To make measurements, you need a system of units, that is, a set of magnitudes with which to compare those things that you want to measure.

The decimal metric system is a system of units in which the multiples and sub-multiples of the unit of measurement are interrelated by multiples or sub-multiples of $$10$$.

For example, the following belong to metric units: gram and kilogram (to measure the mass), meter and centimeter (for measuring length) or liter (to measure capacity).

Apart from the metric system, there are other systems of units: the Anglo-Saxon system, the so-called traditional measurements, etc.

## Measurements of length

The unit for measuring length is the meter. However, there are other units:

Name | Symbol | Equivalence |
---|---|---|

kilometer | km | 1000 m |

hectometer | hm | 100 m |

decameter | dam | 10 m |

meter | m | 1 m |

decimeter | dm | 0.1 m |

centimeter | cm | 0.01 m |

millimeter | mm | 0.001 m |

To convert an amount from one unit into another:

- If the original unit is less than the one we want to get, the amount will be divided by $$10$$ as many times as the number of rows that have to be “climbed” in the table above.
- If the original unit is larger than the one we want to get, the amount will be multiplied by $$10$$ as many times as the number of rows that have to be “gone down” in the table above.

If you want to convert $$1400$$ meters into decameters: One meter is less than a decameter therefore we have to divide $$1400$$ by $$10$$ once (because we have to go up once from the meter to decameter)

$$\dfrac{1400}{10}=140$$ decameters

That is, $$1400$$ meters are $$140$$ decameters.

## Measurements of mass

The unit for measuring mass is the gram. The other units that exist are:

Name | Symbol | Equivalence |
---|---|---|

kilogram | kg | 1000 g |

hectogram | hg | 100 g |

decagram | dag | 10 g |

gram | g | 1 g |

decigram | dg | 0.1 g |

centigram | cg | 0.01 g |

milligram | mg | 0.001 g |

To convert an amount from one unit into another one:

- If the original unit is less than the one we want to get, the amount will be divided by $$10$$ as many times as the rows that have to be "climbed" in the table above.
- If the original unit is larger than the one we want to get, the amount will be multiplied by $$10$$ as many times as the rows that have to be "gone down" in the table above.

If we want to convert $$23,4$$ hectograms into decigrams:

An hectogram is greater than a decigram, therefore we have to multiply $$23,4$$ by $$10$$ three times given that in the above table we have to move down three rows to go from hectograms to decigrams.

Therefore:

$$$23,4 \cdot 10 \cdot 10 \cdot 10 = 23.400$$$ decigrams.

Namely, $$23,4$$ hectograms are $$23.400$$ decigrams.

## Measurements of capacity

To measure capacity the unit used is the liter. The following table shows other common measurements of capacity:

Name | Symbol | Equivalence |
---|---|---|

kiloliter | kl | 1000 l |

hectoliter | hl | 100 l |

decaliter | dal | 10 l |

liter | l | 1 l |

deciliter | dl | 0.1 l |

centiliter | cl | 0.01 l |

milliliter | ml | 0.001 l |

To convert a number from one unit to another:

- If the original unit is less than the one we want to get, the amount will be divided by $$10$$ the same number of times as the number of rows that have to be “climbed” in the table above.
- If the original unit is larger than the one we want to get, the amount will be multiplied by $$10$$ the same number of times as the number of rows that have to be “gone down” in the table above.

If you want to convert $$400$$ milliliters to liters:

If we go from milliliters to liters we have to go up three rows, then we must divide by $$10$$ three times (which is the same as dividing by $$1000$$). Therefore:

$$400:1000=0,4$$ liters.

Namely $$400$$ milliliters are $$0,4$$ liters.

## Measurements of surface

To measure surfaces, the basic unit is the square meter, although the following units are also used:

Name | Symbol | Equivalence |
---|---|---|

square kilometer | km^{2} |
1.000.000 m^{2} |

square hectometer | hm^{2} |
10.000 m^{2} |

square decameter | dam^{2} |
100 m^{2} |

square meter | m^{2} |
1 m^{2} |

square decimeter | dm^{2} |
0.01 m^{2} |

square centimeter | cm^{2} |
0.0001 m^{2} |

square millimeter | mm^{2} |
0.000001 m^{2} |

To switch a number from one unit to another:

- If the original unit is less than the one we want to get, the amount will be divided by $$100$$ the same number of times as the number of rows that have to be “climbed” in the table above.
- If the original unit is larger than the one we want to get, the amount will be multiplied by $$100$$ the same number of times as the number of rows that have to be “gone down” in the table above.

If you want to convert $$0,003$$ square kilometers to square decameters, then, in order to pass from square kilometers to square decametres, we move down two rows in the table above, therefore we must multiply by $$100$$ twice (or what is the same, per $$10.000$$) . Therefore:

$$0,003\cdot10000=30$$ square decameters.

Namely, $$0,003$$ square kilometers are $$30$$ square decameters.

## Measurements of volume

The most commonly used unit for measuring volume is the cubic meter. Other units commonly used are:

Name | Symbol | Equivalence |
---|---|---|

cubic kilometero | km^{3} |
1.000.000.000 m^{3} |

cubic hectometer | hm^{3} |
1.000.000 m^{3} |

cubic decameter | dam^{3} |
1000 m^{3} |

cubic meter | m^{3} |
1 m^{3} |

cubic decimeter | dm^{3} |
0.001 m^{3} |

cubic centimeter | cm^{3} |
0.000001 m^{3} |

cubic millimeter | mm^{3} |
0.000000001 m^{3} |

To switch a number from one unit to another:

- If the original unit is less than the one we want to get, the amount will be divided by $$1000$$ the same number of times as the number of rows that have to be “climbed” in the table above.
- If the original unit is larger than the one we want to get, the amount will be multiplied by $$1000$$ the same number of times as the number of rows that have to be “gone down” in the table above.

If you want to convert $$6.000.000$$ cubic centimeters into cubic decimeters, you have to climb only one row, then it must divid it once by $$1.000$$:

$$6.000.000:1.000=6.000$$ cubic decimeters.

Therefore $$6.000.000$$ cubic centimeters are $$6.000$$ cubic decimeters.