Scientific Notation — Mathematics Knowledge System

Overview

Scientific notation writes a number as a decimal between 1 and 10 multiplied by a power of 10, making it compact to write and compare extremely large or small numbers — the mass of the sun (about 1.989 × 10³⁰ kg) or the size of an atom (about 1 × 10⁻¹⁰ m) would otherwise require writing out dozens of digits.

Formal Definition

Definition

A number is in scientific notation when written as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer.

a \times 10^n, \quad 1 \le |a| < 10, \quad n \in \mathbb{Z}

Scientific notation form

Properties

Positive exponent

n > 0 \text{ corresponds to numbers} \geq 10

Negative exponent

n < 0 \text{ corresponds to numbers between 0 and 1}

Worked Examples

Move the decimal point left until only one nonzero digit remains before it; count the moves.
$45{,}000{,}000 = 4.5 \times 10^7$

Answer: 4.5 × 10⁷

Practice Problems

Difficulty 2/10

Write 0.00032 in scientific notation.

Common Mistakes

Common Mistake

Getting the sign of the exponent backwards, e.g. writing 0.0005 as 5 × 10³ instead of 5 × 10⁻⁴.

Numbers smaller than 1 get NEGATIVE exponents (moving the decimal point right to standard form), while numbers larger than 10 get positive exponents (moving left).

Summary

Scientific notation writes a number as a × 10ⁿ with 1 ≤ |a| < 10.
Large numbers (≥10) get positive exponents; small numbers (between 0 and 1) get negative exponents.
It makes comparing and computing with very large or very small quantities far more manageable.

References

WebsiteWikipedia — Scientific notation