As of October, 2016, Embarcadero is offering a free release
of Delphi (Delphi
10.1 Berlin Starter Edition ). There
are a few restrictions, but it is a welcome step toward making
more programmers aware of the joys of Delphi. They do say
"Offer may be withdrawn at any time", so don't delay if you want
to check it out. Please use the
link to let me know if the link stops working.
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The "Proof by Contradiction" is also known as reductio ad absurdum,
which is probably Latin for "reduce it to something absurd".
Here's the idea:
- Assume that a given proposition is untrue.
- Based on that assumption reach two conclusions that contradict each other.
This is based on a classical formal logic construction known as Modus
Tollens: If P implies Q and Q is false, then P is false.
In this case, Q is a proposition of the form (R and not R) which
is always false. P is the negation of the fact that we
are trying to prove and if the negation is not true then the original
proposition must have been true. If computers are not "not
stupid" then they are stupid. (I hear that "stupid
computer!" phrase a lot around here.)
Lets prove that there is no largest prime number (this is the idea of
Euclid's original proof). Prime numbers are integers with no exact integer
divisors except 1 and themselves.
- To prove: "There is no largest prime number" by
- Assume: There is a largest prime number, call it p.
- Consider the number N that is one larger than the product of
all of the primes smaller than or equal to p. N=2*3*5*7*11...*p
+ 1. Is it prime?
- N is at least as big as p+1 and so is larger than p and
so, by Step 2, cannot be prime.
- On the other hand, N has no prime factors between 1 and
p because they would all leave a remainder of 1. It has no
prime factors larger than p because Step 2 says that there are no
primes larger than p. So N has no prime factors and
therefore must itself be prime (see note below).
- We have reached a contradiction (N is not prime by Step 4, and N
is prime by Step 5) and therefore our original assumption that
there is a largest prime must be false.
Note: The conclusion in Step 5 makes
implicit use of one other important theorem: The Fundamental Theorem of
Arithmetic: Every integer can be uniquely represented as the product of
primes. So if N had a composite (i.e.
non-prime) factor, that factor would itself have prime factors which would also
be factors of N.