In our physical world we’ve identified 5 basic dimensions: length (L), mass (M), time (T), temperature (Θ), & electric charge (Q).1 An object has each of these 5 physical quantities. Each of these quantities is distinct from and orthogonal to the others. These are the base dimensions.
Closely related to, but not the same as, the base dimensions are the base units: meter (m), kilogram (kg), second (s), Kelvin (K), Ampere (A), mole (mol), and candela (cd). The units we measure in are defined and maintained by International Organization for Standardization, and are the basis of the Metric system.
Base Dimensions & Units
| Base Dimension | Quantity | Common Variable | SI Unit | SI Symbol |
|---|---|---|---|---|
| L | length | l | meter | m |
| M | mass | m | kilogram | kg |
| T | time | t | second | s |
| Q | electric charge | C | ||
| electric current | i | ampere | A | |
| Θ | temperature | T | Kelvin | K |
| amount of substance | n | mole | mol | |
| luminous intensity | lv | candela | cd |
The ISO Units differ from the base dimensions in a few cases. Electric current is used rather than electric charge.
There are a bunch of other common quantities and units, such as force. However those are derived from combinations of the base set.
force=mass×(length/time/time)
When doing problems in physics and engineering, it’s sometimes handy to just check out the dimensions of the problem. If the answer to a problem doesn’t have the correct configuration of base dimensions… you got it wrong. This is called dimensional analysis. See the page on converting units for some additional detail.
What’s the Big Deal
Units are essential. Without units, numbers have no physical meaning or context. The number 2 does not exist somewhere in space. It is a concept. It is a concept within a rigid framework, granted, but it is still abstract in nature. I cannot find “2” apart from some other thing. I can find two cars. I can find two chairs. I can find two frogs, given enough time. I cannot, however, just find “2”.
Units with meanings that all humans agree upon are absolutely _vital_for economics, science, and everything stemming from those fields. Without the concept of a meter, we’d never have made it to the moon, we’d probably have never made it to any place to which we could walk.
👉 Units are what turns math into science & engineering
So. Always be very clear with what units you’re using. Without units, nothing means anything.
Everything is Arbitrary, Nothing is Sacred
One interesting thing to note is that units are an entirely human invention… and they derive from completely arbitrary origins. There is nothing sacred about the length of a meter, other than it’s what we all agree a certain length in space is. There’s nothing saying a second had to be exactly as long as a second is today. If our ancestors had decided to use a base-10 system for time, we’d be referring to some other unit of time as our “de facto standard”. The second would be the second division of the day by 10, I suppose. That would equivalent to 864 seconds (24 hrs / 100). We’d probably have decided that we wanted a more fine-grain unit of time and then went with “thirds”, which would be the 3rd division of the day by 10: 8.64 of our seconds. Then “fourths” would be slightly less than a our second in duration.
Footnotes
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I think there are actually more if you get into string theory, but I’m definitely not getting into that. ↩