Scientific Notation: How to Express Very Large and Very Small Numbers
Blog › Math · 7 min read · Published 2026-03-06
Master scientific notation with examples from physics, chemistry, and astronomy. Learn conversion, arithmetic rules, and E notation.
Why Scientific Notation Exists
The observable universe is approximately 93 billion light-years across. The mass of an electron is 0.00000000000000000000000000000091 kilograms. Writing these numbers in standard form is impractical — it's error-prone, hard to compare, and wastes space. Scientific notation solves this by expressing any number as a coefficient between 1 and 10 multiplied by a power of 10.
The format is a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. The speed of light becomes 3 × 10⁸ m/s. The electron mass becomes 9.1 × 10⁻³¹ kg. Instantly manageable.
How to Convert to Scientific Notation
The process is straightforward: move the decimal point until you have a number between 1 and 10, then count how many places you moved. Moving the decimal left gives a positive exponent (large numbers); moving right gives a negative exponent (small numbers).
Example 1: 186,000 → Move decimal 5 places left → 1.86 × 10⁵
Example 2: 0.00045 → Move decimal 4 places right → 4.5 × 10⁻⁴
Example 3: 7.3 → Already between 1 and 10 → 7.3 × 10⁰
Arithmetic Rules
Multiplication: Multiply coefficients and add exponents. (3 × 10⁴)(2 × 10³) = 6 × 10⁷.
Division: Divide coefficients and subtract exponents. (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴.
Addition/Subtraction: First align exponents, then add/subtract coefficients. 3.5 × 10⁴ + 2.1 × 10³ = 3.5 × 10⁴ + 0.21 × 10⁴ = 3.71 × 10⁴.
E Notation in Computing
Computers and calculators use E notation: 3.5e4 means 3.5 × 10⁴. This is standard in programming languages (Python, JavaScript, C++), spreadsheets, and scientific software. When you see "6.022e23" in a Python script, it represents Avogadro's number.
Engineering Notation
A variant called engineering notation restricts exponents to multiples of 3, aligning with SI prefixes: 10³ (kilo), 10⁶ (mega), 10⁹ (giga), 10⁻³ (milli), 10⁻⁶ (micro), 10⁻⁹ (nano). Engineers prefer this because 47 × 10³ ohms is immediately recognizable as 47 kΩ.
Famous Numbers in Scientific Notation
Speed of light: 2.998 × 10⁸ m/s. Gravitational constant: 6.674 × 10⁻¹¹ N⋅m²/kg². Avogadro's number: 6.022 × 10²³ mol⁻¹. Planck's constant: 6.626 × 10⁻³⁴ J⋅s. Earth's mass: 5.972 × 10²⁴ kg. Bohr radius: 5.292 × 10⁻¹¹ m.
Significant Figures and Precision
Scientific notation naturally communicates precision. Writing 1.50 × 10³ (three significant figures) conveys more precision than 1.5 × 10³ (two significant figures), even though both equal 1500. The trailing zero in 1.50 is significant and intentional.
Common Mistakes
The most common error is reversing the exponent sign: 0.005 is 5 × 10⁻³, not 5 × 10³. Another mistake is having a coefficient ≥ 10: writing 15 × 10³ instead of the proper 1.5 × 10⁴. Always verify your coefficient falls between 1 and 10.
FAQ
What's the difference between scientific and standard notation?
Standard notation is the regular way of writing numbers (1,500,000). Scientific notation expresses the same number compactly as 1.5 × 10⁶, making it easier to work with in calculations and comparisons.
Can negative numbers use scientific notation?
Yes. A negative coefficient indicates a negative number: -3.2 × 10⁵ = -320,000. Don't confuse this with a negative exponent, which indicates a small positive number: 3.2 × 10⁻⁵ = 0.000032.