Math was, and is, so often taught poorly by people unable to instill the joy and beauty of it. The diagram is evocative (and I'm a big believer in visualization) but it doesn't scale generally to The Theorem. It only works for integer values of the sides so you can draw the squared values as equal-sized square tiles in a square. (There is no way, for example, to evenly subdivide the box with area two in the header diagram.) In a way, the diagram demonstrates the problem!
As you may know, specific integer solutions for right triangles were known in antiquity. The ancient Egyptians knew about the 3-4-5 solution and used it to create right angles. All you need is a long rope with 12 evenly spaced knots or marks on it. Stake it out taut in a 3-4-5 spacing (as in the diagram), and there's your right angle.
Heh. I've long had a thing for number bases, and it wasn't long after I read "Hitchhiker" that I tried to find a base in which the phrase Arthur picked out with his homemade Scrabble tiles actually did give the answer 42. I got a kick out of it turning out to be 13, an infamous number.
And it suggests the Creator must have 13 fingers, leaving theologians to argue incessantly about which hand has five and which has six. :)
I mean in contrast with the common philosophical meaning of rational and irrational. As in irrational meaning not comprehensible by reason. If you can pinpoint quite comprehensible what the properties are of a certain kind of numbers, it's not legitimate anymore to call these numbers irrational and so we should find a better suited name for them :-)
Ah, yes, indeed. It leads to some interestingly twisted language. If the real numbers do underlie reality (a debatable point), then reality is real but irrational (since nearly all real numbers are irrational). Or if (as I suspect) it's the rational numbers that underlie reality, then reality is rational but not real. 😁
It helps to remember that "rational" comes from "ratio" which meant to calculate, reason, or plan. Later it came to mean the relative magnitudes of two things, a ratio. The rational numbers are all ratios, a/b, so it's this modern meaning that's being referenced. In this context, "irrational" means they're NOT ratios.
And the Greeks thought they were irrational in the philosophical sense, too. Supposedly, people got killed over this discovery.
Ok, rational in the meaning of “ratio” meaning a ratio of two integers still makes a lot of sense. For now let’s keep calling it rational numbers then! ;-)
Also a very good reminder for myself since in my language (Dutch) the common word for “ratio” isn’t “ratio”. But in English it makes total sense.
Why the heck Geometry I in 1962 didn’t use that visual illustration of the P-Theorem is beyond me.
Math was, and is, so often taught poorly by people unable to instill the joy and beauty of it. The diagram is evocative (and I'm a big believer in visualization) but it doesn't scale generally to The Theorem. It only works for integer values of the sides so you can draw the squared values as equal-sized square tiles in a square. (There is no way, for example, to evenly subdivide the box with area two in the header diagram.) In a way, the diagram demonstrates the problem!
As you may know, specific integer solutions for right triangles were known in antiquity. The ancient Egyptians knew about the 3-4-5 solution and used it to create right angles. All you need is a long rope with 12 evenly spaced knots or marks on it. Stake it out taut in a 3-4-5 spacing (as in the diagram), and there's your right angle.
I can't believe someone would make jokes in base 13.
Heh. I've long had a thing for number bases, and it wasn't long after I read "Hitchhiker" that I tried to find a base in which the phrase Arthur picked out with his homemade Scrabble tiles actually did give the answer 42. I got a kick out of it turning out to be 13, an infamous number.
And it suggests the Creator must have 13 fingers, leaving theologians to argue incessantly about which hand has five and which has six. :)
A proof of numbers being definitely something different than the other kind of numbers and at the same time being all but irrational?
I’m sorry, I don’t understand your question. 😞
I mean in contrast with the common philosophical meaning of rational and irrational. As in irrational meaning not comprehensible by reason. If you can pinpoint quite comprehensible what the properties are of a certain kind of numbers, it's not legitimate anymore to call these numbers irrational and so we should find a better suited name for them :-)
Ah, yes, indeed. It leads to some interestingly twisted language. If the real numbers do underlie reality (a debatable point), then reality is real but irrational (since nearly all real numbers are irrational). Or if (as I suspect) it's the rational numbers that underlie reality, then reality is rational but not real. 😁
It helps to remember that "rational" comes from "ratio" which meant to calculate, reason, or plan. Later it came to mean the relative magnitudes of two things, a ratio. The rational numbers are all ratios, a/b, so it's this modern meaning that's being referenced. In this context, "irrational" means they're NOT ratios.
And the Greeks thought they were irrational in the philosophical sense, too. Supposedly, people got killed over this discovery.
Ok, rational in the meaning of “ratio” meaning a ratio of two integers still makes a lot of sense. For now let’s keep calling it rational numbers then! ;-)
Also a very good reminder for myself since in my language (Dutch) the common word for “ratio” isn’t “ratio”. But in English it makes total sense.