There are seven SI base units - the metre, the kilogram, the mole, the candela, the ampere, the second, and Kelvin. These units vary in terms of use from the general public to specific scientific fields, yet they form the foundations of measuring concepts such as length, mass and time. Without these units, standardisation would be rather difficult considering the various units already in use. For crying out loud, the British road signs prefer to use imperial units for length such as miles and feet, whereas most people (and their rulers) speak in (divisions of) metres. People do like to give the US flak for being almost purely imperial, when the UK is still extremely disordered in the units used in day to day life.
SI units are now all defined using physical constants, and sometimes they overlap with each other, as can be seen in the diagram in this post. For example, the second is needed to define the ampere, metre, kilogram, Kelvin, and candela. In this way, every SI unit aside from the mole is linked to each other by the second, which arguably makes it the most important unit.
Second
The SI unit of time, the second is defined as 9,192,631,770 periods of the frequency of the emission of caesium-133 microwaves. Why Cs-133 specifically and not just any element? Cs-133 is a stable isotope, for one, which means the frequency emitted can reliably be predicted, and caesium itself is used in very reliable atomic clocks which very rarely fall a few seconds off the actual time.
Seconds dictate our lives heavily. 86,400 seconds make up one day (partly based on a base-60 system) and is perhaps one of the more used SI units. The second also happens to define a various number of SI units, such as the metre (where the interval takes place during the 9,192,631,770 periods) and the:
Ampere
The ampere is the SI unit of current. It is equal to the flow of one coulomb of charge in one second (yet again defined by the Cs-133 definition), with the charge of one coulomb equivalent to the reciprocal of the elementary charge (that of a proton) - 1/1.6x10-19. From here, the field of electricity can be defined.Metre
The metre is the SI unit for length and has been around for over 200 years, since the French Revolution. The people then in charge wanted consistency and greater standardisation, due to the various different units used in France at the time, so made various changes such as in time and measurement. Some of these changes, such as decimal time and altering the names of the months of the year, didn't stick around for much longer, but the metre certainly has.
The metre is one interval of the speed of light (so about the distance travelled in 1/3x108 seconds) in a vacuum, though this definition hasn't always been set in stone. Originally, it was one ten-millionth of the distance between the Equator and North Pole, and has also been represented as a physical bar the length of a metre. This definition of the metre, however, does depend on the definition of another SI unit, that being the:
Kilogram
It's interesting how the SI unit for mass is not the gram but instead the kilogram. Indeed, its prefixed status makes it a bit of an anomaly amongst the other base units. However, it's much harder to accurately measure one gram of a substance than it would be a kilogram, much like how you can find the width of a sheet of paper by measuring the width of a stack and dividing by the number of sheets.
This one also dates back to the French Revolution, it being first defined as the mass of one litre (1m3) of water. A physical weight was then used to define the kilogram - if its mass changed in any way, the definition of the kilogram was also altered. Now, however, it's defined using the Planck constant, h. h is useful in many ways, particularly as it links energy and frequency: E = hf. As a result, h has SI units of kgm2s-1, so if you had a mass, you could use h to define the kilogram. It's rather interesting, and the NIST website has various articles on how h is used to define the kilogram.
Kelvin
This is the SI unit of temperature, which is merely Celsius but defined using absolute zero; 0K is -273.15℃. At 0K, particles stop moving as there's no energy present. You may thus assume that Kelvin is defined as Celsius is, where 0℃ and 100℃ are the melting and boiling points of water.
But reforms mean even the Kelvin must be defined using a physical unit defining the Kelvin is the already blogged about Boltzmann constant, where kB is the ratio of energy and temperature. In fact, this even means the Celsius scale isn't exactly placed with 0 and 100.
Mole
Moles are defined as the "amount of substance", or the number of atoms in 12g of 12C. More generally, the number of moles can be found using this equation:
mol = # particles/Avogadro's number
Avogadro's number is a constant: 6.02x1023. So moles are constant values as well, and this causes some people to claim that a mole isn't really a unit, though you could use the same argument against every SI unit - after all, they're all constants.
Moles are very specific to chemistry and so have no overlap with the other base units in the diagram.
Candela
The only SI base unit I have never used in school, the candela is the unit for luminous intensity - essentially, how bright is the visible light emitted by a source at a certain angle. Initially it was measured using a physical lamp, but in 1979 was defined using "a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian".
The physical constant that defines a candela is Kcd, where 540×1012 hz = 683 candelas. The frequency corresponds to green, and if using different colours of light, there are various documents explaining how to account for this. A steradian, meanwhile, is a unit formed by the angle between the radius of a sphere and a resultant circle of area radius2; this website explains the concept better. In 2018, the definition was refined, though still using Kcd along with other SI units - the kilogram, meter and second to be exact.
Epilogue
From all these units, you get various other SI units that are defined through the bases. For example, acceleration has units ms-2, the volt has units kgm2A-1s-3, and the katal is defined as mols-1. In much the same way, imperial units are also defined through the SI system.
As such, one could argue we have worked out all units, which is lovely considering science is far from complete.
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