This page builds every letter, symbol, and idea the parent note leans on — starting from a smart 12-year-old who has never seen any of it. Read top to bottom; each block is a brick for the next.
Why does the topic need both? Because the whole story is about a change: we compare "now" (N) to "the beginning" (N0). Without a fixed starting mark you cannot say how much has shrunk.
Figure 1. Two jars. On the left, at t=0, every dot is present — that whole count is N0. On the right, later in time, the orange crosses have already decayed and left; only the blue dots remain, and their count is N. Notice the blue count on the right is smaller — that shrinkage is the entire topic.
The Greek letter λ (say "lambda") looks like an upside-down y. It stands for one number.
What picture? Imagine each atom rolls a weighted die every second. λ is how loaded the die is toward "break now." Crucially, the die does not remember past rolls — the chance is the same on roll one and roll one million. Atoms do not age or wear out.
Why the topic needs λ: it is the single knob that sets everything — the speed of decay, the half-life, and the activity all come straight from λ.
Step A — count the decays in the slice. Turning a proportion into an equation just needs a multiplier — and that multiplier is exactly λ:
decays in dt=λNdt
Step B — every decay removes one atom. The change in the count, dN, is minus the number that decayed (minus, because they leave):
dN=−(decays in dt)=−λNdt
This is the master equation of the whole chapter. Divide both sides by dt to read it as a rate:
Figure 2. Two jars watched over the same tiny slice dt. The fuller jar (bigger N) loses more atoms (red X's) than the half-full jar, and it loses them in the same proportion. This is dN=−λNdt made visible: same λ, same dt, but more decays where there is more N.
Why the topic needs it: the decay curve does not care about the absolute number of atoms, only about the fraction. A jar of a million and a jar of a trillion lose the same proportion in the same time. Ratios let us forget the raw size and see the universal shape.
The master equation dtdN=−λN already used the derivative; here is what the two calculus symbols are and why this tool and not another.
Rearranging the master equation to prepare for adding up the slices. Starting from dN=−λNdt, divide both sides by N to gather all the "N stuff" on the left and all the "time stuff" on the right:
Figure 3. Three shapes over the same time span. The red dashed line is a straight drop — wrong, because it crosses zero and keeps going negative (you cannot have fewer than zero atoms). The blue curve is the true e−λt: steep at first, then flattening, always staying above zero. The green dotted steps mark one half-life after another — each equal-length step halves the height, and the curve never lands on zero.
These are the three "named landmarks" of the curve. The parent note derives them; here we name what each is.
Figure 4. All three landmarks on one decay curve. The green mark is T1/2, where the height has dropped to 0.5. The orange mark is the mean life τ (further along), where the height is about 0.37 — proof that τ>T1/2. The two short tangent lines show activity as the steepness of the curve: steep and loud early on, gentle and quiet later.
Every arrow says "you need the left box before the right box makes sense." The map bottoms out at the boxed decay law, which then branches into the three landmarks.
