# Understanding the weird error in thinking

Discussion in 'Recreational Math' started by James Harris, Oct 16, 2004.

1. ### James HarrisGuest

Consider

x^2 + 4x + 3 = (x+3)(x+1)

where you have the unit 1 paired with 3, which, of course is not a
unit.

Now consider

x^2 + Ax + 3 = (x + 3u_1)(x + u_2)

where A is some integer other than 4 such that u_1 and u_2 are
irrational, and u_1 u_2 = 1, but guess what?

Both u_1 and u_2 cannot be algebraic integers.

Now that's a specific that's forced upon them based on the definition
of algebraic integers as roots of monic polynomials with integer
coefficients.

With

x^2 + Ax + 3 = (x + 3u_1)(x + u_2)

only u_2 can be such a root, so u_1 cannot be an algebraic integer.

Mathematically that's what follows, but some people take a leap from
there, and go from the proper conclusion that u_1 cannot be an
algebraic integer, to the position that the unit case, easily seen
with

x^2 + 4x + 3 = (x+3)(x+1)

has no corollary with irrationals!!!

It's an assertion of a fundamental difference between rational and
irrational numbers based on one simple thing: a particular number not
being the root of a monic polynomial with integer coefficients.

Once you get started on that path, it's easy to continue, as you can
find algebraic integers 'b', and 'a', such that b/a = u_1, and find
that 'a' is a factor within the ring of 3.

So it seems simple, u_1 is some kind of fraction, right?

Well, it *seems* simple enough, and people begin working from that
apparent simplicity to complexity building what are today called the
modern math tools, including those that Andrew Wiles used in his most
famous work.

The trouble is, the idea is easily proven to be wrong with some basic
algebra.

That is, the idea that because a particular irrational number cannot
be an algebraic integer, you know something about the factors of that
number, is just wrong.

It makes sense then to not try to create an artificial grab-bag, like
the ring of algebraic integers, and instead go down to the deeper
principle, which is to consider a ring which includes integers where
no integer is a factor of any other integer that it's not a factor of
in the ring of integers.

That is managed with two requirements:

1. No rational unit other than 1 or -1 exists in the ring.

2. No non-unit member of the ring is a factor of any two integers
that are coprime in the ring of integers.

Now then, with those requirements if you go back to

x^2 + Ax + 3 = (x+3u_1)(x+u_2)

then u_1 and u_2 may both be in that ring depending on the value of A.

Notice then they are units and the integer case

x^2 + 4x + 3 = (x + 3)(x + 1)

has an irrational corrollary.

What makes the story here sad is that many people have put a lot of
time and energy in the wrong idea, but mathematics is a hard
discipline.

None of that time and effort matters mathematically.

So why is it a big deal?

Well mathematics as a discipline is full of proud people who pride
themselves on work and ideas they long have thought were correct.
One of the more potent ideas within the math community itself is the
belief in their own perfection that the foundations of mathematics are
so well-worked that there are no errors in thinking.

You might say, since that idea has been out there for so long that
it's ironic that lurking within what math people teach is an error in
thinking older than any of them today, as it has sat there for
generations.

Now, for a while, yeah, they'll fight it. But they'll eventually
lose. Kids growing up today who've yet to learn the wrong way may
through various means learn of this problem and when the old guard
tries to teach them the wrong way--reject it and them.

Intriguingly, there should be a rather steady decline in math students
worldwide, which I'd think would follow a particular curve, like a
power curve, until there's something that pushes that society to
accept the truth.

Like a precipitating event might be required. I don't know what that
would be, but my guess is that it would have to be fairly huge.

James Harris

James Harris, Oct 16, 2004

2. ### John Roberts-JonesGuest

If A is divisible by 3 in the ring of rational integers, then any
ring containing u_1 also contains u_1(A/3 - u_1) = 1/3?
So what? In general they can't both be objects.
Oooh, a new long word!

But I can't begin to guess what you think it means.

John Roberts-Jones, Oct 16, 2004