Complex number | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-25

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Complex_number


00:01:15 1 Definition
00:05:00 2 Notation
00:05:19 3 Visualisation
00:05:52 3.1 Cartesian complex plane
00:06:06 3.2 Polar complex plane
00:06:59 3.2.1 Modulus and argument
00:07:30 3.3 Complex graphs
00:08:43 4 History in brief
00:09:22 5 Relations and operations
00:11:06 5.1 Equality
00:11:15 5.2 Ordering
00:12:12 5.3 Conjugate
00:16:05 5.4 Addition and subtraction
00:17:59 5.5 Multiplication
00:19:19 5.6 Reciprocal and division
00:19:29 5.7 Multiplication and division in polar form
00:20:43 5.8 Square root
00:21:14 5.9 Exponentiation
00:21:17 5.9.1 Euler's formula
00:21:26 5.9.2 Natural logarithm
00:26:59 5.9.3 Integer and fractional exponents
00:30:04 6 Properties
00:34:19 6.1 Field structure
00:37:55 6.2 Solutions of polynomial equations
00:38:16 6.3 Algebraic characterization
00:42:40 6.4 Characterization as a topological field
00:45:10 7 Formal construction
00:45:19 7.1 Construction as ordered pairs
00:50:33 7.2 Construction as a quotient field
00:53:05 7.3 Matrix representation of complex numbers
00:56:59 8 Complex analysis
00:57:08 8.1 Complex exponential and related functions
00:58:23 8.2 Holomorphic functions
00:58:44 9 Applications
01:00:35 9.1 Control theory
01:01:39 9.2 Improper integrals
01:03:26 9.3 Dynamic equations
01:03:35 9.4 Signal analysis
01:04:19 9.5 Geometry
01:07:13 9.5.1 Shapes
01:09:11 9.5.2 Fractal geometry
01:09:59 9.5.3 Triangles
01:16:14 9.6 Algebraic number theory
01:17:38 9.7 Analytic number theory
01:17:41 9.8 In Physics
01:17:50 9.8.1 Electromagnetism and electrical engineering
01:18:11 9.8.2 Fluid dynamics
01:19:08 9.8.3 Quantum mechanics
01:19:31 9.8.4 Relativity
01:19:53 10 History
01:20:17 11 Generalizations and related notions
01:24:57 12 See also
01:25:05 13 Notes
01:26:32 14 References
01:27:25 14.1 Mathematical references
01:27:49 14.2 Historical references
01:28:41 15 Further reading
01:29:49 16 External links
01:30:21 In Physics
01:30:30 Electromagnetism and electrical engineering
01:33:06 Fluid dynamics
01:33:21 Quantum mechanics
01:33:52 Relativity
01:34:23 History
01:45:25 Generalizations and related notions
01:49:29 See also



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SUMMARY
=======
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1.  Because no real number satisfies this equation, i is called an imaginary number.  For the complex number a + bi, a is called the real part, and b is called the imaginary part.  Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

  
    
      
        (
        x
        +
        1
        
          )
          
            2
          
        
        =
        −
        9
      
    
    {\displaystyle (x+1)^{2}=-9}
  has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) that is taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1:

  
    
      
        (
        (
        −
        1
        +
        3
        i
        )
        +
        1
        
          )
          
            2
          
        
        =
        (
        3
        i
        
          )
          
            2
          
        
        =
        
          (
          
            3
           ...