Decimal prefixes

Two components are labelled 4 мкФ and 4 мФ. The number is the same, yet their capacitances differ by a thousandfold. How do a few letters convey this difference, and how can we check without a conversion table?

A millimetre ruler shows more divisions along the same segment than a metre ruler. Milli- means the unit is a thousand times smaller: 1\,\text{мм}=10^{-3}\,\text{м}. It multiplies the unit; it is not a property of the object. Reduce the unit by another thousandfold, and the number of units along the same segment grows a thousandfold. Prefixes let us write both small and large quantities conveniently.

A negative exponent does not mean a negative length: 10^{-3}=1/1000 is a positive fraction. Milli, centi and deci correspond to 10^{-3}, 10^{-2} and 10^{-1}. Thus 3 dm is three tenths of a metre, or 0.3 m. Imagine a metre divided into ten parts: three parts really are less than the whole. This picture checks both the exponent's sign and the decimal point.

For smaller scales, use micro, 10^{-6}; nano, 10^{-9}; and pico, 10^{-12}. Their Russian length symbols are мкм, нм and пм. Adjacent units in this sequence differ by a thousand: 1 μm contains 1000 nm. We compare unit sizes, not object lengths. Thus 1000 nm and 1 μm describe the same segment despite very different numbers.

In mm², the square applies to the entire length unit, including its prefix: 1\,\text{мм}^2=(10^{-3}\,\text{м})^2=10^{-6}\,\text{м}^2. Picture a square and convert each side separately: the result is the same. For a cubic millimetre, the factor is 10^{-9}. The prefix remains milli; the square or cube tells us how many times its factor appears in the product.

When a small unit makes a number unwieldy, group units together. Deca means 10 units, hecto 100, and kilo 1000. Two kilometres are two groups of a thousand metres, or 2000 m. Reverse the question: how many thousands are in 3000 m? Three, so it is 3 km. A larger unit gives a smaller numerical value; the route itself never changes.

Mega, giga and tera mean 10^6, 10^9 and 10^{12}; their Russian symbols are М, Г and Т. It is especially easy to confuse мВт and МВт: the first is a thousandth of a watt, the second a million watts. Their ratio is 10^6/10^{-3}=10^9, a billionfold difference. Letter case conveys scale just as importantly as the digits.

To convert between any two prefixes, first express the quantity in a common unit, then divide by the new unit's size. For example, 4 μF is 4\cdot10^{-6} F, while one mF is 10^{-3} F. The result is 4\cdot10^{-6}/10^{-3}=0{,}004 mF. Thus 4 mF is a thousand times the capacitance of 4 μF. Identical prefixes have a factor ratio of one, leaving the notation unchanged.

Express 6 μm in millimetres.

Micro means 10⁻⁶: 6 мкм = 6·10⁻⁶ м.

A millimetre equals 10⁻³ m. The number of millimetres is (6·10⁻⁶)/(10⁻³).

Subtract the exponents: 6·10⁻³ мм = 0,006 мм.

Check: a micrometre is smaller than a millimetre, so the number decreases when converting to millimetres.

6 мкм = 0,006 мм. This is six thousandths of a millimetre: a millimetre contains a thousand micrometres.

A prefix states the unit size: smaller units give a larger number for the same quantity.

Convert between prefixes by comparing their factors through a common unit, rather than guessing which way to move the decimal point.

A power of a unit includes its prefix: a square millimetre contains two factors of 10⁻³.

Next, we will use units to describe motion.