What are complex numbers and how do they differ from real numbers?

AI Thread Summary
Complex numbers are defined as pairs of real numbers, represented as (a, b), where addition and multiplication are performed using specific rules. The discussion clarifies that the imaginary unit i, often defined as the square root of -1, cannot be simply understood in the same way as real numbers due to the ambiguity of square roots in complex numbers. Unlike real numbers, complex numbers do not have a natural ordering, which complicates their interpretation. The proper definition allows for the representation of complex numbers in the form a + bi, where i is identified with the pair (0, 1). This framework resolves the paradoxes associated with the square root of negative numbers by establishing a consistent arithmetic for complex numbers.
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actually what is complex number...
i know it's root of the -1
but how can we imagine the kind of number exist??
 
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Mathematics news on Phys.org
Please read this article:
http://physicspost.com/articles.php?articleId=118
 
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Of course, you can't DEFINE i as "square root of -1", not because -1 doesn't have a square root, but because, like any number, it has TWO.

Silly "paradoxes" like: i= [sqrt](-1) so i*i= i2= [sqrt](-1)*[sqrt](-1)= [sqrt](-1*-1)= [sqrt](1)= 1 depend on that ambiguity.

When we are working in the real numbers, we can specify sqrt[x] as meaning the POSITVE root. In complex numbers, we don't have any way of distinguishing "positive" or "negative" (the complex numbers cannot be an ordered field).

The way complex numbers are properly defined is as PAIRS of real numbers (a,b) with addition defined as (a,b)+ (c,d)= (a+b, c+d) and multiplication defined as (a,b)*(c,d)= (ac-bd,ad+bc). It then follows that numbers of the form (a,0) act like real numbers while (0,1)*(0,1)= (0*0-1*1,0*1+1*0)= (-1,0). If we identify (0,1) with i (having dodged the question of how to distinguish between roots), we can write any complex number as (a, b)= a+ bi and have i*i= (-1,0)= -1.
 
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