Why are numbers squared in the Pythagorean Theorem?

Answer 1

If the numbers that represent the lengths of the sides of a right triangle were

not squared, then the statement would not be true.

Coincidentally, Pythagoras did have a brother-in-law ... Murray ... who was

also something of a math buff. Murray published a paper ... well, in those days,

"publish" meant copying your ideas down by hand onto 75 or a hundred pieces

of parchment, and nailing them all up on trees around your neighborhood ... but

Murray figured out that if you had any triangle, with sides labeled 'a', 'b', and 'c',

then the lengths were related by the equation [ a + b = c ]. Exactly the same

formula as his brother-in-law's, but in Murray's equation, nothing was squared.

This story is just to show you what can happen if your work is careless and

inadequately researched. All of the land surveyors in Murray's town immediately

began to use his equation, to cut down on the volume of work required on two

new big contracts that they had just landed ... one to lay out plots for the entire

olive industry in Pelloponesia, and the other to mark out the foundations for a

whole new subdivision of condos and apartment buildings in Troy, the next

city over. Well, if you know your history, then you're way ahead of us. Murray's

formula was a disaster, the land surveyors totally screwed up everything they

touched, the Pelloponesians were so riled up that they touched off a series of

wars, and the developers from Troy, having taken a bath and lost a bundle,

headed across the bay to yell and holler on Murray, but they were not

accustomed to the cold maritime climate in Murray's town so as soon as they

arrived they all became hoarse. In any event, the whole project was a total

disaster as we said. In later years, it was discovered that Murray's equation

[ a + b = c ] was true only for isosceles triangles with base angles of zero.

But the damage was done. Greece was never anything more after that than a

lame early John Travolta vehicle and a failed economy, and nobody ever heard

of Murray again. And that's what can happen if you use Pythagoras' equation

but you don't square your numbers.