What Radioactive Decay and Compound Interest Have in Common

Though they are both metals, gold and uranium have little in common.

Gold is permanent. It is made into jewelry. For many centuries, it was used as currency. Uranium, on the other hand, is not permanent. It is radioactive, and it decays (albeit very, very slowly) into lead. If you owned an ounce of gold and an ounce of uranium, after about 32 billion years¹, you would still have an ounce of gold but the uranium would turn into an ounce of lead.

Compound Interest

Let's pretend that you no longer want your ounce of gold. You sell it and put $1,000 into the bank at 5% interest. If your bank only compounds the interest once per year, then after one year, you would have $1,000 + $50 = $1,050. But, banks are typically a little more generous than that, so they compound the interest monthly. In that case, you would have $1,051.16 after one year.

Interest doesn't have to be just compounded annually or monthly. Any time interval is allowable. Banks could compound the interest weekly, daily, hourly, or by the second. In fact, they could compound it continuously. Using the above example, $1,000 at 5% interest that is compounded continuously would be worth $1,051.27 after one year².

The formula for continuously compounded interest is:

A = Pe^rt

where A = amount of money at the end of the time period;

P = amount of money at the beginning of the time period;

e = a magical number that is roughly equivalent to 2.72;

r = interest rate; and

t = length of the time period.

That "magical number" e is technically known as "Euler's number" or the "base of the natural logarithm." Like the other better known magical number pi (π), which is roughly 3.14, e is found everywhere in nature, sometimes in totally unexpected places³. Not only is it in finance, but e also pops up in statistics, population growth, physics, and various bizarre mathematical phenomena⁴. And yes, it plays a role in radioactive decay.

Radioactive Decay

While the gold that you sold is making money, the uranium is slowly decaying away. As it turns out, all radioactive elements follow the formula for exponential decay:

N_t = N₀e^-λt

where N_t = number of atoms at the end of the time period;

N₀ = number of atoms at the beginning of the time period;

e = a magical number that is roughly equivalent to 2.72;

λ = decay constant⁵; and

t = length of the time period.

Uranium decays very slowly. It would take 4.5 billion years for half of the ounce of uranium to decay into lead.

What Radioactive Decay and Compound Interest Have in Common

Notice how similar the formula for radioactive decay is to the formula for continuously compounded interest. They're nearly identical. They both involve some stuff (money or atoms) that is either growing or shrinking, a time period over which the stuff grows/shrinks, a rate at which the stuff grows/shrinks, and the magical number e. In fact, we could re-write the radioactive decay formula using the letters from the compound interest formula:

A = Pe^-rt

Now, it's perfectly clear: The formulas are almost identical. The only difference is that in the case of compound interest, e is raised to a positive power, while in the case of radioactive decay, e is raised to a negative power. That makes sense. In the compound interest example, the money is growing; in the radioactive decay example, the original atoms are disappearing.

What's Up with e?

Indeed, e is a very strange number. Unlike π, it's not based on anything geometrical. Instead, it is a number involved in rates of change, which is why formulas that describe growth/decay often contain it. The Numberphile, a mathematician and YouTube sensation, calls e "the natural language of calculus."⁶

So, tuck this little number away in your memory. The next time somebody discusses the wonders of π, wow them with the even greater wonders of e.

Notes/Sources

(1) The most common isotope is uranium-238, which has a half-life of roughly 4.5 billion years. After 7 half-lives (~32 billion years), less than 1% of the original uranium would be left.

(2) The formula for compounding interest over various time intervals is: A = P(1 + r/n)^nt, where A = amount of money at the end of the period, P = amount of money at the beginning of the period, r = interest rate, n = number of compounds per time period, and t = the length of the time period.

(3) This is an excellent video about where e can be found.

(4) The best example of a bizarre mathematical phenomenon is Euler's identity: e^iπ + 1 = 0. This formula is so incredibly bizarre because it contains two irrational and seemingly unconnected numbers (e and π), along with the number "i," which is imaginary. (Don't ask.) When e is raised to the power of iπ, the result is -1.

(5) The decay constant is essentially a rate, and it is inversely proportional to the half-life of the isotope.

(6) Here is Numberphile's video.