Theory — SI prefixes and base-unit scaling
SI prefixes are shorthand labels for powers of ten. They allow the same physical unit to be written at very different scales
without changing the underlying dimension. For example, a millimeter and a kilometer are both still units of length. The only
thing that changes is the factor of ten attached to the meter. This is what makes SI-prefix conversion much simpler than a general
unit-conversion problem: the base unit stays the same, so only the prefix factor must be adjusted.
General prefix idea.
\[
\begin{aligned}
\text{prefixed unit} &= 10^n \cdot \text{base unit}
\end{aligned}
\]
Common examples include kilo \((10^3)\), milli \((10^{-3})\), micro \((10^{-6})\), mega \((10^6)\), and giga \((10^9)\).
So a kilometer is \(10^3\) meters, a millisecond is \(10^{-3}\) seconds, a microampere is \(10^{-6}\) amperes, and a megagram
is \(10^6\) grams. Because the system is decimal, moving between prefixes is always a matter of adding or subtracting exponents.
Selected prefixes.
\[
\begin{aligned}
\mathrm{k} &= 10^3 \\
\mathrm{m} &= 10^{-3} \\
\mu &= 10^{-6} \\
\mathrm{M} &= 10^6 \\
\mathrm{G} &= 10^9
\end{aligned}
\]
How a prefix conversion works
Suppose a source unit uses prefix exponent \(n_{\text{from}}\) and the target unit uses prefix exponent \(n_{\text{to}}\).
Since both share the same base unit, the conversion factor is just the ratio
\(10^{n_{\text{from}}} / 10^{n_{\text{to}}}\). That simplifies to \(10^{n_{\text{from}} - n_{\text{to}}}\).
The numerical value is then multiplied by that factor.
General conversion rule.
\[
\begin{aligned}
x_{\text{out}}
&= x_{\text{in}} \cdot \frac{10^{n_{\text{from}}}}{10^{n_{\text{to}}}} \\
&= x_{\text{in}} \cdot 10^{\,n_{\text{from}} - n_{\text{to}}}
\end{aligned}
\]
This formula is why prefix conversion can be done so quickly. There is no change of physical dimension, only a change of notation.
A quantity written in a smaller prefix will often have a larger numeric value, while the same quantity written in a larger prefix
will often have a smaller numeric value.
Example 1: \(2500\ \mu\mathrm{m}\) to \(\mathrm{m}\)
The micro prefix means \(10^{-6}\), while the plain meter has exponent \(0\). So the factor is \(10^{-6}\).
Micrometers to meters.
\[
\begin{aligned}
2500\ \mu\mathrm{m}
&= 2500 \cdot 10^{-6}\ \mathrm{m} \\
&= 2.5 \cdot 10^{-3}\ \mathrm{m} \\
&= 0.0025\ \mathrm{m}
\end{aligned}
\]
Example 2: \(1.5\ \mathrm{Mg}\) to \(\mathrm{kg}\)
The source prefix is mega, which means \(10^6\), and the target prefix is kilo, which means \(10^3\). The difference is
\(10^{6-3} = 10^3\). So the value must be multiplied by one thousand.
Megagrams to kilograms.
\[
\begin{aligned}
1.5\ \mathrm{Mg}
&= 1.5 \cdot 10^{6}\ \mathrm{g} \\
&= 1.5 \cdot 10^{3}\ \mathrm{kg} \\
&= 1500\ \mathrm{kg}
\end{aligned}
\]
This example also illustrates a common source of confusion: the SI base unit for mass is the kilogram, but SI prefixes are normally
attached to the gram when writing symbolic mass conversions. That is why \(1\ \mathrm{Mg}\) means one megagram, not one milligram.
Engineering perspective
Engineers and physicists often choose a prefix so that the resulting number is convenient to read. A current of \(0.0033\ \mathrm{A}\)
is often written as \(3.3\ \mathrm{mA}\), while a capacitance of \(4.7 \cdot 10^{-6}\ \mathrm{F}\) is more naturally written as
\(4.7\ \mu\mathrm{F}\). The physical quantity has not changed. Only the notation has been rescaled to make the number easier to interpret.
Engineering-style rescaling.
\[
\begin{aligned}
0.0033\ \mathrm{A} &= 3.3\ \mathrm{mA} \\
4.7 \cdot 10^{-6}\ \mathrm{F} &= 4.7\ \mu\mathrm{F}
\end{aligned}
\]
Because prefix conversion is purely decimal, it is also a useful way to build intuition about order of magnitude. Moving from
milli to micro makes the numerical value one thousand times larger. Moving from mega to kilo makes it one thousand times smaller.
A prefix ladder therefore provides a quick visual reminder of how many powers of ten separate two notations.
The main safety rule is simple: convert only between tokens that share the same base unit. A length can convert to another length,
a pressure to another pressure, and a frequency to another frequency. Prefix conversion never changes the physical type of the quantity.