C.0 Arithmetic and numerical computation

Treat every unit as part of the calculation. Substitute units with the numerical values, simplify powers and quotients, and check that the final unit matches the physical quantity being calculated.

Convert to compatible units before substitution. For example, $5.5\,\mathrm{mA}=5.5\times10^{-3}\,\mathrm{A}$, so a current calculated in amperes can be compared fairly with the measurement.

When a length is squared or cubed, its conversion factor is also powered: $1\,\mathrm{cm^3}=(10^{-2}\,\mathrm{m})^3=10^{-6}\,\mathrm{m^3}$.

Standard form writes a number as $a\times10^n$, where $1\leq |a|<10$ and $n$ is an integer. It preserves significant figures while making very large and very small physical quantities easier to combine.

Use physical constants with their stated precision and units. For light, $c=3.00\times10^8\,\mathrm{m\,s^{-1}}$. A time of $2.20\,\mathrm{\mu s}$ is $2.20\times10^{-6}\,\mathrm{s}$.

At $0.980c$, the distance travelled in $2.20\,\mathrm{\mu s}$ is $s=0.980(3.00\times10^8)(2.20\times10^{-6})=6.47\times10^2\,\mathrm{m}$.

A ratio compares quantities in the same unit; a fraction expresses one quantity relative to a whole or reference; multiplying that fraction by $100\%$ gives a percentage. State the denominator because it defines the comparison.

If a device receives $240\,\mathrm{J}$ and transfers $180\,\mathrm{J}$ usefully, $\eta=180/240=0.75=75\%$. The remaining fraction is not automatically all one named loss unless the energy pathways are known.

A reciprocal is also a fraction: if $\lambda=5.00\times10^{-7}\,\mathrm{m}$, then $1/\lambda=2.00\times10^6\,\mathrm{m^{-1}}$. Transform the unit as well as the number.

An estimate predicts magnitude or direction of change from the governing relationship before detailed calculation. Round inputs sensibly, keep the dominant powers and use proportional scaling to expose impossible results.

For radiation intensity $I\propto1/d^2$, doubling distance gives $I_2/I_1=(d_1/d_2)^2=(1/2)^2=1/4$. This estimate predicts the direction and factor without knowing the luminosity.

A change that increases the measured value while leaving a similar absolute reading uncertainty reduces percentage uncertainty. This can justify changing an experimental parameter, provided the model and apparatus remain valid.

Use brackets and the calculator power key to evaluate transformed quantities such as $1/d^2$ or $x^{3/2}$. A negative exponent means a reciprocal; a fractional exponent means a root combined with a power.

For exponential decay, the ratio $N/N_0$ is dimensionless and lies between 0 and 1, so its natural logarithm is negative; the leading minus sign then gives a positive time.

Enter the complete exponent in brackets and verify whether the problem requires $\ln$ or $\log_{10}$. Substitute the result back into the original expression as a check.

Match calculator mode to the angle unit: degrees for angles marked $^\circ$, radians for angles stated in rad or produced by angular relationships. The same numerical input gives different results in the two modes.

For resultant components $R_x=4.0\,\mathrm{N}$ and $R_y=3.0\,\mathrm{N}$, the direction is $\theta=\tan^{-1}(3.0/4.0)=36.9^\circ$ above the positive $x$ direction.

The inverse-tangent value alone can be ambiguous. Use the signs of both components—or a calculator's two-argument angle function when available—to place the vector in the correct quadrant and state the reference direction.

Replace the prefix by its power of ten, then convert. For example, $3.6\,\mathrm{MW}=3.6\times10^6\,\mathrm{W}$ and $420\,\mathrm{nm}=420\times10^{-9}\,\mathrm{m}=4.20\times10^{-7}\,\mathrm{m}$.

For powered units, power the factor: $1\,\mathrm{cm^2}=10^{-4}\,\mathrm{m^2}$ and $1\,\mathrm{cm^3}=10^{-6}\,\mathrm{m^3}$.

Prefix symbols are case-sensitive: $\mathrm{M}$ means mega while $\mathrm{m}$ means milli and also serves as the unprefixed symbol for metre depending on position. Keep the quantity unit visible when interpreting the symbol.