Complex Number

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.^{[note 1][1]}

Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

${\displaystyle (x+1)^{2}=-9}$

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:

${\displaystyle ((-1+3i)+1)^{2}=(3i)^{2}=\left(3^{2}\right)\left(i^{2}\right)=9(-1)=-9,}$

${\displaystyle ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}\left(i^{2}\right)=9(-1)=-9.}$

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.^[2]

Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.^[3] This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable i, with the rule i² = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.

Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.

The geometric identification of the complex numbers with the complex plane, which is a Euclidean plane (${\displaystyle \mathbb {R} ^{2}}$), makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis. The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space; e.g., the multiplication of two complex numbers always yields again a complex number, and should not be mistaken for the usual "products" involving vectors, like the scalar multiplication, the scalar product or other (sesqui)linear forms, available in many vector spaces; and the broadly exploited vector productexists only in an orientation-dependent form in three dimensions.

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Best theory notes on complex numbers

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