Foundations — Physical quantities — fundamental and derived

This page assumes you have seen nothing. We will earn every symbol on the parent note — $n$, $u$, $[]$, $L$, $M$, $T$, the exponents like ${\text{s}}^{-2}$, and the arithmetic of units — one at a time, each anchored to a picture. When you finish, re-reading the parent should feel obvious.

1. Measuring = counting copies of a standard

Look at the figure. The blue bar is the thing we measure. The short orange tile is the standard — one unit. We slide the tile along and count: it fits 5 times. So the length is "5 standard-tiles".

We now have our first two symbols, each earned by the picture:

• $n$ = the count. Picture: how many tiles fit.
• $u$ = the standard tile itself. Picture: the single orange tile.

2. Why smaller unit ⇒ bigger number (the $n_1{u}_1=n_2{u}_2$ picture)

The same blue bar doesn't change when you switch measuring tiles. But a smaller tile fits more times.

WHY this happens: the actual amount of length is the product $n\times u$, not $n$ or $u$ alone. If you shrink $u$, then $n$ must grow to keep the product fixed. That is exactly the parent's equation:

3. The bracket notation $[]$ — "the unit of"

The parent writes things like $[F]$ and $[{\text{m s}}^{-2}]$. This is not a new operation — it is shorthand.

Why we need it: when deriving a unit we don't care about "5" or "72"; we only track the standards. The bracket lets us do algebra on units alone.

4. Exponents on units: what ${\text{s}}^{-2}$ actually says

Before force can appear, you must be comfortable with ${\text{s}}^2$ and ${\text{s}}^{-2}$.

Why physics loves the negative-exponent form: writing $\text{kg}\text{m}{\text{s}}^{-2}$ on one line keeps everything as a single product, so the exponent-arithmetic rules below apply cleanly.

5. The base symbols $L,M,T$ (and the other four)

Now the letters in the parent's table make sense.

Why letters instead of "metre"? Because a rule like "speed = length ÷ time" is true no matter which length-standard you pick. The letter $L$ captures "length-ness" without committing to metres, feet, or miles.

The figure shows the LEGO idea literally: three base bricks ($L$, $M$, $T$) snapping together to form derived words like speed ($L/T$) and force ($ML/T^2$).

6. Units multiply because quantities multiply

This is the single fact that powers every derivation in the parent.

7. Dimensionless quantities — when the units cancel

The parent's angle-mistake needs this idea.

Why it matters: dimensionless things (angles, refractive index, strain) are not new base quantities — nothing new was measured, just a comparison of two lengths. The Vectors you'll meet later use angles freely for this reason.

Prerequisite map

Equipment checklist

Cover the right side and test yourself before moving on.