move the decimal point left.
Answer: 0.071
Multiply (2 × 10³) × (3 × 10²)
💡 Hint: Multiply the decimal parts and add the exponents.
Answer: 6 × 10⁵
Multiply (4 × 10⁵) × (5 × 10⁻²) and write the result in scientific notation
💡 Hint: The decimal parts multiply to 20, which needs to be renormalized.
Answer: 2 × 10⁴
Divide (9 × 10⁶) ÷ (3 × 10²)
💡 Hint: Divide the decimal parts and subtract the exponents.
Answer: 3 × 10⁴
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❓Scientific Notation FAQ
Why use scientific notation instead of writing the full number?
It keeps very large or very small numbers short and easy to compare, and it makes multiplying and dividing them far less error-prone than working with long strings of zeros.
Can the coefficient be exactly 10?
No. By definition the coefficient a must satisfy 1 ≤ a < 10, so a value of 10 is rewritten as 1.0 × 10 with the exponent adjusted up by one.
How do calculators display scientific notation?
Most calculators show it with an E or a small exponent, such as 4.5E3 or 4.5⁰³, both meaning 4.5 × 10³.