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

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SUMMARY

Complex numbers are defined as pairs of real numbers (a, b), where addition and multiplication are performed using specific rules: (a, b) + (c, d) = (a + c, b + d) and (a, b) * (c, d) = (ac - bd, ad + bc). The imaginary unit i is identified with the pair (0, 1), leading to the conclusion that i * i = -1. Unlike real numbers, complex numbers do not have an inherent order, making them unsuitable for comparison as an ordered field. This distinction is crucial for understanding their properties and applications.

PREREQUISITES
  • Understanding of real numbers and their properties
  • Familiarity with basic algebraic operations
  • Knowledge of mathematical notation for pairs and functions
  • Concept of ordered fields in mathematics
NEXT STEPS
  • Study the properties of complex numbers in detail
  • Learn about the geometric representation of complex numbers on the complex plane
  • Explore applications of complex numbers in engineering and physics
  • Investigate the concept of complex conjugates and their significance
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Students of mathematics, engineers, physicists, and anyone interested in advanced algebra and its applications in various scientific fields.

newton1
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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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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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