What is Scientific notation? — Physics

Definition 1.5 High School Physics · Chapter 1 — Orders of Magnitude: Measuring the Universe

A number is in scientific notation when written as

a \times 10^{n}, \qquad 1 \leq a < 10, \quad n \text{ an integer.}

The factor $a$ is the mantissa, the integer $n$ the exponent. Thus $149\,600\,000 = 1.496 \times 10^{8}$ and $0.000\,52 = 5.2 \times 10^{-4}$ .

Examples

Example 1.7 (Multiplying in scientific notation)

How many Earths fit in the Sun, by mass? With $m_{\text{Sun}} = 1.99 \times 10^{30}\,\mathrm{kg}$ and $m_{\text{Earth}} = 5.97 \times 10^{24}\,\mathrm{kg}$ :

\frac{1.99 \times 10^{30}}{5.97 \times 10^{24}} = \frac{1.99}{5.97} \times 10^{30-24} = 0.333 \times 10^{6} = 3.33 \times 10^{5}.

Mantissas and exponents are handled separately; the final answer is re-normalized so that the mantissa lands back in $[1, 10)$ .

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