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Section2Ancient 8

At the dawn of civilization, \(8\) was a little known integer, nestled between 7 and 9. It led a quiet life, occasionally making itself useful to hunters as they checked their digits after run-ins with mastodons and rival clans. The oldest known joke involving \(8\) begins with a concussed cave man whose rival Thok stands before him, arms raised. Thok asks him how many fingers he's holding up. The groggy man furrows his great brow and answers 10, whereupon Thok says, “No. Eight. These are thumbs!” which he wiggles and then pokes into the poor man's eyes.

All joking aside, numbers began to be studied in earnest in many ancient civilizations, perhaps nowhere more seriously than in Greece. Around 500 B.C., the Greek philosopher Pythagoras and his disciples vigorously studied the counting numbers (1,2,3,...). By making careful observations it appeared to them that the workings of the universe could be understood via these numbers.

Perhaps inevitably, the Pythagoreans began to ascribe particular qualities to the counting numbers. Odd numbers were considered female, the even numbers male. The number 10 gained special prominence among the Pythagoreans because of the fact (coincidence, let's face it) that 1 + 2 + 3 + 4 = 10.

But the Pythagoreans are best known for thrusting a different number onto the world stage, and they did so by the tried and true method of trying to suppress it. Imagine the scandal that ensued on a particularly fine fall day on the island of Samos about 2500 years ago...

A pleasant breeze blew in from the Aegean Sea as Pythagoras air-dried after a morning swim. A young Pythagorean waited respectfully for the master to finish his morning routine before approaching. As great as Pythagoras was, he welcomed young and old, man and woman, to approach with questions. He encouraged communal thought and dialogue. After his swim, of course. And who could argue with the results? His school had established the basic tools with which one could explain everything in the universe: the counting numbers.

The young philosopher spoke: “Good morning, sir. How was your swim?”

“Invigorating. I believe I swam for five thirds of one hour. A fine ratio.”

“Indeed, sir. And I have been waiting to speak to you for four hours. I believe this makes our meeting auspicious.”


“Because of your theorem, sir.”


“Because \(3^2 + 4^2 = 5^2\text{,}\) sir.”

“Yes. Well, what is it, young man?”

“I just love your triangle theorem, sir.”

“It is a theorem for us all.”

“Of course, sir. But it rightfully bears your name. The Pythagorean Theorem. That will endure to the end of time.”

“As all truths will. Your question?”

“Yes. I have two sticks here. Notice they are the same length. Let me place them on the ground at right angles to each other with their tips touching. How far apart are their ends?”

“But you know the answer, by the theorem about which you spoke. It will be the number \(c\) such that \(c^2 = 1^2 + 1^2\text{,}\) if we assume the length of each stick is 1 unit.”

“Yes, but what value does \(c\) assume? I'm afraid it may not be a ratio of counting numbers.”

Did Pythagoras laugh heartily at this remark? Or did a kernel of fear take root in his stomach? History does not record his initial reaction. One wonders whether he instantly intuited the toppling of a central tenant of his school of thought at the hands of his own theorem. It is true that the value of \(c\text{,}\) which today we denote as \(\sqrt{2}\text{,}\) is not expressible as the ratio of counting numbers, and the Pythagoreans proved it by contradiction, thus giving the world one of the first and most famous uses of this proof technique.

The Pythagoreans also offer the first known effort to suppress a number's greatness. That a quantity physically constructed could not be represented as a ratio of whole numbers was a devastating result to them, one which may have precipitated the murder of an individual who let slip this fact to the outside world. But that's a story for another time. For all these reasons \(\sqrt{2}\) appears on most top 10 lists for the all time great numbers. But is it greater than \(8\text{?}\) Not a chance!

The Pythagoreans may not have done much for \(8\) directly beyond giving it the dubious distinction of being male, but they did introduce to the world the idea that some numbers are more dangerous than others. Numbers, up to the time of Pythagoras, were viewed for their quantity. Now they were encouraged to assert their quality; \(8\) could begin in earnest its ascent to a position of dominance in the number line. It did not take long.