Scientific notation writes a number as a coefficient between 1 and 10 times a power of ten: the speed of light is 3.00 × 10⁸ m/s, a hydrogen atom’s mass is about 1.67 × 10⁻²⁷ kg. Arithmetic with numbers like these is mostly bookkeeping of exponents, and that’s exactly where errors creep in. This calculator performs addition, subtraction, multiplication and division on two numbers in scientific notation, normalizes the answer, and lays out each coefficient and exponent step.
How to use the scientific notation calculator
- Enter the first number’s coefficient and power of 10. For 3.2 × 10⁵, type 3.2 and 5.
- Pick the operation.
- Enter the second number the same way. Negative exponents are fine: 4.5 × 10⁻³ is 4.5 and −3.
- Optionally round the answer to a number of significant figures.
- The tape shows the normalized result, the E-notation your calculator would display, engineering notation and the ordinary decimal form.
Coefficients don’t have to be normalized when you enter them — 32 × 10⁴ is converted to 3.2 × 10⁵ automatically, and the steps say so.
The rules
After any operation, normalize: if the coefficient is 10 or more, divide it by 10 and add 1 to the exponent; if it is less than 1, multiply by 10 and subtract 1.
Worked examples
Multiplication
(3.2 × 10⁵) × (4.5 × 10⁻³)
Coefficients: 3.2 × 4.5 = 14.4
Exponents: 5 + (−3) = 2
Combine and normalize: 14.4 × 10² = 1.44 × 10³ (that is, 1,440)
Addition
(3.2 × 10⁵) + (4.5 × 10⁴)
Match exponents: 4.5 × 10⁴ = 0.45 × 10⁵
Add coefficients: 3.2 + 0.45 = 3.65
Answer: 3.65 × 10⁵
A physics example
How heavy is a mole of protons? Multiply Avogadro’s number, 6.022 × 10²³, by the proton mass, 1.673 × 10⁻²⁷ kg: 6.022 × 1.673 ≈ 10.07, and 23 + (−27) = −4, so the product is 10.07 × 10⁻⁴ = 1.007 × 10⁻³ kg — about one gram, which is why hydrogen’s molar mass is close to 1 g/mol.
Significant figures in scientific notation
Every digit of the coefficient counts as significant, which makes scientific notation the cleanest way to report precision. When you multiply or divide measurements, round the answer to the fewest significant figures among the inputs — choose that count in the rounding menu. The significant figures calculator explains the rules in detail.
Reading calculator displays
Most calculators and spreadsheets show 1.44 × 10³ as 1.44E3 or 1.44e+3, and 4.5 × 10⁻³ as 4.5E-3. The E stands for “times ten to the power”. To convert between scientific, E-notation, engineering notation and ordinary decimals for a single number, use the scientific notation converter.
Common mistakes
- Adding exponents when adding numbers. Exponents are added only when multiplying. For addition, match them first.
- Forgetting to normalize. 14.4 × 10² is correct in value but not in standard scientific form.
- Sign slips with negative exponents. Dividing by 10⁻³ means subtracting −3, which adds 3.
- Losing the small term. Adding 10⁻³ to 10⁵ changes only the ninth significant digit; with rounding, the sum may look identical to the larger number.
Frequently asked questions
How do you multiply numbers in scientific notation?
Multiply the coefficients and add the exponents, then normalize. (3.2 × 10⁵) × (4.5 × 10⁻³) = 14.4 × 10² = 1.44 × 10³.
How do you divide numbers in scientific notation?
Divide the coefficients and subtract the exponents. (8.4 × 10⁶) ÷ (2.1 × 10²) = 4 × 10⁴. If the new coefficient is less than 1, move the decimal point and lower the exponent to normalize.
How do you add numbers in scientific notation?
Rewrite them with the same power of ten first, then add the coefficients. 3.2 × 10⁵ + 4.5 × 10⁴ = 3.2 × 10⁵ + 0.45 × 10⁵ = 3.65 × 10⁵.
What does it mean to normalize scientific notation?
Adjust the number so the coefficient is at least 1 and less than 10. 14.4 × 10² becomes 1.44 × 10³ by moving the decimal point one place left and adding 1 to the exponent.
Can the exponents be very large?
Yes. This calculator handles exponents into the thousands without overflowing, because it tracks the coefficient and the power of ten exactly rather than converting to an ordinary computer number.