Scientific Notation
Scientific notation writes very large or very small measurements as a number between 1 and 10 multiplied by a power of ten.
Formula
a \times 10^{n}, \quad 1 \le a < 10
Definition
Scientific notation is a short way to write really big or really small numbers: a number between $1$ and $10$ multiplied by $10$ raised to a power that says how many places the decimal moved. Formally a value is written $a \times 10^{n}$ with $1 \le a < 10$ and $n$ an integer, which keeps measurements of any size readable, makes the number of significant figures explicit, and simplifies multiplication and division because exponents add and subtract. The mantissa $a$ carries the precision ($3.00 \times 10^8$ claims three significant figures while $3 \times 10^8$ claims one, a distinction $300{,}000{,}000$ cannot make) and the exponent carries the order of magnitude. Comparing orders of magnitude gives quick estimates of whether a result is plausible.
Example
The Sun is about $150{,}000{,}000\ \text{km}$ away, or $1.5 \times 10^8\ \text{km}$, and a grain of salt about $0.0005\ \text{m}$ wide is $5 \times 10^{-4}\ \text{m}$. The speed of light is $3.0 \times 10^8\ \text{m/s}$ and a red blood cell about $7 \times 10^{-6}\ \text{m}$ across; light takes $\frac{1.5 \times 10^{11}\ \text{m}}{3.0 \times 10^8\ \text{m/s}} = 5.0 \times 10^2\ \text{s}$, about $8$ minutes, to reach Earth. The mass of one water molecule is $\frac{18.0\ \text{g/mol}}{6.022 \times 10^{23}\ \text{mol}^{-1}} = 2.99 \times 10^{-23}\ \text{g}$, and a quick check ($18 / 6 = 3$, $10^0 / 10^{23} = 10^{-23}$) confirms the order of magnitude before any calculator is touched.
Key Insight
A positive power means a big number and a negative power a small one, with the power counting how many places the decimal moved. Scientific notation and metric prefixes say the same thing two ways: $7 \times 10^{-6}\ \text{m}$ is $7\ \mu\text{m}$. Order-of-magnitude reasoning is the scientist's first sanity check; if a calculation gives a cell mass of $10^{-3}\ \text{g}$ when cells are near $10^{-9}\ \text{g}$, the exponent arithmetic went wrong somewhere and no amount of decimal precision will save it.