By Kristen Dennis (updated Aug 30, 2022)
Why Use Scientific Notation?
Scientists and engineers routinely work with numbers that are extremely large or tiny. Writing these numbers in conventional decimal form is cumbersome and error‑prone. Scientific notation reduces any real number to a concise expression: a coefficient between 1 and 10 multiplied by a power of ten.
Converting to Scientific Notation
To convert an integer or decimal to scientific notation, place the decimal point immediately after the first non‑zero digit. The number of positions the decimal moved becomes the exponent.
- 987 000 000 000 → 9.87 × 10¹¹
- 0.00000000001 → 1.0 × 10⁻¹¹
Multiplying Numbers in Scientific Notation
When multiplying two terms, follow these steps:
- Multiply the coefficients.
- Sum the exponents (treat negative exponents normally).
- Adjust the coefficient to lie between 1 and 10, carrying over any excess to the exponent.
Examples:
- (2 × 10⁶) × (4 × 10⁸) = 8 × 10¹⁴
- (6 × 10⁸) × (9 × 10⁴) = 5.4 × 10¹³ (after adjusting the coefficient)
- (3 × 10⁻⁴) × (3 × 10⁻³) = 9.0 × 10⁻⁷
- (2 × 10⁻⁷) × (3 × 10¹¹) = 6.0 × 10⁴
Key Points to Remember
- The coefficient must always be ≥ 1 and < 10.
- When the product of coefficients exceeds 10, shift the decimal left by one place and add 1 to the exponent.
- Negative exponents denote numbers smaller than 1.
