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TRUTH AND PROOF: AN EXAMPLE FROM EASY NUMBER THEORY

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:

statement

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:

sums.

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.

product

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.

2 comments on “TRUTH AND PROOF: AN EXAMPLE FROM EASY NUMBER THEORY

  1. sabresoftware
    November 2, 2016

    Here’s one that I figured out myself (although I am sure that there probably is a long dead mathematician with his name on it already).

    The sum of any string of numbers, i.e. 1, 2, 3, 4 . . n is (n * (n+1))/2

    Sum 1 + 2 + 3 = 6
    3 * 4 / 2 = 12/2 = 6

    Sum 1 + 2 + 3 + 4 + 5 = 15
    5 * 6 / 2 = 30/2 = 15

    You can use Excel to test a much larger range of numbers.

    Also the sum of numbers between any two numbers can easily be determined using the formula twice to subtract the summation of 1-to-n of the lower limit from the summation of the upper limit of 1-to-m.

    Useful? Well it was the day that I figured it out for some specific task that I was doing.

  2. simanaitissays
    November 3, 2016

    Good for you, sabresoftware! You’re in very good company. This summing a series of numbers was formulated by Carl Friedrich Gauss, 1777-1855, known to many as the Prince of Mathematicians. See http://www.coolmath.com/algebra/19-sequences-series/06-gauss-problem-arithmetic-series-03 for full details. Loosely, it’s recognizing pairs of numbers at either end of the string, working toward the middle.

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