x2 + 2x + 2 = 0She discovered that if it had a solution then there would be a number a such that a2=-1. Since there clearly was no such number, the equation had not solution. She missed her chance to (depending on your viewpoint) discover or invent complex numbers.
Fast Forward 6012 years.
In the year 2012 I wondered: is there a probability p such that if you flip a coin that has prob(H)=p twice the prob that you get HT is 1/2? This leads to
p(1-p)=1/2If you solve this you get p=(1+i)/2. Hence there is no such coin. WAIT A MINUTE! I don't want to miss the chance that my great...great grandmother missed! In the real world you can't have a coin with prob(H) = (1+i)/2. But is there some meaning to this?
More generally, for any 0 ≤ d ≤ 1 there is a p ∈ C (the complex numbers) such that ``prob(HT)=d.'' The oddest case (IMHO) was to take d=1. You then get that if a coin has prob(H)=(1+\sqrt(-3))/2 then prob(HT)=1. Does that mean it always happens? No since prob(TH)=1. Do the probs of HH, HT, TH, TT add up to 1? Yes they do since some are negative.
Is there an interpretation or use for this? I know that quantum mechanics uses stuff like this. Could examples like this be good for education? Are there non-quantum examples of the uses of this thatcould be taught in a discrete math course?