Q: What is the difference between scientific notation and engineering notation?

Scientific notation restricts the coefficient to 1 ≤ |a| < 10 with any integer exponent (e.g. 4.7 × 10^4). Engineering notation restricts the exponent to multiples of 3 (…10^-6, 10^-3, 10^0, 10^3, 10^6…) so that every representation lines up with an SI prefix (µ, m, unit, k, M) per the BIPM SI brochure §3.1 Table 5 and NIST SP 330 (2019 revision). Example: 47,000 in scientific notation is 4.7 × 10^4, but in engineering notation it is 47 × 10^3 which reads directly as 47 kilo-units. Engineering notation is the default on most scientific calculators (TI-30X, Casio fx-115) in ENG mode.

Q: How many significant figures should the coefficient carry?

The coefficient carries exactly as many significant figures as the measurement warrants. NIST SP 811 §7 and the ISO/IEC Guide 98-3:2008 (GUM, JCGM 100:2008) both require reporting no more precision than the underlying measurement supports: 6.022 × 10^23 has four sig figs (Avogadro's number, CODATA 2018 value 6.02214076 × 10^23 as fixed by the SI redefinition of 20 May 2019 has nine); 3.14 × 10^0 has three; 3.14159265 × 10^0 has nine. Trailing zeros in the coefficient are significant when the number is written in scientific notation, which resolves the ambiguity of writing '3,000' in decimal.

Q: How do I multiply and divide numbers in scientific notation by hand?

For multiplication (a × 10^m) × (b × 10^n) = (a × b) × 10^(m+n) — multiply the coefficients, add the exponents; then normalize so the new coefficient falls in [1, 10). For division (a × 10^m) / (b × 10^n) = (a / b) × 10^(m-n) — divide the coefficients, subtract the exponents; then normalize. Example: (3.0 × 10^8) × (2.5 × 10^-3) = 7.5 × 10^5. Example: (4.2 × 10^6) / (1.4 × 10^2) = 3.0 × 10^4. This algebraic identity follows directly from the exponent laws (Cauchy, Cours d'analyse 1821), taught in every algebra II curriculum aligned with the Common Core Standards (CCSS.MATH.CONTENT.HSN.Q.A.3).

Q: How is scientific notation written in programming languages?

Nearly every modern programming language accepts E-notation (also called exponent notation): a coefficient followed by the letter e or E and an integer exponent. Examples: 6.022e23 (Python, JavaScript, C, C++, Java, Rust, Go, Swift, SQL) all parse to the double-precision IEEE 754-2019 floating-point value ≈ 6.022 × 10^23. The Python float and JavaScript Number types are both IEEE 754 binary64, which can represent exponents from 10^-308 to 10^308 with ~15-17 decimal digits of precision (Goldberg 1991, 'What Every Computer Scientist Should Know About Floating-Point Arithmetic', ACM Computing Surveys 23(1), pp. 5-48). For higher precision use Python's Decimal module (PEP 327, following IBM's General Decimal Arithmetic specification) or fractions.Fraction for exact rational arithmetic.

Question 1

What is scientific notation?

Accepted Answer

Scientific notation is a way of writing very large or very small numbers as the product of a coefficient (1 ≤ |a| < 10) and a power of 10, i.e. a × 10^n where n is an integer. The convention was standardized by the International System of Units brochure (BIPM Le Système international d'unités, 9th edition 2019, §5.4.1) and is taught as the primary form for scientific measurement per NIST Special Publication 811 (2008 revision, §6.1.5). Example: 299,792,458 m/s (the speed of light in vacuum, IUPAC/NIST 2019 fixed constants) is written as 2.99792458 × 10^8 m/s.

Question 2

What is the difference between scientific notation and engineering notation?

Accepted Answer

Scientific notation restricts the coefficient to 1 ≤ |a| < 10 with any integer exponent (e.g. 4.7 × 10^4). Engineering notation restricts the exponent to multiples of 3 (…10^-6, 10^-3, 10^0, 10^3, 10^6…) so that every representation lines up with an SI prefix (µ, m, unit, k, M) per the BIPM SI brochure §3.1 Table 5 and NIST SP 330 (2019 revision). Example: 47,000 in scientific notation is 4.7 × 10^4, but in engineering notation it is 47 × 10^3 which reads directly as 47 kilo-units. Engineering notation is the default on most scientific calculators (TI-30X, Casio fx-115) in ENG mode.

Question 3

How many significant figures should the coefficient carry?

Accepted Answer

The coefficient carries exactly as many significant figures as the measurement warrants. NIST SP 811 §7 and the ISO/IEC Guide 98-3:2008 (GUM, JCGM 100:2008) both require reporting no more precision than the underlying measurement supports: 6.022 × 10^23 has four sig figs (Avogadro's number, CODATA 2018 value 6.02214076 × 10^23 as fixed by the SI redefinition of 20 May 2019 has nine); 3.14 × 10^0 has three; 3.14159265 × 10^0 has nine. Trailing zeros in the coefficient are significant when the number is written in scientific notation, which resolves the ambiguity of writing '3,000' in decimal.

Question 4

How do I multiply and divide numbers in scientific notation by hand?

Accepted Answer

For multiplication (a × 10^m) × (b × 10^n) = (a × b) × 10^(m+n) — multiply the coefficients, add the exponents; then normalize so the new coefficient falls in [1, 10). For division (a × 10^m) / (b × 10^n) = (a / b) × 10^(m-n) — divide the coefficients, subtract the exponents; then normalize. Example: (3.0 × 10^8) × (2.5 × 10^-3) = 7.5 × 10^5. Example: (4.2 × 10^6) / (1.4 × 10^2) = 3.0 × 10^4. This algebraic identity follows directly from the exponent laws (Cauchy, Cours d'analyse 1821), taught in every algebra II curriculum aligned with the Common Core Standards (CCSS.MATH.CONTENT.HSN.Q.A.3).

Question 5

How is scientific notation written in programming languages?

Accepted Answer

Nearly every modern programming language accepts E-notation (also called exponent notation): a coefficient followed by the letter e or E and an integer exponent. Examples: 6.022e23 (Python, JavaScript, C, C++, Java, Rust, Go, Swift, SQL) all parse to the double-precision IEEE 754-2019 floating-point value ≈ 6.022 × 10^23. The Python float and JavaScript Number types are both IEEE 754 binary64, which can represent exponents from 10^-308 to 10^308 with ~15-17 decimal digits of precision (Goldberg 1991, 'What Every Computer Scientist Should Know About Floating-Point Arithmetic', ACM Computing Surveys 23(1), pp. 5-48). For higher precision use Python's Decimal module (PEP 327, following IBM's General Decimal Arithmetic specification) or fractions.Fraction for exact rational arithmetic.

Scientific Notation Calculator