| There is a lot more to mathematics than you believe.
Consider the Isosceles
triangle theorem.
Three mathematicians attacked it with different goals.
Euclid was building a consistent body of theorems. He started from a handful of
axioms and added theorems one by one such that each one only depended on what he
had already proved. His proof of the Isosceles triangle theorem is therefore
rather complex. Standard textbooks have different goals, to educate. They
therefore use a theorem that Euclid proved later in order to simplify the proof
so that high-school students understand it. Thirdly, Pappus was looking for
simplicity and beauty. His proof is the simplest of all and far more
beautiful... once you manage to wrap your head around it.
Quite often I
while away long car journeys with maths problems. Since I'm driving, I can't use
a calculator. Say I want to multiply 15 x 28 in my head. That's fairly hard to
do, so I take the factors: 3x5 x 7x2x2. Then I rearrange them: 7x3 x 2 x 5x2. So
that's 21x2x10 which I can easily calculate in my head: 420.
Another trick I
use: say I get a 95 in my calculation. I will use 100 instead, but remember that
it is 5% too large. So when I get the final number I will take 10% of it by
shifting the decimal place, halve that to get 5% and subtract it to get an
accurate figure.
There is no one true way to do mathematics, there are
always an infinite number of ways to design an algorithm or a proof. Depending
on their goals, mathematicians and programmers choose one way or another. [ Reply to This | Parent | # ]
| 
