On cars, old, new and future; science & technology; vintage airplanes, computer flight simulation of them; Sherlockiana; our English language; travel; and other stuff
THIS PARTICULAR election campaign–thank goodness it’s almost over!– has been littered with innuendo, accusation and downright prevarication. As a counterpoint, I offer an example of truth, verified by genuine, unadulterated proof.
What’s more, this is not only mental relief from campaign fatigue. It’s also a brief sales pitch of a logic exercise for those who remember a tad of middle-school algebra. My example involves the collection of natural numbers, the sort we count with, 1, 2, 3, ….
The beauty of mathematics, even in basic number theory, is its generality: getting a lot from a little. Here’s my theorem to be proved, my statement of truth:
It’s no big deal to guess this is true. For instance, 3 + 5 = 8; 3 x 5 = 15. But the word “any” raises this guess from arithmetic into the generality of algebra.
As with Euclid and his geometry, we start with definitions. An even number is a multiple of 2, 2N, where N is any natural number. An odd number is one more than an even number: 2M + 1, M any natural number.
Here comes the proof. (Watch my hands; they never leave my wrists.)
The sum of any two odd numbers can be written (2N + 1) + (2M + 1). This is the same algebraically as 2N + 2M + 1 + 1 which is the same as 2N + 2M + 2, which, with easy factoring, is the same as 2(N + M + 1). Hence, this is twice the number N + M + 1, and thus it’s an even number. Q.E.D. (Latin: Quod erat demonstrandum, that which was to be shown.)
All these words can be summarized into pure algebraic format:
The second assertion, the product of two odd numbers is odd, uses just a tad more algebra: (a + b) x (c + d) = ac + ad + bc + bd.
Nothing beyond middle-school algebra, but it’s logical, verifiable truth. Forgive me for finding this so elegant, especially in comparison with the last year.
If you too would seek some added relief, prove that both the sum and product of any two even numbers are always even. ds
© Dennis Simanaitis, SimanaitisSays.