Irrational, Imaginary, and Transcendental
( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17) ( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) (20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125) (88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149) (85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181) (57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197) (84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221) (60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257) (23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277) (160, 231, 281) (161, 240, 289) (68, 285, 293)
The more mathematically minded among you may recognize the above as a list of pythagorean triples. If you’ve forgotten what that means, I offer the below animated .gif to jog your memory.
Of course most right triangles don’t have sides that can be represented by whole numbers; fractions abound, and many commonly encountered right triangles in construction, fabrication, and other enterprises have hypotenuses that cannot even be expressed as a ratio of integers.
For example, a right triangle with two sides that are equal has a long side (hypotenuse) of a length that cannot be expressed as the ratio of two integers. This number is a familiar and indispensable friend to photographers. There are other examples these sorts of numbers that have proved to be useful; the Golden Ratio, Pi, and others.
You’d think that the discovery of these sorts of very useful numbers would have been met with great acclaim, but in fact legend says that the first person to prove/discover these sorts of numbers, Hippasus of Metapontum, was either murdered or exiled by his fellows.
My nephew is a smart lad. I know this from talking to him, but it is confirmed by the fact that he is presently pursuing a Ph.D in economics at MIT.
I saw him over Thanksgiving, and because of my earlier posts and the ensuing comment threads, I asked him if his work ever called for the use of i.
“I hate i!” came his reply without a moment’s hesitation.
This puts my nephew in good company. René Descartes was suspicious of i, and the numbers that that spill forth from i.
And my experience as a math teacher suggests that discomfort with i is not confined to the brilliant. Average students often (usually) find i discomfiting.
In reading up to make this post, I came across a category of numbers I’d never heard of before: Transcendental Numbers. As this is new information to me, I hope I’m correct when I say that transcendental numbers are numbers that are not algebraic numbers. Pi is a transcendental number; so it e. 4 is not. The ratio of the hypotenuse of a right triangle with two equal shorter sides to those shorter sides is not a transcendental number.
Color me not well informed about what transcendental numbers are, or why this property of transcendentalism is important.
Thanksgiving morning, as I was doing my research, I made a tweet or two, as mental markers, and as it happens, an iFriend who is a mathematician saw my tweets and that that prompted a short conversation where I learned (if I understood him correctly) that the aspects of mathematics that seem most self-evident, as in 1,2,3,4… is actually the least well-founded.
Today, as I talked about this with my wife on our long drive back from Flatbush to Montauk, I tried to conceive of a race of beings who apprehended the ratio of a circles diameter to its circumference, ratio of the hypotenuse of a right triangle with two equal shorter sides to those shorter sides as fundamental concepts, and have to build upon such a foundation novel ideas like whole numbers. I doubt it’s an original thought, but my brain couldn’t take the idea any further.
As mentioned in the comment thread of a previous post, the Help Wanted post has yield a hire. We’ve managed to poach a young artist and craftsman away from the Smithsonian and soon he and I and the rest of our build team will be making practical application of the properties of triangles; right, left, and other.
I’ve also just learned that before he was a US Supreme Court Justice, Oliver Wendell Holmes Jr. practiced admiralty law. In relation to the concept of property and property rights and how and why we build the social systems we build, I’ve been thinking about Ecuador and the “tuna wars” of the 1970s. More on that soon.