Scientific Notation, SI Units, and Measurement Error

Why measurement precision matters

Applied mathematics almost always begins with a measured or reported quantity, and real measurements are never perfectly exact. A biologist estimating the number of bacteria in a culture, an engineer stating the mass of a bridge cable, and an economist quoting a nation's GDP all work with numbers that are large, small, or uncertain. Scientific notation, SI units, and error analysis are the tools that let us handle these numbers honestly and communicate them clearly.

Scientific notation

A number is in scientific notation when it is written as $a \times 10^{k}$, where $1 \le |a| < 10$ and $k$ is an integer. This form compresses awkward magnitudes: the speed of light $299{,}792{,}458$ m/s becomes $2.998 \times 10^{8}$ m/s, and the mass of a hydrogen atom $0.0