Re: i think this is interesting

In article <4389fafb$1_4@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jo Ling <donald.duck@xxxxxxxxxx> wrote:
>Simon Singh's book on Fermat's Last Theorem states that "i" was invented out
>of necessity,
>because mathematicians needed to define the square root of -1.

I think this is a half-truth. Up to the time of Cardan, an
equation such as x^2 + 1 = 0 was simply "impossible", and there was
no need to think about its formal solution, x = +- sqrt (-1). But
then the algebraic way to solve cubics was discovered, and turned out
to depend on an auxiliary quadratic. In many cases, the quadratic
was "impossible", but the results of using it were "possible", as
long as you "pretended" that sqrt (-1) was a perfectly normal number.

A specific example is x^3 = 15x + 4 [solution x = 4]. We try
x = p+q, so p^3 + 3pq(p+q) + q^3 = 15(p+q) + 4, satisfied by pq = 5
and p^3 + q^3 = 4, so that p^3 and q^3 are the roots of the quadratic
y^2 - 4y + 5^3 = 0. So p^3, q^3 are 2 +- sqrt (-121), which we can
[formally] solve for p and q, and hence find x = p+q.

So it wasn't so much sqrt (-1) that needed definition as rules
by which "impossible numbers" can be manipulated. Cardan [in 1540-odd]
just took a very simplistic and algebraic view; by 1813, we had the
Argand diagram, after which complex numbers could be understood as plane
co-ordinates and sqare/cube roots in terms of angle bi/trisection, so
that "complex" numbers had a very real interpretation, and their rules
were just traditional geometry.

>It set me thinking ... what's the square root of i?

Possibly curiously, this is a problem that defeated Leibniz,
who thought that the Fundamental Theorem of Algebra was false, on the
[incorrect] grounds that x^4 + 1 has no real factors. [Robert can
redeem his mistake somewhere around here! Of course, in Leibniz's
time, it was the Fundamental *Conjecture* of Algebra!]

--
Andy Walker, School of MathSci., Univ. of Nott'm, UK.
anw@xxxxxxxxxxxxxxxx
.

Relevant Pages

  • Re: i think this is interesting
    ... Fascinating! ... clearly has been done), and then the sqrt of that, ad infinitum. ... >>because mathematicians needed to define the square root of -1. ... > who thought that the Fundamental Theorem of Algebra was false, ...
    (uk.education.maths)
  • Re: Factoring problem solution
    ... >> indeed compute a square root in that algebraic number field. ... > was one of the most difficult parts of the complete algorithm... ... Linear algebra: `The linear algebra was performed on the MSRC cluster using ... were found on the seventh dependency. ...
    (sci.crypt)
  • Re: i think this is interesting
    ... > Simon Singh's book on Fermat's Last Theorem states that "i" was invented out ... > of necessity, ... > because mathematicians needed to define the square root of -1. ...
    (uk.education.maths)
  • Re: SIMPLE algorithm that took me by surprise.
    ... in the algebra, like with the proof that 1=2. ... But when you cancel out (a-b) ... the square root of an algebraic expression which is negative, ...
    (comp.programming)
  • Re: i think this is interesting
    ... Simon Singh's book on Fermat's Last Theorem states that "i" was invented out ... : of necessity, ... because mathematicians needed to define the square root of -1. ... on the complex plane." ...
    (uk.education.maths)