Sunday, March 27, 2011

Sunday Numbers 2.0, Vol. 6:
Solving quadratics and the imaginary number i.

You might dimly remember the quadratic formula from a class you took in high school or college. One memorization technique is to sing the formula to the tune of Pop Goes The Weasel. This week, we are going to look at the formula, notice a problem that arises and see if we can get around it.

The original impetus of the problem was this. We have two numbers j and k, but instead of giving you the two numbers directly I tell you their sum j + k and their product jk. For example, if I tell you the sum is 10 and the product 21, with a little guess and check you should be able to figure out on of the numbers is 3 and the other 7. It doesn't really matter which one we call j and which one we call k, since 7 + 3 = 3 + 7 and 7 * 3 = 3 * 7. In terms of our variables a, b and c, the sum is -b and the product is c, while we set a = 1.

The problem is that guess and check can come up craps. If say the sum is 2 and the product is 5, you won't be able to guess the answer. Even more annoying, no pair of real numbers works. In this case, consider the quadratic formula where a = 1, b = 2 and c = 5. The quadratic formula has a square root in it sqrt( - 4ac), and the mathematical expression b² - 4ac is called the discriminant. Remember that we are not supposed to take the square roots of negative numbers, but in this case 2² - 4*1*5 = -16. We know the square root of 16 is 4, but what is the square root of -16? It was decided that is was 4 times the square root of -1. This means the two numbers that add up to 2 and multiply to 5 are 1 + 2*sqrt(-1) and 1 - 2*sqrt(-1).

Just because they wanted to solve all problems of this form and not for any practical application, mathematicians started working with the square root of -1, which the called the imaginary number and was eventually symbolized by the letter i. Little did they realized this new system of real numbers combined with imaginary numbers, which together are called complex numbers, would have real world applications in fields such as electrical engineering and quantum mechanics that were not even discovered when this algebra problem was kicking around. This is not the last time some mathematician works on a problem for his or her own amusement only to have practical applications discovered for this branch of math many years later, sometimes decades and sometimes centuries after the fact.

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