How To Find The Conjugate Of A Complex Number

Mathematics concept

Geometric representation (Argand diagram) of ${\displaystyle z}$ and its conjugate ${\displaystyle {\overline {z}}}$ in the circuitous plane. The complex conjugate is institute by reflecting ${\displaystyle z}$ across the real axis.

In mathematics, the complex cohabit of a complex number is the number with an equal real function and an imaginary part equal in magnitude but reverse in sign. That is, (if ${\displaystyle a}$ and ${\displaystyle b}$ are real, then) the complex conjugate of ${\displaystyle a+bi}$ is equal to ${\displaystyle a-bi.}$ The complex conjugate of ${\displaystyle z}$ is frequently denoted as ${\displaystyle {\overline {z}}.}$

In polar course, the conjugate of ${\displaystyle re^{i\varphi }}$ is ${\displaystyle re^{-i\varphi }.}$ This tin can exist shown using Euler'south formula.

The production of a complex number and its conjugate is a real number: ${\displaystyle a^{2}+b^{2}}$  (or ${\displaystyle r^{2}}$ in polar coordinates).

If a root of a univariate polynomial with existent coefficients is circuitous, and then its complex conjugate is also a root.

Annotation [edit]

The complex conjugate of a circuitous number ${\displaystyle z}$ is written equally ${\displaystyle {\overline {z}}}$ or ${\displaystyle z^{*}.}$ The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which tin be thought of equally a generalization of the complex cohabit. The 2d is preferred in physics, where dagger (†) is used for the cohabit transpose, every bit well as electrical engineering and computer technology, where bar note can be confused for the logical negation ("Non") Boolean algebra symbol, while the bar notation is more than mutual in pure mathematics. If a circuitous number is represented as a ${\displaystyle 2\times 2}$ matrix, the notations are identical.^{[ clarification needed ]}

Properties [edit]

The post-obit properties apply for all complex numbers ${\displaystyle z}$ and ${\displaystyle west,}$ unless stated otherwise, and can be proved by writing ${\displaystyle z}$ and ${\displaystyle west}$ in the form ${\displaystyle a+bi.}$

For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and sectionalisation:^[i]

$${\displaystyle {\begin{aligned}{\overline {z+due west}}&={\overline {z}}+{\overline {due west}},\\{\overline {z-west}}&={\overline {z}}-{\overline {west}},\\{\overline {zw}}&={\overline {z}}\;{\overline {westward}},\quad {\text{and}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}west\neq 0.\end{aligned}}}$$

A circuitous number is equal to its complex cohabit if its imaginary part is zip, or equivalently, if the number is real. In other words, real numbers are the just fixed points of conjugation.

Conjugation does not change the modulus of a complex number: ${\displaystyle \left|{\overline {z}}\right|=|z|.}$

Conjugation is an involution, that is, the conjugate of the conjugate of a complex number ${\displaystyle z}$ is ${\displaystyle z.}$ In symbols, ${\displaystyle {\overline {\overline {z}}}=z.}$ ^[1]

The product of a complex number with its conjugate is equal to the square of the number's modulus. This allows easy computation of the multiplicative inverse of a circuitous number given in rectangular coordinates.

$${\displaystyle {\begin{aligned}z{\overline {z}}&={\left|z\right|}^{2}\\z^{-1}&={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad {\text{ for all }}z\neq 0\terminate{aligned}}}$$

Conjugation is commutative nether composition with exponentiation to integer powers, with the exponential part, and with the natural logarithm for nonzero arguments:

${\displaystyle {\overline {z^{north}}}=\left({\overline {z}}\right)^{north},\quad {\text{ for all }}northward\in \mathbb {Z} }$

${\displaystyle \exp \left({\overline {z}}\right)={\overline {\exp(z)}}}$

${\displaystyle \ln \left({\overline {z}}\right)={\overline {\ln(z)}}{\text{ if }}z{\text{ is non-zero }}}$

If ${\displaystyle p}$ is a polynomial with real coefficients and ${\displaystyle p(z)=0,}$ then ${\displaystyle p\left({\overline {z}}\right)=0}$ as well. Thus, non-real roots of existent polynomials occur in circuitous cohabit pairs (run into Circuitous conjugate root theorem).

In general, if ${\displaystyle \varphi }$ is a holomorphic role whose restriction to the real numbers is real-valued, and ${\displaystyle \varphi (z)}$ and ${\displaystyle \varphi ({\overline {z}})}$ are defined, then

$${\displaystyle \varphi \left({\overline {z}}\correct)={\overline {\varphi (z)}}.\,\!}$$

The map ${\displaystyle \sigma (z)={\overline {z}}}$ from ${\displaystyle \mathbb {C} }$ to ${\displaystyle \mathbb {C} }$ is a homeomorphism (where the topology on ${\displaystyle \mathbb {C} }$ is taken to be the standard topology) and antilinear, if one considers ${\displaystyle \mathbb {C} }$ as a complex vector infinite over itself. Fifty-fifty though it appears to exist a well-behaved role, it is not holomorphic; information technology reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. Equally it keeps the real numbers fixed, it is an element of the Galois group of the field extension ${\displaystyle \mathbb {C} /\mathbb {R} .}$ This Galois grouping has just 2 elements: ${\displaystyle \sigma }$ and the identity on ${\displaystyle \mathbb {C} .}$ Thus the only 2 field automorphisms of ${\displaystyle \mathbb {C} }$ that get out the real numbers stock-still are the identity map and complex conjugation.

Use equally a variable [edit]

In one case a complex number ${\displaystyle z=10+yi}$ or ${\displaystyle z=re^{i\theta }}$ is given, its cohabit is sufficient to reproduce the parts of the ${\displaystyle z}$ -variable:

Furthermore, ${\displaystyle {\overline {z}}}$ can be used to specify lines in the plane: the set

$${\displaystyle \left\{z:z{\overline {r}}+{\overline {z}}r=0\correct\}}$$

is a line through the origin and perpendicular to ${\displaystyle {r},}$ since the real part of ${\displaystyle z\cdot {\overline {r}}}$ is zero only when the cosine of the bending between ${\displaystyle z}$ and ${\displaystyle {r}}$ is null. Similarly, for a fixed complex unit ${\displaystyle u=eastward^{ib},}$ the equation

$${\displaystyle {\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{ii}}$$

determines the line through ${\displaystyle z_{0}}$ parallel to the line through 0 and ${\displaystyle u.}$

These uses of the conjugate of ${\displaystyle z}$ every bit a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.

Generalizations [edit]

The other planar real algebras, dual numbers, and separate-circuitous numbers are also analyzed using circuitous conjugation.

For matrices of circuitous numbers, ${\textstyle {\overline {\mathbf {AB} }}=\left({\overline {\mathbf {A} }}\right)\left({\overline {\mathbf {B} }}\right),}$ where ${\textstyle {\overline {\mathbf {A} }}}$ represents the element-past-chemical element conjugation of ${\displaystyle \mathbf {A} .}$ ^[2] Contrast this to the property ${\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*},}$ where ${\textstyle \mathbf {A} ^{*}}$ represents the conjugate transpose of ${\textstyle \mathbf {A} .}$

Taking the cohabit transpose (or adjoint) of circuitous matrices generalizes complex conjugation. Fifty-fifty more general is the concept of adjoint operator for operators on (perhaps space-dimensional) complex Hilbert spaces. All this is subsumed past the *-operations of C*-algebras.

One may also define a conjugation for quaternions and split-quaternions: the conjugate of ${\textstyle a+bi+cj+dk}$ is ${\textstyle a-bi-cj-dk.}$

All these generalizations are multiplicative only if the factors are reversed:

$${\displaystyle {\left(zw\right)}^{*}=w^{*}z^{*}.}$$

Since the multiplication of planar real algebras is commutative, this reversal is not needed in that location.

There is too an abstract notion of conjugation for vector spaces ${\textstyle V}$ over the circuitous numbers. In this context, any antilinear map ${\textstyle \varphi :5\to Five}$ that satisfies

1. ${\displaystyle \varphi ^{two}=\operatorname {id} _{5}\,,}$ where ${\displaystyle \varphi ^{ii}=\varphi \circ \varphi }$ and ${\displaystyle \operatorname {id} _{V}}$ is the identity map on ${\displaystyle 5,}$
2. ${\displaystyle \varphi (zv)={\overline {z}}\varphi (v)}$ for all ${\displaystyle 5\in V,z\in \mathbb {C} ,}$ and
3. ${\displaystyle \varphi \left(v_{1}+v_{2}\right)=\varphi \left(v_{1}\correct)+\varphi \left(v_{2}\correct)\,}$ for all ${\displaystyle v_{ane}v_{2},\in V,}$

is chosen a complex conjugation, or a real structure. As the involution ${\displaystyle \varphi }$ is antilinear, information technology cannot be the identity map on ${\displaystyle V.}$

Of form, ${\textstyle \varphi }$ is a ${\textstyle \mathbb {R} }$ -linear transformation of ${\textstyle V,}$ if ane notes that every complex infinite ${\displaystyle V}$ has a real class obtained past taking the same vectors as in the original space and restricting the scalars to be existent. The above properties actually define a real structure on the complex vector space ${\displaystyle V.}$ ^[3]

One example of this notion is the cohabit transpose performance of complex matrices defined to a higher place. However, on generic circuitous vector spaces, at that place is no canonical notion of complex conjugation.

Come across also [edit]

• Accented square
• Complex conjugate line
• Complex conjugate representation
• Complex conjugate vector space – Mathematics concept
• Limerick algebra – Type of algebras, possibly non associative
• Conjugate (foursquare roots)
• Hermitian role – Type of complex function
• Wirtinger derivatives

References [edit]

1. ^ ^{a b} Friedberg, Stephen; Insel, Arnold; Spence, Lawrence (2018), Linear Algebra (v ed.), ISBN978-0134860244 , Appendix D
2. ^ Arfken, Mathematical Methods for Physicists, 1985, pg. 201
3. ^ Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29

Bibliography [edit]

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in department 3.iii).

Source: https://en.wikipedia.org/wiki/Complex_conjugate

Posted by: benitohoure1990.blogspot.com

Share this post