Scientific Notation Calculator

How it works

x = m × 10ᵉ with 1 ≤ |m| < 10 · (a × 10ᵐ)(b × 10ⁿ) = ab × 10ᵐ⁺ⁿ · (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ · a × 10ᵐ + b × 10ⁿ = (a + b × 10ⁿ⁻ᵐ) × 10ᵐ for m ≥ n · (a × 10ᵐ)ᵏ = aᵏ × 10ᵐᵏ · √(a × 10²ᵐ) = √a × 10ᵐ

Scientific notation writes a number as a coefficient (mantissa) between 1 and 10 times a power of ten. The exponent counts how many places the decimal point moved: positive for big numbers, negative for small ones. E notation is the same thing typed on a keyboard (4.5e+6). Engineering notation keeps the exponent a multiple of 3 so it lines up with SI prefixes such as kilo, mega, micro and nano. To multiply, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents. To add or subtract, first rewrite both numbers with the same power of ten. A power multiplies the exponent, and a square root needs an even exponent, which can then be halved. After each operation the answer is normalized back to one non-zero digit in front of the point. Under More options you can round to a fixed number of significant figures, or apply the usual rules: the fewest significant figures for × and ÷, and the coarsest decimal place for + and −. The coefficient and exponent are kept apart, so answers far beyond a pocket calculator’s range (such as 10⁵⁰⁰) still work.

Worked example

4,500,000 = 4.5 × 10⁶: the decimal point moves 6 places left to turn 4,500,000 into 4.5. That is 4.5 mega in engineering terms, or 4.5e+6 in E notation. Going small, 0.00052 = 5.2 × 10⁻⁴. To add 1.225 × 10⁵ + 3.655 × 10³, rewrite the second number as 0.03655 × 10⁵, then add the coefficients: 1.26155 × 10⁵ (122,500 + 3,655 = 126,155). To multiply 3 × 10⁴ by 5 × 10³, multiply 3 × 5 = 15 and add 4 + 3 = 7, giving 15 × 10⁷ = 1.5 × 10⁸.

Frequently asked questions