Scientific Notation Calculator
Convert any number into scientific notation, e-notation, and engineering notation at once — or do arithmetic on two numbers already in m × 10ⁿ form, with the mantissa/exponent working shown at each step.
Three notations, one number
Scientific notation (3.45 × 10⁻⁴) keeps one digit before the point — ideal for math class and significant figures. E-notation (3.45e-4) is the same thing made typeable, used by every spreadsheet and programming language. Engineering notation forces the exponent to a multiple of three (345 × 10⁻⁶) so the value maps straight onto SI prefixes: 345 µ, 4.08 k, 2.998 × 10⁸ m/s → 299.8 × 10⁶. This page always shows all three, so you can move between a lab report, a datasheet, and code without slips.
How it’s calculated
Conversion slides the decimal point until exactly one nonzero digit leads, counting the slides as the exponent (left slides positive, right slides negative). Multiplication multiplies mantissas and adds exponents; division divides and subtracts; addition and subtraction first rewrite both terms over the larger exponent, then combine mantissas and re-normalize if needed.
Mantissas display up to 10 significant digits; the decimal-form line is omitted beyond 10±²⁰ to avoid absurdly long strings.
Worked example
0.000345: slide the decimal four places right to get one leading nonzero digit — 3.45 × 10⁻⁴, or 3.45e-4, or 345 × 10⁻⁶ in engineering form. Arithmetic: (1.2 × 10⁵) × (3.4 × 10⁻²) multiplies mantissas (4.08) and adds exponents (5 + (−2) = 3) for 4.08 × 10³; and 5.2 × 10³ + 3.1 × 10² = 5.2 × 10³ + 0.31 × 10³ = 5.51 × 10³.
Common mistakes
- Adding exponents when adding numbers — exponents only add under multiplication.
- Leaving the mantissa outside 1–10 (34.5 × 10⁻⁵ is the right value but not normalized scientific form).
- Reading 3.45e-4 as involving Euler’s e — in e-notation the letter means “×10^”.
- Flipping the exponent sign: small numbers get negative exponents, large numbers positive.
Where it is used
- Chemistry and physics: Avogadro’s number (6.022 × 10²³), Planck’s constant, atomic scales.
- Engineering: component values that map to SI prefixes (kΩ, µF, nm).
- Astronomy: distances and masses spanning dozens of orders of magnitude.
- Spreadsheets and code that print 1.23E+08 and need translating.
Frequently asked questions
What is scientific notation?
A number written as m × 10ⁿ where the mantissa m has exactly one nonzero digit before the decimal point (1 ≤ |m| < 10). So 0.000345 becomes 3.45 × 10⁻⁴ and 299,800,000 becomes 2.998 × 10⁸. It keeps very large and very small values readable and comparable at a glance.
What is the difference between e-notation and scientific notation?
They are the same idea in different clothes: 3.45 × 10⁻⁴ is written 3.45e-4 in e-notation, the form calculators, spreadsheets, and programming languages use because superscripts are hard to type. The letter e here means “times ten to the”, not Euler’s number.
What is engineering notation?
A variant where the exponent must be a multiple of 3, so the mantissa runs from 1 to 999.99. 0.000345 becomes 345 × 10⁻⁶, which reads directly as 345 micro-units — engineering notation lines up with SI prefixes (kilo, mega, milli, micro).
How do I multiply numbers in scientific notation?
Multiply the mantissas and add the exponents: (1.2 × 10⁵) × (3.4 × 10⁻²) = 4.08 × 10³. For division, divide mantissas and subtract exponents. Re-normalize if the mantissa leaves the 1–10 range.
How do I add or subtract in scientific notation?
First rewrite both numbers with the same exponent, then add the mantissas: 5.2 × 10³ + 3.1 × 10² = 5.2 × 10³ + 0.31 × 10³ = 5.51 × 10³. The calculator shows the alignment step explicitly.