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Could Pythagoras do your Mathematics?


Before we answer that question, let’s see who Pythagoras was! Actually, there is so much fact and fiction surrounding Pythagoras that it is sometimes difficult to locate the real hypotenuse, but at least we can try!

Most Greeks at this time (6th Century BC) believed that gods and spirits moved in the trees and the wind and the lightning. And cults were popular all over the Greek world – led by mystics who promised to bring their members close to the gods in secret rites. Few Greeks could read or write. Pythagoras had a natural thirst for geometric proofs and rules, mainly based on lines and ratios. His “cult” was called the Pythagoreans.

Pythagoras was a native of the large Greek island of Samos (which has a plain in the southeast called “Pythagorio”). He gained such a reputation for wisdom and the magical arts that people began to whisper that he was son of the god Apollo. However, the political situation in Samos became oppressive under a local tyrant, Polycrates, who ruled harshly and wanted more and more taxation. So Pythagoras emigrated, as did many other refugees. He settled to the west, in Croton, a little island off the tip of Italy.

Pythagoras was a younger contemporary of the Greek philosopher Thales. Perhaps rumors of Thales’ exciting new game of string, straight edge, and shadows had spread, and people came from the neighbouring cities and islands to take part. Anyway, the young Pythagoras visited Thales, to learn this new way of thinking. The aging Thales must have been pleased with the young man’s keen interest and he taught Pythagoras all he knew. He also encouraged him to travel to the ancient lands and study the development of learning at its source.

Pythagoras followed this advice. Fired with enthusiasm by the stories of Babylon, he visited that fabulous city to absorb the learning of the Chaldean stargazers. Naturally, he also wanted to see the ancient pyramids, obelisks, and impressive temples of Egypt. In addition, he learned a great deal just by traveling to all the known parts of the Mediterranean world. During his long sea voyages, the Phoenician sailors taught him much about the importance of stars in navigation.
PythagWorldOn the open sea, he realized that the surface of the waters was not flat but curved. He could “see” this whenever another ship appeared in the distance. At first, only the top of its mast was visible over the horizon; then gradually the whole vessel would come into view as it sailed toward them. Surely then, the earth must be round! And what about the other heavenly bodies?

The moon, when it was full, was a round disk in the sky. As it waxed and waned, you could imagine that its surface was curved too, and partly in light and partly in shade.

So, this was the world of Pythagoras. Would he have coped with our modern Mathematics? Yes, I don’t doubt it! He would be quite surprised that there was such a concept as zero, and that we could have decimals and computers and calculators to ease the difficulty of working with numbers. And, he would have remembered the “Rule of Pythagoras” a different way. Still, he would have felt at home in a classroom of buzzing discovery – where students were engaged in trying to figure out the world around them, rather than just answering Exercise 7 in a textbook. However, he might have looked a bit “alternative”!

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NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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