# Why not quarter-life?

The number of nuclei left after time $t$ in radioactive decay is given by:

$$N(t) = N_0 e^{-t/ \tau}$$

Now if we put $N(t)$ as $\dfrac{N_0}2$, we get half-life. But, if we had put $\dfrac{N_0}4$, we would have quarter-life, which is also independent of $N_0$.

Is there anything special about half-life as opposed to quarter-life

• Not really, $1/2$ is just a nice number. – knzhou Feb 26 '18 at 10:51
• Note also that if you go for $\frac{N_0}{e}$ then the time $t=\tau$ is the time it takes to become one-eeeth of its original value. – Farcher Feb 26 '18 at 11:48
• There is the very practical reason that if it were "quarter" you wouldn't know whether it referred to a quarter gone or a quarter left. – Hot Licks Feb 26 '18 at 13:15

The decay time $t_{1/2}$ of half the given number $N_0$ of atoms atoms is just convenient and visually appealing. Of the unit fractions it is also nearest to the decay time constant (mean lifetime) $\tau$ $t_{1/2}=0.6931 \tau$. The decay time to a unit fraction $1/n$ given by the positive integer $n$ is $$t_{1/n}=\tau \cdot ln(n)$$
The half life $t_{1/2}$ is the median. The time constant $\tau$ is the mean. There's nothing particularly meaningful about the quarter life.