Scientific Notation: Measurement in Scientific Work - Physics (Grade 10)

When Zeros Get in the Way

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, but an electron mass of about $0.000000000000000000000000000000911 \text{ kg}$ makes it easy to lose count of the zeros.

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 $a$ carries the significant figures. The exponent $n$ carries the scale.

The Decimal Point Moves, the Value Stays

The safest way to read scientific notation is to imagine moving the decimal point until the coefficient is between $1$ and $10$ .

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}

The Exponent Is Not a Significant Figure

In scientific notation, the number of significant figures is read from the coefficient, not from the power of $10$ .

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

That is why scientific notation is useful for measurement. The value $1000 \text{ m}$ can be ambiguous, but $1.000 \times 10^3 \text{ m}$ clearly has $4$ significant figures.

The Order for Reporting Measurements

When scientific notation is used for a measurement result, do not reverse the order.

\text{round to significant figures} \to \text{convert units} \to \text{write scientific notation}

For example, the calculated area of a bottle cap is $7.641504 \text{ cm}^2$ . If the original diameter supports only $3$ significant figures, the area is written as:

7.641504 \text{ cm}^2 \to 7.64 \text{ cm}^2

Only after that do we convert the area to the International System of Units (SI). SI is the international measurement unit standard used in science.

\begin{aligned} 1 \text{ cm} &= 10^{-2} \text{ m} \\ 1 \text{ cm}^2 &= (10^{-2} \text{ m})^2 = 10^{-4} \text{ m}^2 \\ 7.64 \text{ cm}^2 &= 7.64 \times 10^{-4} \text{ m}^2 \end{aligned}

So the SI scientific-notation result is $7.64 \times 10^{-4} \text{ m}^2$ .

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 $10^{-4}$ .

Calculating Without Long Strings of Zeros

Scientific notation also keeps operations tidy because powers of $10$ can be grouped.

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 \\ &= 1.00 \times 10^{-12} \text{ m}^2 \end{aligned}

The last step matters: the coefficient $10.0$ is not in standard form because it is not less than $10$ . We rewrite it as $1.00 \times 10^{-12}$ while preserving $3$ significant figures.

Measurement-value writing references from NIST Guide to the SI and OpenStax Significant Figures can be opened through nist.gov and openstax.org.