Scientific Notation Calculator
Use this free scientific notation calculator to convert between standard form and scientific notation (a × 10ⁿ), or to add, subtract, multiply, and divide numbers written in scientific notation. Get the mantissa, the exponent, the engineering notation form, the number of significant figures, and step-by-step working — perfect for chemistry, physics, engineering, and math.
How to Use the Scientific Notation Calculator
Convert mode
- Enter a number in any format — plain decimal, scientific notation (
4.5e3), or with an explicit power (4.5 × 10^3). - Choose your target format: scientific, engineering, standard decimal, or significant figures only.
- Optionally enter a number of significant figures to round the output.
- Click Calculate to see the mantissa, exponent, and step-by-step normalization.
Arithmetic mode
- Enter the first number (any supported format).
- Choose an operation: add, subtract, multiply, divide, or power.
- Enter the second number and click Calculate.
Formula & Calculation Method
Scientific notation
N = a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer
The mantissa a must satisfy 1 ≤ |a| < 10 (so negative numbers use −10 < a ≤ −1). The exponent n indicates the power of ten.
Normalization
To convert a number x to scientific notation:
- If x = 0 → result is 0 (with exponent 0).
- Otherwise compute
n = ⌊log₁₀|x|⌋. - Then
a = x / 10ⁿ, which always lies in [1, 10) in absolute value.
Engineering notation
Same idea, but the exponent must be a multiple of 3, so the mantissa satisfies 1 ≤ |a| < 1000.
Arithmetic
Let x = a × 10ᵐ and y = b × 10ⁿ.
| Operation | Rule |
|---|---|
| Multiply | (a × b) × 10^(m + n) |
| Divide | (a ÷ b) × 10^(m − n) |
| Add / Subtract | Align exponents to the smaller of m, n, then add mantissas: (a × 10^(m−n) ± b) × 10ⁿ |
| Power | (a × 10^m)^b — computed directly, then re-normalized |
After the operation, the result is re-normalized so the mantissa is again in [1, 10).
Worked Example
Example 1 — Convert 45,000 to scientific notation:
| Step | Calculation | Result |
|---|---|---|
| log₁₀ | log₁₀(45000) | ≈ 4.653 |
| Exponent n | ⌊4.653⌋ | 4 |
| Mantissa a | 45000 / 10⁴ | 4.5 |
| Result | 4.5 × 10⁴ | |
Example 2 — Multiply (3.2 × 10⁵) × (1.5 × 10⁴):
| Step | Calculation | Result |
|---|---|---|
| Multiply mantissas | 3.2 × 1.5 | 4.8 |
| Add exponents | 5 + 4 | 9 |
| Result | 4.8 × 10⁹ | |
Example 3 — Add (3 × 10⁵) + (5 × 10⁴):
| Step | Calculation | Result |
|---|---|---|
| Align exponents | 3 × 10⁵ = 30 × 10⁴ | — |
| Add mantissas | 30 + 5 | 35 |
| Combine | 35 × 10⁴ | — |
| Normalize | 3.5 × 10⁵ | |
Interpreting the Result
- Positive exponent (n > 0) — the number is large (≥ 10).
- Negative exponent (n < 0) — the number is small (0 < |x| < 1).
- Zero exponent (n = 0) — the number is in [1, 10).
- Zero — represented as 0 (with exponent 0); the mantissa is not normalized in the usual sense.
- Negative sign is carried by the mantissa, not the exponent: −4.5 × 10³, not 4.5 × 10⁻³.
- Engineering notation prefers exponents like 0, 3, 6, 9, −3, −6, −9 — useful for SI prefixes (kilo = 10³, milli = 10⁻³, etc.).
- Significant figures indicate precision — 3.20 × 10⁴ has 3 sig figs; 3.2 × 10⁴ has 2.
Assumptions and Limits
- Input numbers must be finite (no Infinity or NaN). JavaScript's double-precision floating point is used for all calculations.
- Supported input formats:
4500,4500.5,4.5e3,4.5E3,4.5 × 10^3,4.5*10^3,4.5x10^3. - Division by zero produces an error; multiplication or addition involving very large numbers (|x| > ~1e308) results in Infinity and is rejected.
- Rounding to significant figures uses standard half-up rounding; precision loss is possible with very small values.
- Exponents are displayed as Unicode superscripts for readability, but internally use standard floating-point representation.
- For very large or very small numbers, the standard form may lose precision — the scientific notation is the accurate representation.
Frequently Asked Questions
What is scientific notation?
Scientific notation expresses a number as a × 10ⁿ, where the mantissa a satisfies 1 ≤ |a| < 10 and n is an integer. It is used to handle very large or very small numbers compactly — e.g., 0.0000045 = 4.5 × 10⁻⁶.
How do I convert a number to scientific notation?
Move the decimal point so that exactly one non-zero digit precedes it; count how many places you moved. If you moved left, the exponent is positive; right, it is negative. For example: 45000 → 4.5 × 10⁴; 0.00023 → 2.3 × 10⁻⁴.
What is the difference between scientific and engineering notation?
Both use the form a × 10ⁿ. In scientific notation, 1 ≤ |a| < 10 and n can be any integer. In engineering notation, 1 ≤ |a| < 1000 and n is always a multiple of 3 — this aligns with SI prefixes (kilo, mega, milli, micro, etc.).
How do I add or subtract numbers in scientific notation?
Rewrite both numbers with the same exponent (usually the smaller one), then add or subtract the mantissas. Finally, re-normalize the result so the mantissa is in [1, 10). Example: 3×10⁵ + 5×10⁴ = 30×10⁴ + 5×10⁴ = 35×10⁴ = 3.5×10⁵.
How do I multiply or divide numbers in scientific notation?
Multiply or divide the mantissas separately, then add or subtract the exponents: (a × 10ᵐ) × (b × 10ⁿ) = (a·b) × 10^(m+n). Example: (3×10⁴) × (2×10⁵) = 6 × 10⁹.
Can scientific notation have a negative exponent?
Yes. A negative exponent means the value is a small positive (or negative) number less than 1 in absolute value. For example, 1.5 × 10⁻³ = 0.0015.
What are significant figures in scientific notation?
Significant figures indicate the precision of a measurement. In scientific notation, all digits of the mantissa are significant (except trailing zeros in integers without a decimal point). For example, 3.20 × 10⁴ has 3 sig figs; 3.2 × 10⁴ has 2 sig figs.
What is the scientific notation for zero?
Zero cannot be written in normalized scientific notation because there is no non-zero mantissa. It is conventionally written as 0 × 10⁰ or simply 0.