Complex number/Citable Version: Difference between revisions
imported>Greg Woodhouse (definition of division) |
imported>Aleksander Stos (reworked) |
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The '''complex numbers''' <math>\mathbb{C}</math> are obtained by adjoining the [[imaginary unit]] <math> i = | The '''complex numbers''' <math>\mathbb{C}</math> are numbers of the form ''a+bi'', | ||
obtained by adjoining the [[imaginary unit]] ''i'' to the [[real number]]s (here ''a'' and ''b'' are reals). The number ''i'' can be thought of as a solution of the equation <math>x^2+1=</math>. In other words, its basic property is <math>i^2=1</math>. Of course, since the square root of any real number is positive, <math>i\notin \mathbb{R}</amth>. ''A priori'', it is not even clear whether such an object exists and that it ''deserves'' be called 'a number', i.e. whether we can associate with it some natural operations as addition or multiplication. Admitting for a moment that the positive answer is given for granted, we define | |||
<math>\mathbb{C} = \{ a + bi | a, b \in \mathbb{R} \}</math> | <math>\mathbb{C} = \{ a + bi | a, b \in \mathbb{R} \}</math> | ||
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<math>\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}</math> | <math>\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}</math> | ||
It turns out that with addition and multiplication defined this way, <math>\mathbb{C}</math> satisfies the [[axiom]]s for a [[field]], and is called the field of complex numbers. If <math>c = a + bi</math> is a complex number, we call <math>a</math> the real part of <math>c</math> and write <math>a = Re (c)</math>. Similarly, <math>b</math> is called the imaginary part of <math>c</math> and we write <math>b = Im (c)</math>. If the imaginary part of a complex number is <math>0</math>, the number is said to be real, and we write <math>a</math> instead of <math>a + 0i</math>. We thus identify <math>\mathbb{R}</math> with a subset (and, in fact, a subfield) of <math>\mathbb{C}</math>. | It turns out that with addition and multiplication defined this way, <math>\mathbb{C}</math> satisfies the [[axiom]]s for a [[field]], and is called the field of complex numbers. If <math>c = a + bi</math> is a complex number, we call <math>a</math> the real part of <math>c</math> and write <math>a = Re (c)</math>. Similarly, <math>b</math> is called the imaginary part of <math>c</math> and we write <math>b = Im (c)</math>. If the imaginary part of a complex number is <math>0</math>, the number is said to be real, and we write <math>a</math> instead of <math>a + 0i</math>. We thus identify <math>\mathbb{R}</math> with a subset (and, in fact, a subfield) of <math>\mathbb{C}</math>. | ||
Revision as of 03:01, 2 April 2007
The complex numbers are numbers of the form a+bi, obtained by adjoining the imaginary unit i to the real numbers (here a and b are reals). The number i can be thought of as a solution of the equation . In other words, its basic property is . Of course, since the square root of any real number is positive,
We then define addition and multiplication in the obvious way, using to rewrite results in the form :
To handle division, we simply note that , so
and, in particular,
It turns out that with addition and multiplication defined this way, satisfies the axioms for a field, and is called the field of complex numbers. If is a complex number, we call the real part of and write . Similarly, is called the imaginary part of and we write . If the imaginary part of a complex number is , the number is said to be real, and we write instead of . We thus identify with a subset (and, in fact, a subfield) of .
Algebraic Closure
An important property of is that it is algebraically closed. This means that any non-constant real polynomial must have a root in .
A Note on Notation
This article follows the usual convention in mathematics (and physics) of using as the imaginary unit. Complex numbers are frequently used in electrical engineering, but in that discipline it is usual to use instead, reserving for electrical current. This usage is found in some programming languages, notably Python.
