Scientific Notation Explained: How to Read, Write & Convert

Scientific notation is a compact way to write very large or very small numbers by expressing them as a coefficient multiplied by a power of 10.

a × 10ⁿ

Where a is the coefficient — a number greater than or equal to 1 and less than 10 — and n is any positive or negative integer.

Why it exists: numbers in science, astronomy, chemistry, and physics can be astronomically large (the number of atoms in a gram of carbon is about 5 × 10²²) or vanishingly small (a hydrogen atom is about 5.3 × 10⁻¹¹ metres in radius). Writing these in full decimal form is error-prone and impractical. Scientific notation handles both extremes with equal ease.

Valid scientific notation must satisfy two rules:

1. The coefficient (a) must be greater than or equal to 1 and less than 10.
   ✓ 4.75 × 10³  |  ✗ 47.5 × 10² (coefficient is not in range)
2. The exponent (n) must be a whole integer — positive, negative, or zero.
   ✓ 6.02 × 10²³  |  ✓ 1.6 × 10⁻¹⁹  |  ✓ 2.0 × 10⁰ (= 2.0)

Common notation variants you will encounter:

• a × 10ⁿ — standard mathematical notation
• aEn — calculator and programming notation (e.g., 6.02E23)
• a × 10^n — plain text version when superscripts are unavailable

Step-by-step for large numbers (moving decimal left):

1. Locate the decimal point (or the rightmost digit for whole numbers).
2. Move the decimal point left until the number is between 1 and 10.
3. Count the places moved — that is your positive exponent.

Example: 4,500,000

• Move decimal 6 places left: 4,500,000 → 4.5
• Result: 4.5 × 10⁶

Step-by-step for small numbers (moving decimal right):

1. Locate the decimal point.
2. Move the decimal point right until the number is between 1 and 10.
3. Count the places moved — that is your negative exponent.

Example: 0.000073

• Move decimal 5 places right: 0.000073 → 7.3
• Result: 7.3 × 10⁻⁵

To convert from scientific notation back to standard decimal form, move the decimal point:

• Positive exponent → move right (number gets larger).
  3.2 × 10⁴ = 3.2000 → move 4 right → 32,000
• Negative exponent → move left (number gets smaller).
  5.6 × 10⁻³ = 5.6 → move 3 left → 0.0056

Fill with zeros as needed when moving past the existing digits.

Multiplication

Multiply coefficients, add exponents:

(a × 10ⁿ) × (b × 10ᵐ) = (a × b) × 10ⁿ⁺ᵐ

Example: (4.0 × 10³) × (3.0 × 10⁵) = 12.0 × 10⁸ → adjust to 1.2 × 10⁹

Division

Divide coefficients, subtract exponents:

(a × 10ⁿ) ÷ (b × 10ᵐ) = (a ÷ b) × 10ⁿ⁻ᵐ

Example: (9.0 × 10⁶) ÷ (3.0 × 10²) = 3.0 × 10⁴

Addition & Subtraction

Adjust to the same exponent first, then add or subtract coefficients:

Example: 2.0 × 10⁴ + 3.0 × 10³

• Convert 3.0 × 10³ = 0.30 × 10⁴
• Add: (2.0 + 0.30) × 10⁴ = 2.30 × 10⁴

A negative exponent does not mean a negative number — it means a fraction.

10⁻ⁿ = 1 ÷ 10ⁿ

• 10⁻¹ = 0.1
• 10⁻² = 0.01
• 10⁻³ = 0.001
• 10⁻⁶ = 0.000001 (one millionth)
• 10⁻⁹ = 0.000000001 (one billionth = 1 nanometre scale)

Common small-number scientific notation values:

• Charge of an electron: 1.6 × 10⁻¹⁹ coulombs
• Mass of a proton: 1.67 × 10⁻²⁷ kg
• Diameter of a virus: ~1 × 10⁻⁷ m (100 nanometres)

• Speed of light: 3 × 10⁸ m/s (300,000,000 m/s)
• Distance from Earth to Sun: ~1.5 × 10¹¹ m
• Avogadro's number: 6.022 × 10²³ molecules per mole
• US national debt: ~3.5 × 10¹³ dollars (as of 2025)
• Diameter of a human hair: ~7 × 10⁻⁵ m (70 micrometres)
• Wavelength of visible light: 4–7 × 10⁻⁷ m (400–700 nm)

Use the scientific notation calculator to convert any number to or from scientific notation, and to perform multiplication, division, addition, and subtraction directly in scientific notation format.