I understand that dydx=k∗y and when k>0 this is the law of natural growth and when k<0 this is the law of natural decay, but my textbook gives an example of radioactive decay as follows which confuses me:
Radioactive substances decay by spontaneously emitting radiation. If is the mass remaining from an initial mass of the substance after time t, then the relative decay rate
−1mdmdt (1)
has been found experimentally to be constant. (Since dmdt is negative, the relative decay rate is positive.) It follows that:
dmdt=km (2)
where k is a negative constant. In other words, radioactive substances decay at a rate proportional to the remaining mass. This means that we can use to show that the mass
decays exponentially:
m(t)=m(0)ekt (This equation I understand and accept, the above two confuses me)
Now eventually when the example becomes numerical, then of course k becomes a negative number since it is natural decay. However, the above general notation explanation confuses me because it keeps changing the sign of k=relative decay rate(from negative(1) to positive(2), but eventually when numerically worked it turns out to be a negative constant since it's natural decay). I know that k must be less than zero for decay but I'm just trying to fully grasp the notation signs that the textbook uses in the explanation above. Please clarify this. Thank you.