Scientific Notation Converter

Converts numbers between standard decimal notation and scientific notation (m × 10^n where 1 ≤ |m| < 10). Engineering notation, where the exponent is constrained to multiples of three, is also supported and aligns with SI prefixes (kilo, mega, micro).

Handles very small numbers (Avogadro's reciprocal: 1.66 × 10^−24), very large numbers (mole quantity: 6.022 × 10^23), and produces appropriate significant-figure rendering.

Scientific notation removes ambiguity in two ways. First, magnitude is immediately visible — 6.022 × 10^23 is unambiguously 24 digits, while 602200000000000000000000 requires counting. Second, significant figures are explicit — 1.500 × 10^3 indicates four significant figures (the trailing zeros are deliberate), while "1500" is ambiguous about whether the trailing zeros are significant or padding. The notation has been the standard for scientific communication since the 19th century.

Engineering notation aligns the exponent with SI prefix boundaries. 1.2 × 10^3 in scientific is 1.2k (kilo) in engineering. 4.7 × 10^−6 is 4.7µ (micro). The constraint that exponents are multiples of three slightly reduces mantissa precision (mantissas can range up to 999.99 instead of 9.99) but produces values that map cleanly to physical units. Electrical engineering, where component values follow E12 or E96 series and are quoted in standard SI prefixes, uses engineering notation almost exclusively.

A worked example: convert 0.0000456 to both notations. Scientific: 4.56 × 10^−5. Engineering: 45.6 × 10^−6 = 45.6µ. The engineering form maps directly to "45.6 microns" or "45.6 micrograms" depending on the underlying quantity. Convert 12,500,000 to both. Scientific: 1.25 × 10^7. Engineering: 12.5 × 10^6 = 12.5M (mega). Engineering notation matches conversational expression for these magnitudes.

Significant figure handling is a common pitfall. The conversion preserves the input precision: 1.50 × 10^3 has three significant figures and converts to 1500 with the trailing zero being significant. 1.5 × 10^3 has two significant figures and converts to 1500 where the trailing zeros are not significant. Plain decimal "1500" is ambiguous and the converter defaults to two significant figures unless the user flags otherwise. Best practice in scientific writing is to use scientific notation specifically to remove this ambiguity.

Limitations: complex numbers are not handled; only real values. Very large exponents (above 10^308) overflow IEEE 754 double precision and require arbitrary-precision arithmetic. The converter handles exponents within the practical scientific range (-323 to +308) cleanly. Outside this range, the input must be manipulated symbolically before conversion.