The short answer
Scientific notation writes a number as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. It's a compact way to express very large or very small numbers — Avogadro's number is 6.022 × 10²³, and an electron's mass is about 9.11 × 10⁻³¹ kg. Converting to standard form just means shifting the decimal point n places: right for a positive exponent, left for a negative one.
Key takeaways
- Multiplying and dividing in scientific notation combines mantissas and adds or subtracts exponents; adding and subtracting requires matching exponents first.
- A positive exponent means a large number (decimal point shifts right); a negative exponent means a small number (decimal point shifts left).
- The mantissa must stay between 1 and 10 — if a calculation pushes it outside that range, shift the decimal and adjust the exponent to renormalize.
- Scientific notation and standard decimal notation always represent the exact same value — converting between them changes the appearance, not the number itself.
Converting between forms
| Standard form | Scientific notation |
|---|---|
| 4,500 | 4.5 × 10³ |
| 0.045 | 4.5 × 10⁻² |
| 602,200,000,000,000,000,000,000 | 6.022 × 10²³ |
The four operations in scientific notation
| Operation | Rule | Example |
|---|---|---|
| Multiply | Multiply mantissas, add exponents | (2×10³)(3×10⁴) = 6×10⁷ |
| Divide | Divide mantissas, subtract exponents | (6×10⁷) ÷ (3×10⁴) = 2×10³ |
| Add / Subtract | Match exponents first, then add/subtract mantissas | (3×10⁵)+(2×10⁴) = 3.2×10⁵ |
Worked example: adding numbers with different exponents
3×10⁵ + 2×10⁴
Rewrite 2×10⁴ as 0.2×10⁵ (same exponent as the first term)
3×10⁵ + 0.2×10⁵ = 3.2×10⁵
Checking in standard form: 300,000 + 20,000 = 320,000 = 3.2 × 10⁵ — matching exactly, since matching exponents first is really just finding a common "place value" before adding, the same idea as a common denominator for fractions.
Common mistakes to avoid
- Adding or subtracting mantissas without matching exponents first — this is the single most common scientific notation error.
- Leaving the mantissa outside the 1-to-10 range after a calculation instead of renormalizing — write 6.5×10⁷, not 65×10⁶.
- Mixing up which direction the decimal moves for positive vs. negative exponents — positive shifts right (bigger number), negative shifts left (smaller number).
- Treating scientific notation as a rounding technique — it's just a different way to write the exact same value, not an approximation by itself.
Related calculators
- Exponent Calculator — check the powers of 10 behind every scientific notation exponent.
- Big Number Calculator — get exact, full-precision results for numbers scientific notation would round.
- Rounding Calculator — round a mantissa to a specific number of significant figures.
- Log Calculator — connect base-10 logarithms to orders of magnitude.