Scientific Notation: Writing Huge and Tiny Numbers

The distance to the Sun is about 150,000,000 km. A typical bacterium is about 0.000002 m long. Scientific notation writes both in a compact, comparable way.

The format

• 150,000,000 = 1.5 × 10^8
• 0.000002 = 2 × 10^−6

The exponent n counts how many places the decimal point moves: positive for big numbers, negative for small ones.

Converting into scientific notation

1. Move the decimal point until exactly one non-zero digit sits to its left.
2. Count how many places you moved it. That's n.
3. Moved it left (a big number)? n is positive. Moved it right (a small number)? n is negative.

Example: 0.00045 → move 4 places right → 4.5 × 10^−4.

Multiplying and dividing

Handle the numbers and the powers separately:

(3 × 10^5) × (4 × 10^3) = 12 × 10^8, which you then tidy to 1.2 × 10^9.

Adding and subtracting

Rewrite both numbers with the same power of ten first: 3 × 10^4 + 5 × 10^3 = 30 × 10^3 + 5 × 10^3 = 35 × 10^3 = 3.5 × 10^4.

Significant figures

Scientific notation makes precision visible. 1.50 × 10^8 claims three significant figures; 1.5 × 10^8 claims two. Keep the same number of significant figures as your least precise measurement.

E notation

Calculators and spreadsheets often show 1.5E8 or 1.5e8, which means the same as 1.5 × 10^8.

Orders of magnitude

Comparing exponents gives a quick sense of scale: the Sun (about 1.5 × 10^8 km away) is almost 400 times farther from us than the Moon (about 3.84 × 10^5 km).