Unit Conversion: Factors, Dimensions and SI Prefixes

What is a conversion factor?

A conversion factor is a ratio equal to one. Because 1 in = 25.4 mm, the ratio 25.4 mm / 1 in equals one, and multiplying by it changes the unit without changing the quantity.

The two sides of that ratio measure the same thing, so the ratio itself has no size. It is one, written in a form that carries a unit. Multiplying by one is always safe, which is why the number on the page changes and the thing being measured does not.

Every conversion in this family is that single move, repeated. Choose the ratio, point it the right way round, multiply. If you have another dimension to convert, the index lists every quantity the family covers.

How do you know whether to multiply or divide?

Write the factor as a fraction and put the unit you are leaving underneath. It cancels, and the arithmetic decides for you whether that was a multiplication or a division.

Write the factor so the unit you are leaving cancels: 12 in x (25.4 mm / 1 in) = 304.8 mm. If the unwanted unit does not cancel, the factor is the wrong way up.

1. Write the value with its unit attached: 12 in.
2. Write the factor as a fraction, with the unit you are leaving underneath: 25.4 mm over 1 in.
3. Multiply, and strike out the unit that appears above and below the line.
4. Read what survived. The unit left standing is the unit of the answer.

Result: 12 in = 304.8 mm

Going the other way turns the same fraction upside down, so the inch ends up underneath and the millimetres cancel instead. The factor never changes; only which half of it sits below the line.

What is dimensional analysis?

Dimensional analysis is carrying the units through the arithmetic as though they were algebra. Units above and below the line cancel, and whatever is left is the unit of the answer. It is also called the factor-label method.

It doubles as a check on the whole calculation. Multiply a speed in metres per second by a time in seconds and the seconds cancel, leaving metres, which is what a distance should be. If the surviving unit is not the one you wanted, the arithmetic went wrong before the number did.

The same habit handles a unit built out of several others. A watt is a joule per second and a joule is a newton metre, so a power quoted in horsepower reaches watts through a chain of ratios rather than one.

What do the SI prefixes mean?

An SI prefix multiplies a unit by a power of ten: k is 10^3, M is 10^6, G is 10^9, m is 10^-3, micro is 10^-6 and n is 10^-9. The prefix belongs to the unit, so 1 kV is 1000 V.

A prefix and its unit behave as one symbol. Nothing separates them, which is why a prefix is squared along with the unit whenever the unit is squared.

Every electrical unit takes the same ladder, from microvolts to megawatts, and the electrical unit index groups the ones built on the ampere.

What are the seven SI base units?

The SI has seven base units: the second (s), metre (m), kilogram (kg), ampere (A), kelvin (K), mole (mol) and candela (cd). Every other SI unit is a product of powers of these seven.

A derived unit is a product of powers of those seven, sometimes under a name of its own. The newton is the kilogram metre per second squared and the pascal is the newton per square metre. Giving it a name changes nothing about what it is made of, which is why the cancelling above always ends.

Why does a squared unit need a squared factor?

When a unit is squared or cubed, its factor is squared or cubed with it. 1 m = 100 cm, so 1 m2 = 10 000 cm2 and 1 m3 = 1 000 000 cm3.

An area is a length multiplied by a length, so both lengths carry the factor and the factor appears twice. A volume is three lengths, so it appears three times.

This is the mistake that hides best. Converting an area with the length factor leaves an answer that is short by that factor all over again, and the result still looks like a plausible number.

Why does temperature need an offset instead of a factor?

Most units convert by a ratio alone, so zero maps to zero. Celsius and Fahrenheit do not: their scales start at different points, so each conversion needs an offset as well as a ratio.

Kelvin and Celsius use the same size of degree and differ only by an offset: T/K = t/degC + 273.15. Absolute zero is 0 K, which is -273.15 degC.

A ratio scale has a true zero: no length, no mass, no time. Celsius and Fahrenheit have zeros that were chosen rather than found, so a conversion has to move the zero before it scales the degree. You can run a temperature through the offset formula and watch both steps happen.

What makes a conversion factor exact?

An exact factor is a definition rather than a measurement. 1 in = 25.4 mm and 1 lb = 0.45359237 kg were fixed by the 1959 international yard and pound agreement, so they never gain another digit.

An exact factor is written out in full because there is nothing to round: the digits stop where the definition stops. A measured factor carries an uncertainty instead, and rounding it is a decision about how much of that uncertainty to keep.

One inch is exactly 25.4 mm, one foot is exactly 0.3048 m, one yard is exactly 0.9144 m and one mile is exactly 1609.344 m.

One avoirdupois pound is exactly 0.45359237 kg. One ounce is a sixteenth of a pound, 0.028349523125 kg, and one stone is fourteen pounds, 6.35029318 kg.

One constant is fixed the same way: the standard acceleration of free fall is 9.80665 m/s2, set by the 3rd General Conference on Weights and Measures in 1901 and used wherever a mass has to be read as a force.

The length values and the pound come from the same 1959 agreement, so you can convert a length with the exact inch factor or convert a mass with the exact pound factor and get digits that will not move.