http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/ Learning shouldn't hurt. Let's share the insights that made difficult ideas click. Mon, 15 Mar 2010 12:04:03 +0000 http://wordpress.org/?v=2.8.4 hourly 1 http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/#comment-274357 Dan Sat, 27 Feb 2010 14:39:09 +0000 http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/#comment-274357 Err, quite elegant for the most part, however I'm hung up on the division. Why do we take the conjugate of the denominator and not the numerator (obviously, algebraically this is obvious, but intuitively?) And why do we WANT to shrink the denominator by its modulus, why not shrink it by 56 or something arbitrary? Err, quite elegant for the most part, however I’m hung up on the division. Why do we take the conjugate of the denominator and not the numerator (obviously, algebraically this is obvious, but intuitively?) And why do we WANT to shrink the denominator by its modulus, why not shrink it by 56 or something arbitrary?

]]> http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/#comment-272109 Carl Tue, 02 Feb 2010 16:13:14 +0000 http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/#comment-272109 Thank you for your explanations. Can you do an article on imaginary exponentiation? I get that multiplication by an imaginary is like rotation, but what about multiplying a real by itself an imaginary number of times!? Thank you for your explanations. Can you do an article on imaginary exponentiation? I get that multiplication by an imaginary is like rotation, but what about multiplying a real by itself an imaginary number of times!?

]]> http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/#comment-264160 Abe Dalis Fri, 04 Dec 2009 13:22:43 +0000 http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/#comment-264160 The traditional difficulty in understanding the complex numbers is a man-made one. It reflects intellectual shortage in the course of definitions and thought construction. The term "imaginary" component in the definition of complex numbers is misleading. Furthermore, the definition of radicals or square roots in the field of Algebra has its own flaw where "negative" numbers become hard to define a radical for. Much of that is due to the historic and gradual accumulation of mathematical knowledge which characterizes mathematicians more as a cult---a culture with rich inheritance of thought, history, and terminology. The term "imaginary" for instance should have been called "rotational" instead. Imaginary things are understood as things that has no real reflection and exist only in the human mind. This is not a true case with complex numbers and their "imaginary" components. Complex numbers have real applications in physics when rotational physical phenomena arises---things that has a cyclic nature such as an alternative current in electricity. All what an imaginary component in a complex number means is that the magnitude of the complex number is specially aligned at a certain angle relative to the conventional positive real line axis. Again the real line is a misname for real numbers. So really, the complex numbers are nothing but real numbers in two-dimensional space with some closure field property pertaining to the roots of negative numbers. With this idea of rotation in mind, the study of complex numbers can be much easily understood. The traditional difficulty in understanding the complex numbers is a man-made one. It reflects intellectual shortage in the course of definitions and thought construction. The term “imaginary” component in the definition of complex numbers is misleading. Furthermore, the definition of radicals or square roots in the field of Algebra has its own flaw where “negative” numbers become hard to define a radical for. Much of that is due to the historic and gradual accumulation of mathematical knowledge which characterizes mathematicians more as a cult—a culture with rich inheritance of thought, history, and terminology. The term “imaginary” for instance should have been called “rotational” instead. Imaginary things are understood as things that has no real reflection and exist only in the human mind. This is not a true case with complex numbers and their “imaginary” components. Complex numbers have real applications in physics when rotational physical phenomena arises—things that has a cyclic nature such as an alternative current in electricity. All what an imaginary component in a complex number means is that the magnitude of the complex number is specially aligned at a certain angle relative to the conventional positive real line axis. Again the real line is a misname for real numbers. So really, the complex numbers are nothing but real numbers in two-dimensional space with some closure field property pertaining to the roots of negative numbers. With this idea of rotation in mind, the study of complex numbers can be much easily understood.

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