Scientific Notation Converter

Understanding the scientific notation converter — concept and principle

Scientific notation writes any number as a × 10ⁿ, where the coefficient a has exactly one non-zero digit before the decimal point (1 ≤ |a| < 10) and n is a whole number. It makes very large and very small quantities easy to read and compare, and it is standard in science and engineering. This converter shows the number in scientific notation, in engineering notation and in the E-notation used by calculators and programming languages, rounded to the number of significant figures you choose.

How it is calculated

• Exponent: n = ⌊log₁₀|x|⌋, the power of ten that puts the decimal point after the first significant digit.
• Coefficient: a = x ÷ 10ⁿ, rounded to the chosen number of significant figures.
• Engineering notation: the exponent is lowered to the nearest multiple of 3 (⌊n ÷ 3⌋ × 3), so the coefficient lies between 1 and 999.
• E-notation: the same as scientific notation, with "× 10^" replaced by "e" (4.50e-5 means 4.50 × 10⁻⁵).

The principle

Our number system is positional in base 10, so moving the decimal point one place to the left divides by 10 and one place to the right multiplies by 10. Scientific notation records how many places the point was moved in the exponent, and keeps the meaningful digits in the coefficient. To multiply, multiply the coefficients and add the exponents. Engineering notation follows the SI prefixes, which step in powers of 1,000: 10³ is kilo (k), 10⁻³ milli (m), 10⁻⁶ micro (µ), 10⁻⁹ nano (n).

Worked example

1. Inputs: number 0.000045, significant figures 3.
2. Move the decimal point 5 places right to reach 4.5, so n = −5.
3. With 3 significant figures the coefficient is 4.50: scientific form 4.50 × 10^-5.
4. Engineering: the nearest lower multiple of 3 is −6, and 0.000045 ÷ 10⁻⁶ = 45, shown with 3 significant figures as 45.0 × 10^-6 (45.0 µ in SI prefixes; the engineering display keeps the significant zero).
5. E-notation: 4.50e-5.

How to read the result

The exponent tells you the order of magnitude at a glance: 10⁻⁵ is a hundred-thousandth, 10⁶ is a million. To compare two numbers, compare exponents first and coefficients only when the exponents are equal.

Common mistakes

• Leaving the coefficient outside the range 1 to 10, such as 45 × 10⁻⁶ in scientific notation (that form is engineering notation).
• Dropping meaningful trailing zeros: 4.50 × 10⁻⁵ states three significant figures, 4.5 × 10⁻⁵ only two.
• Reading E-notation 4.50e-5 as 4.50 × e⁻⁵; the "e" means "times ten to the power", not Euler's number.

Frequently asked questions

How do I convert scientific notation back to a normal number?

Move the decimal point n places: right for a positive exponent, left for a negative one. 4.50 × 10⁻⁵ becomes 0.0000450.

Why is engineering notation useful?

Its exponents match SI prefixes, so 45 × 10⁻⁶ A can be read directly as 45 µA.

Sources: BIPM, The International System of Units (SI Brochure), 9th edition, 2019 — SI prefixes · NIST Special Publication 811, Guide for the Use of the International System of Units (SI)