Scientific Notation

Writing Numbers with Many Zeros

Physics often uses values that are too large or too small to write comfortably. An area of $0.000764 \text{ m}^2$ is still readable. Writing the approximate electron mass in kilograms is harder: after the decimal point, write $30$ zeros, then the digits $911$ . It is easy to lose count.

Scientific notation writes a value as a coefficient multiplied by a power of $10$ .

a \times 10^n

For positive values, the standard form is:

1 \le a < 10 \quad \text{and} \quad n \in \mathbb{Z}

The coefficient $a$ contains the significant figures. The exponent $n$ sets the order of magnitude.

The electron mass above can now be written more compactly:

9.11 \times 10^{-31} \text{ kg}

Moving the Decimal Point without Changing the Value

Move the decimal point until the coefficient is between $1$ and $10$ . The number and direction of the moves determine the exponent.

Ordinary value	Decimal movement	Scientific notation
$7\,640\,000 \text{ m}$	$6$ places left	$7.64 \times 10^6 \text{ m}$
$0.000764 \text{ m}^2$	$4$ places right	$7.64 \times 10^{-4} \text{ m}^2$
$0.000000911 \text{ kg}$	$7$ places right	$9.11 \times 10^{-7} \text{ kg}$

If the decimal point moves left, the exponent is positive. If the decimal point moves right, the exponent is negative.

\begin{aligned} 7\,640\,000 &= 7.64 \times 10^6 \\ 0.000764 &= 7.64 \times 10^{-4} \end{aligned}

Significant Figures in the Coefficient

In scientific notation, read the significant figures from the coefficient. The power of $10$ sets the scale.

Scientific notation	Significant figures	Reason
$7.64 \times 10^{-4} \text{ m}^2$	$3$	The coefficient is $7.64$
$1.0 \times 10^{-7} \text{ m}$	$2$	The zero after the decimal point in $1.0$ is intentional
$1 \times 10^3 \text{ m}$	$1$	The coefficient is only $1$
$1.000 \times 10^3 \text{ m}$	$4$	The zeros in $1.000$ express precision

Scientific notation records the intended precision of a measurement. The value $1000 \text{ m}$ does not state whether its trailing zeros are significant, while $1.000 \times 10^3 \text{ m}$ explicitly has $4$ significant figures.

Keep Guard Digits Until the Final Report

Scientific notation should not force an early rounding step. Calculate and convert with guard digits, then round the final reported value once.

\text{calculate with guard digits} \to \text{convert with exact factors} \to \text{round once} \to \text{write scientific notation}

For example, a bottle cap with diameter $3.12 \text{ cm}$ has the following calculated area when the calculator value of $\pi$ is retained:

A=\frac{1}{4}\pi(3.12 \text{ cm})^2=7.645379881\ldots \text{ cm}^2

Use the exact power-of-ten conversion to the International System of Units (SI) before discarding those guard digits.

\begin{aligned} 1 \text{ cm} &= 10^{-2} \text{ m} \\ 1 \text{ cm}^2 &= (10^{-2} \text{ m})^2 = 10^{-4} \text{ m}^2 \\ 7.645379881\ldots \text{ cm}^2 &= 7.645379881\ldots \times 10^{-4} \text{ m}^2 \end{aligned}

The measured diameter supports $3$ significant figures, so the final SI result is $7.65 \times 10^{-4} \text{ m}^2$ .

Be careful with powered units. The conversion factor is powered too. From $\text{cm}$ to $\text{m}$ , the factor is $10^{-2}$ , but from $\text{cm}^2$ to $\text{m}^2$ , the factor is $10^{-4}$ .

Calculations with Scientific Notation

In scientific notation, multiply the coefficients and add the exponents of $10$ separately.

Suppose a very small object has length $2.5 \times 10^{-6} \text{ m}$ and width $4.0 \times 10^{-7} \text{ m}$ .

\begin{aligned} A &= (2.5 \times 10^{-6})(4.0 \times 10^{-7}) \text{ m}^2 \\ &= (2.5 \times 4.0) \times 10^{-6+(-7)} \text{ m}^2 \\ &= 10.0 \times 10^{-13} \text{ m}^2 \\ &= 1.0 \times 10^{-12} \text{ m}^2 \end{aligned}

The last step does two jobs. The coefficient $10.0$ is not in standard form because it is not less than $10$ , and both measured lengths carry only $2$ significant figures. We therefore report $1.0 \times 10^{-12}$ .

NIST publishes the official value-writing rules in SP 811 Chapter 7 and the unit-conversion factors in SP 811 Appendix B.