Math 2.0

object

Complex numbers

Description

A complex number is a number of the form a+bia + b i, where aa and bb are real numbers and ii is the imaginary unit, with i2=−1i^2 = -1. We write C\mathbb{C} for the set of complex numbers. The real number aa is the real part and bb is the imaginary part of a+bia + b i, see real and imaginary parts. Addition and multiplication follow the usual rules of algebra together with i2=−1i^2 = -1: (a+bi)+(c+di)=(a+c)+(b+d)i,(a+bi)(c+di)=(ac−bd)+(ad+bc)i.(a + b i) + (c + d i) = (a + c) + (b + d) i, \qquad (a + b i)(c + d i) = (a c - b d) + (a d + b c) i. With these operations, C\mathbb{C} is a field that contains R\mathbb{R} as the numbers a+0ia + 0 i.

Remarks

We picture C\mathbb{C} as the complex plane, with the point (a,b)(a, b) for a+bia + b i. Every polynomial equation of positive degree with complex coefficients has a solution in C\mathbb{C}. This is the fundamental theorem of algebra. Unlike R\mathbb{R}, the field C\mathbb{C} cannot be ordered.

Explanations

L1 human

The equation x2=−1x^2 = -1 has no real solution, because a real square is never negative. The complex numbers add a new number ii with i2=−1i^2 = -1, and then all the numbers a+bia + b i that you can build from it. For example, (1+2i)(3−i)=3−i+6i−2i2=5+5i(1 + 2 i)(3 - i) = 3 - i + 6 i - 2 i^2 = 5 + 5 i.

What this node needs for W2
  • section Invariants: a table of the main invariants (0 of 1)
  • edge instance_of out (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What depends on this

121 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Complex_number, lead section
  2. Web https://en.wikipedia.org/wiki/Complex_number, Definition and basic operations

What links here

Used by