Comments on: Bombelli father of complex arithmetic? http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic/ somewhere near the beginning. Thu, 20 Nov 2008 21:32:27 +0000 http://wordpress.org/?v=2.5 By: Alex http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic/#comment-1409 Alex Fri, 27 May 2005 19:16:35 +0000 http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic#comment-1409 Excellent point--- that is certainly the natural way of thinking. And after this discussion, I realize I did a poor job expressing why Bombelli could in a sense be considered the father of complex arithmetic; I added that onto the post. Excellent point— that is certainly the natural way of thinking.

And after this discussion, I realize I did a poor job expressing why Bombelli could in a sense be considered the father of complex arithmetic; I added that onto the post.

]]> By: Mikael Johansson http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic/#comment-1406 Mikael Johansson Fri, 27 May 2005 16:28:48 +0000 http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic#comment-1406 I'd doubt that the origin of complex arithmetic would have been to start multiplying pairs of numbers - which can be made in a number of ways admittedly - but on the other hand I haven't really delved deep into mathematical history. My comment was more towards that if you instead of just taking [tex]R\times R[/tex] and give it an algebra structure start out by saying "Well, we seem to get negative square roots. Suppose that isn't just a weird impossible idea and let's see what happens." Given the general outlook during 15-18th century on mathematics and numbers, this is a slightly more plausible mindset for originators - and in that mindset defining [tex]i[/tex] by [tex]i^2=-1[/tex] and then calculating [tex](a+bi)(c+di)[/tex] is a more natural way of thinking than putting algebra structures on ordered pairs. I’d doubt that the origin of complex arithmetic would have been to start multiplying pairs of numbers - which can be made in a number of ways admittedly - but on the other hand I haven’t really delved deep into mathematical history. My comment was more towards that if you instead of just taking R\times R and give it an algebra structure start out by saying “Well, we seem to get negative square roots. Suppose that isn’t just a weird impossible idea and let’s see what happens.”

Given the general outlook during 15-18th century on mathematics and numbers, this is a slightly more plausible mindset for originators - and in that mindset defining i by i^2=-1 and then calculating (a+bi)(c+di) is a more natural way of thinking than putting algebra structures on ordered pairs.

]]> By: Alex http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic/#comment-1398 Alex Thu, 26 May 2005 21:55:58 +0000 http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic#comment-1398 Note that you're defining [tex](a,b)(c,d) = (ac - bd, ad + bc)[/tex]; you can't say that is the result of any other axiom. The question is why was multiplication defined that way? As opposed to say [tex](a,b)(c,d) = (ac, 0)[/tex]? The only thing you 'have' to keep is that if you multiply two real numbers together you get another, which the alternative satisfies. So why didn't we choose that? Note that you’re defining (a,b)(c,d) = (ac - bd, ad + bc); you can’t say that is the result of any other axiom. The question is why was multiplication defined that way? As opposed to say (a,b)(c,d) = (ac, 0)? The only thing you ‘have’ to keep is that if you multiply two real numbers together you get another, which the alternative satisfies. So why didn’t we choose that?

]]> By: Mikael Johansson http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic/#comment-1394 Mikael Johansson Thu, 26 May 2005 21:22:05 +0000 http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic#comment-1394 Eeeek. Here's to comment previews over the world. I forgot to slash my [tex] up there. =/ Eeeek. Here’s to comment previews over the world. I forgot to slash my [tex] up there. =/

]]> By: Mikael Johansson http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic/#comment-1393 Mikael Johansson Thu, 26 May 2005 21:21:20 +0000 http://www.tangentspace.net/cz/archives/2005/05/bombelli-father-of-complex-arithmetic#comment-1393 This may just be me being algebraic, arrogant and ignoring the historical part of it all - but if you start out supposing [tex]i^2=-1[/tex], then [tex](a+bi)(c+di)=ac+i^2bd+i(ac+bd)=ac-bd+i(ac+bd)[/tex] as witnessed. This may just be me being algebraic, arrogant and ignoring the historical part of it all - but if you start out supposing i^2=-1, then (a+bi)(c+di)=ac+i^2bd+i(ac+bd)=ac-bd+i(ac+bd) as witnessed.

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