Writing Numbers in Scientific Notation
Simple Explanation
Scientific notation writes a number as a × 10ⁿ, where a is a decimal between 1 and 10 (1 ≤ a < 10) and n is an integer. Move the decimal point until only one nonzero digit remains in front of it, and count how many places you moved it — that count is n.
Why Do We Need It?
Very large numbers (like distances in space) or very small numbers (like the size of an atom) are unwieldy to write and compare in ordinary decimal form — scientific notation keeps them compact and makes their size easy to compare at a glance.
Formula
Scientific Notation
a × 10ⁿ, where 1 ≤ a < 10 and n is an integer
A compact way to write very large or very small numbers as a decimal between 1 and 10, multiplied by a power of 10.
- a
- — the coefficient — a decimal number with 1 ≤ a < 10
- n
- — the exponent (power of 10) — positive for large numbers, negative for small numbers
When to use it: Whenever a number has many digits or many leading/trailing zeros, and you need a compact, easy-to-compare form.
Worked Example
Convert a large number to scientific notation
Write 45,000,000 in scientific notation.
Why Does This Work?
Multiplying by 10ⁿ shifts the decimal point exactly n places, so writing a × 10ⁿ with the decimal point already placed correctly in a, and then recording how many shifts n undoes, reproduces the original number exactly.
Real-Life Example
Astronomical distances
The distance from Earth to the Sun is about 150,000,000 km.
Written as 1.5 × 10⁸ km, the distance is far easier to read, compare to other distances, and use in calculations.
Practice
Write 0.00032 in scientific notation.
EasyCommon mistake
Leaving the coefficient outside the range 1 ≤ a < 10 (e.g. writing 45 × 10⁶ instead of 4.5 × 10⁷) — the coefficient must always have exactly one nonzero digit before the decimal point.
Quick Review
- a × 10ⁿ, with 1 ≤ a < 10 and n an integer.
- Large numbers get a positive exponent; small numbers (less than 1) get a negative exponent.
- Count how many places the decimal point moves to find n.