The principle of scientific notation

Why use scientific notation?

Some numbers are very large (the distance between the Earth and the Sun: approximately 150,000,000 km) or very small (the size of a virus: approximately 0.0000001 m). Writing them out in full is impractical. Scientific notation allows us to write them in a compact form.

The general form

A number is written in scientific notation as:

a × 10^n

where 1 ≤ a < 10 (a has a single non-zero digit before the decimal point) and n is a relative integer (positive, negative or zero).

Examples

• 300,000 = 3 × 10⁵
• 45,000 = 4.5 × 10⁴
• 0.00042 = 4.2 × 10⁻⁴
• 7 = 7 × 10^0

Decimal number	Scientific notation
5,200	5.2 × 10^3
0.08	8 × 10^-2
1,500,000	1.5 × 10⁶

Practical method

To find the exponent n, count how many places you move the decimal point to obtain a number between 1 and 10:

• If the number is large (>= 10), move the decimal point to the left: n is positive.
• If the number is small (< 1), move the decimal point to the right: n is negative.

Common pitfalls

• The coefficient a must always satisfy 1 <= a < 10. Writing 45 x 10^3 is not correct scientific notation (45 has two digits before the decimal point): it should be written as 4.5 x 10^4.
• Ensure the sign of the exponent is correct: numbers greater than or equal to 10 have a positive exponent, whilst numbers between 0 and 1 have a negative exponent.