The metric system measures everything with one set of base units and one set of prefixes that multiply by powers of ten. A kilometer is 1,000 meters, a millimeter is one thousandth of a meter, and the same prefixes work for grams, liters, seconds and watts. That is why converting between metric units is mostly a matter of moving a decimal point.
The modern version is called the International System of Units, abbreviated SI (from the French Système international d'unités). This page covers the seven base units, the prefixes from nano to giga, and a reliable way to convert between them.
Contents
- What the metric system is
- The seven SI base units
- Metric prefixes from nano to giga
- Convert by moving the decimal point
- Worked examples
- Common mistakes
- Practice problems
- Frequently asked questions
- Related calculators
- See also
What the metric system is
The metric system began in France in the 1790s with two ideas: use decimal multiples (powers of ten) and define units from nature instead of from the length of a king's foot. Today it is the official system of measurement in almost every country, and the SI is what scientists use everywhere, including in the United States.
The seven SI base units
Every SI unit is built from seven base units. All other units, such as the newton or the joule, are combinations of them.
| Quantity | Unit | Symbol | How it is defined (since 2019) |
|---|---|---|---|
| Length | meter | m | The distance light travels in a vacuum in 1/299,792,458 of a second |
| Mass | kilogram | kg | A fixed value of the Planck constant, $latex h = 6.62607015\times 10^{-34}$ J·s |
| Time | second | s | 9,192,631,770 periods of radiation from a cesium-133 atom |
| Electric current | ampere | A | A fixed value of the elementary charge, $latex e = 1.602176634\times 10^{-19}$ C |
| Temperature | kelvin | K | A fixed value of the Boltzmann constant, $latex k = 1.380649\times 10^{-23}$ J/K |
| Amount of substance | mole | mol | Exactly $latex 6.02214076\times 10^{23}$ elementary entities |
| Luminous intensity | candela | cd | A fixed luminous efficacy for light of 540 THz |
The kilogram is the odd one out. It is the only base unit whose name already contains a prefix, so the prefixes are applied to the gram: a thousandth of a kilogram is a gram, and a millionth of a kilogram is a milligram, never a "microkilogram".
Derived units combine base units. Speed is measured in m/s. Force uses the newton, $latex 1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2$. Energy uses the joule, $latex 1\ \text{J} = 1\ \text{N}\cdot\text{m}$, and power uses the watt, $latex 1\ \text{W} = 1\ \text{J/s}$. The liter is not an SI unit, but SI accepts it: $latex 1\ \text{L} = 1\ \text{dm}^3 = 0.001\ \text{m}^3$.
Metric prefixes from nano to giga
A prefix is a word that goes in front of a unit to multiply it by a power of ten. Memorize this table and you can read almost any scientific measurement.
| Prefix | Symbol | Factor | Power of 10 | Where you meet it |
|---|---|---|---|---|
| nano | n | 0.000000001 | $latex 10^{-9}$ | A DNA double helix is about 2 nm wide |
| micro | µ | 0.000001 | $latex 10^{-6}$ | A red blood cell is about 7 µm across |
| milli | m | 0.001 | $latex 10^{-3}$ | A credit card is 0.76 mm thick |
| centi | c | 0.01 | $latex 10^{-2}$ | A fingernail is about 1 cm wide |
| deci | d | 0.1 | $latex 10^{-1}$ | Blood sugar is reported in mg/dL |
| (none) | 1 | $latex 10^{0}$ | A long step is about 1 m | |
| deca | da | 10 | $latex 10^{1}$ | Rare in daily life |
| hecto | h | 100 | $latex 10^{2}$ | Weather maps use hectopascals (hPa) |
| kilo | k | 1,000 | $latex 10^{3}$ | A 5 km run |
| mega | M | 1,000,000 | $latex 10^{6}$ | A large wind turbine produces a few MW |
| giga | G | 1,000,000,000 | $latex 10^{9}$ | A phone with 128 GB of storage |
Capitalization matters. A capital M means mega and a lowercase m means milli, so mixing them up is off by a factor of one billion. Symbols are never made plural: write 5 km, not 5 kms.
Convert by moving the decimal point
Every prefix stands for a power of ten, so converting between two prefixes means multiplying by a power of ten, and multiplying by $latex 10^n$ just moves the decimal point n places. Subtract the exponents:
$$\text{shift} = e_{\text{start}} - e_{\text{target}}$$
Here $latex e_{\text{start}}$ is the exponent of the starting prefix and $latex e_{\text{target}}$ is the exponent of the prefix you want.
- If the shift is positive, move the decimal point to the right. The number gets bigger because the target unit is smaller.
- If the shift is negative, move the decimal point to the left. The number gets smaller because the target unit is bigger.
- Fill any empty places with zeros.
Going from kilometers to meters, the shift is $latex 3-0=3$, so the point moves three places to the right. Going from millimeters to meters, it is $latex -3-0=-3$, so the point moves three places to the left. Between two prefixes you do not need to pass through the base unit: just subtract their exponents. The ladder below does the arithmetic and shows where the point lands.
Prefix ladder: slide the decimal point
Type a number and pick the base unit. Then tap one tile for the unit you start from and another for the unit you want. The digits never change, only the decimal point moves.
Worked examples
Example 1: kilometers to meters
A trail is 4.5 km long. How many meters is that? From kilo ($latex 10^3$) to the base unit ($latex 10^0$) the shift is $latex 3-0=3$, so the decimal point moves three places to the right.
$$4.5\ \text{km} = 4500\ \text{m}$$
Check: $latex 4.5\times 1000 = 4500$.
Example 2: millimeters to meters
A board is 250 mm long. The shift is $latex -3-0=-3$, so the decimal point moves three places to the left.
$$250\ \text{mm} = 0.25\ \text{m}$$
Example 3: grams to milligrams
A tablet contains 0.032 g of an active ingredient. The shift is $latex 0-(-3)=3$.
$$0.032\ \text{g} = 32\ \text{mg}$$
Example 4: between two prefixes
A hair is about 7.2 µm wide. To express it in nanometers, the shift is $latex -6-(-9)=3$.
$$7.2\ \mu\text{m} = 7200\ \text{nm}$$
Example 5: squared units
Convert 3.5 m² to cm². One meter is 100 cm, so one square meter is $latex 100^2 = 10{,}000$ square centimeters. The conversion factor is squared, and so is the shift: 4 places instead of 2.
$$3.5\ \text{m}^2 = 35{,}000\ \text{cm}^2$$
Moving the point only two places would give 350 cm², which is wrong because an area has two dimensions.
Common mistakes
- Moving the point the wrong way. Ask yourself: is the new unit smaller? Then you need more of them, so the number grows.
- Confusing M and m. Mega is $latex 10^6$, milli is $latex 10^{-3}$.
- Forgetting to square or cube the factor. For area use $latex 10^{2n}$ and for volume $latex 10^{3n}$, where n is the single-dimension shift.
- Treating the kilogram like a gram. The prefix is applied to the gram, so 5 mg is 5 milligrams, not 5 "millikilograms".
- Writing "Km" or "kms". The symbol for kilo is a lowercase k and symbols have no plural form.
Practice problems
1. Convert 0.75 km to meters.
The shift is $latex 3-0=3$, so move the point three places to the right: 0.75 km = 750 m.
2. Convert 3,400 mg to grams.
The shift is $latex -3-0=-3$, so move the point three places to the left: 3,400 mg = 3.4 g.
3. Convert 62 cm to kilometers.
The shift is $latex -2-3=-5$, so move the point five places to the left: 62 cm = 0.00062 km.
4. Convert 0.0045 mm to micrometers.
The shift is $latex -3-(-6)=3$, so move the point three places to the right: 0.0045 mm = 4.5 µm.
5. Convert 1.2 Gm to kilometers.
The shift is $latex 9-3=6$, so move the point six places to the right: 1.2 Gm = 1,200,000 km.
6. Convert 0.5 m² to cm².
The linear shift is 2, so the area shift is 4: 0.5 m² = 5,000 cm².
Frequently asked questions
Why is the kilogram the base unit and not the gram?
It is a historical choice. The gram was too small to be a practical standard, so the kilogram became the base unit, and the prefixes are attached to the gram for everything else.
Is a liter part of the SI?
Not officially, but it is accepted for use with the SI. One liter equals one cubic decimeter, so 1 L = 1,000 mL = 1,000 cm³.
How many millimeters are in a kilometer?
Go from kilo ($latex 10^3$) to milli ($latex 10^{-3}$): the shift is 6. So 1 km = 1,000,000 mm.
Meter or metre?
Both are correct. "Metre" is the spelling used in the SI brochure and in most English-speaking countries, and "meter" is standard in American English.
Related calculators
- Kilometers to Centimeters Calculator
- Meters to Centimeters Calculator
- Centimeters to Millimeters Calculator
- Scientific Notation Calculator