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Units and Dimension

All measurements have two parts: a quantitative magnitude and a unit. If you want to do mathematical operations using different quantities, the magnitudes should keep the same unit.

Systems of Units

The most common types of systems are shown below: – C.G.S. or French system: this system has for length unit the centimetre (cm), for mass the gram (g) and, finally for time the second (s) – F.P.S. system or British system: the length is expressed in feet (ft), mass in pounds (lb) and time in seconds (s) – M.K.S. system or metric system: length, mass and time are expressed in metre (m), kilogram (kg) and second (s), respectively

International System of Units

The International System of Units, or its acronymic name SI, defines a set of standard units that allow the measurement of all kinds of physical quantities. The SI system was established during the 11th General Conference of Weights and Measures in 1960. The SI has been adopted by the international community. It enlists the magnitudes and basic and derived units.

SI Basic Units

The International System of Units, or SI, defines seven magnitudes or basic units. The quantities and their units in the SI system of units are shown below:
  1. Mass: Expressed in kilograms or kg. The origin of this measurement is explained by the mass contained in a platinum-iridium cylinder with 90% platinum and 10% iridium. This cylinder also has a height and diameter of 39 mm.
  2. Length: The unit defined is the metre or “m”, which is the length travelled by light in a vacuum in 1/1299792458 thousandths of a second.
  3. Time: The base unit of which is the second. This unit refers to the time taken by 9,192,631,770 cycles of radiation emitted by an atom of cesium-133.
  4. Temperature: The unit in this case is the Kelvin or K. 1K is 1/273.16 expresses the temperature of the point where the three states of water (water, ice and steam) coexist.
  5. Current: The unit defined is the ampere or A. This measurement is defined as the direct current between two hypothetical conductors. The conductors are placed straight and parallel at one metre in a vacuum.
  6. Luminous intensity: With the unit defined as Candela or cd. This unit defines the luminous power emitted by a solid angle with a point light source in a certain direction. In fact, a common candle usually has a luminous intensity of 1 cd.
  7. Quantity of substance: Expressed in moles. A mole is the amount of substance expressed in particles. This is derived by using Avogadro’s number, NA = 6.022 141 79 x 1023mol-1.

SI Derived Units

There is a limitless quantity of derived units as they result from different operations using the base units. They are normally expressed in the basic units. Some examples of these units are shown in the following list:
Unit(s) Name SI Unit SI Unit Symbol Expressed in SI Base Unit Expressed in other SI units
Force, Weight newton N kg⋅m⋅s-2 –
Frequency hertz Hz s-1 –
Electric charge coulomb C s⋅A –
Electric potential (Voltage) volt V kg.m2.s-3.A-1 W/A
Inductance henry H kg.m2.s-2.A-2 Wb/A
Capacitance farad F kg-1.m-2.s4.A2 C/V
Resistance, Impedance, Reactance ohm Ω kg.m2.s-3.A-2 V/A
Electrical conductance siemens S kg-1.m-2.s3.A2 Ω−1
Magnetic flux weber Wb kg.m2.s-2.A-1 V⋅s
Magnetic flux density tesla T kg.s-2.A-1 Wb/m2
Energy, Work, Heat joule J kg.m2.s-2 N⋅m = Pa⋅m3
Power, Radiant flux watt W kg.m2.s-3 J/s
Angle radian rad m.m-1 –
Radioactivity becquerel Bq s-1 –
Luminous flux lumen lm cd cd⋅sr

The SI Unit Prefixes

There is a list of prefixes for the units to precisely express quantities. The prefixes multiply or divide by 10 the units. These prefixes are shown in the following table:
Multiples Name deca hecto kilo mega giga tera peta exa zetta yotta
Symbol da h k M G T P E Z Y
Factor 10x 1 2 3 6 9 12 15 18 21 24
Multiples Name deci centi mili micro nano pico femto atto zepto yocto
Symbol d h m μ n p f a z y
Factor 10-x -1 -2 -3 -6 -9 -12 -15 -18 -21 -24

Units and Dimensional Analysis and Unit Conversion

Dimensional Analysis is crucial to define a base unit and operate different quantities. Using conversion technique, a quantity can be expressed in a different unit, resulting in an equivalent result. The operations are simple and only need to apply conversion factors that define the equivalent value of a quantity in a different unit. Original quantity x conversion factor = equivalent quantity.