Imaginary line (mathematics)
Encyclopedia
In geometry
, an imaginary line is a straight line
that only contains one real point
. It can be proven that this point is the intersection point with the conjugated line
.
It is a special case of an imaginary curve
.
It can be proven that there exists no equation of the form
in which a, b and c are all real
coefficients. However there do exist equations of the form
, but at least one of the coefficients need be complex
.
As follows, it can be proven that, if an equation of the form
in which a, b and c are all real coefficients, exist, the straight line is a real line
, and it shall contain an infinite number of real points.
This property of straight lines in the complex projective plane is a direct consequence of the duality principle
in projective geometry
.
In the complex plane
(Argand Plane), we have a term called "imaginary axis".In Argand plane, y-axis is imaginary axis. All numbers in this axis are in form of 0+ib form.
in the Argand plane on the real axis (x-axis), denoted Arg(z). The argument can be easily found by following procedure:
If a+ib is any complex number foming angle A on real axis then ,
cosA = a/√a^2+b^2 sinA= b/√a^2+b^2 tanA=b/a
arg(z)=A
Domain of argument = R
Range
= (-180,180)
mod(z)=√a^2+b^2
In Argand plane, modulus
denotes distance between a complex number and the origin (0,0).
Example:
mod(z)=2 denotes locus
of all complex numbers z lying in circle of radius 2 at centre (0,0)
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, an imaginary line is a straight line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
that only contains one real point
Real point
A point in the complex projective plane is called real if there exists a complex number z such that za, zb and zc are all real numbers.This definition can be widened to complex projective space and complex projective hyperspaces as follows:...
. It can be proven that this point is the intersection point with the conjugated line
Conjugated line
The conjugated line of a straight line is the line that one becomes by taking the complex conjugate of each point on this line. One can prove that this is the same as taking the complex conjugates of the coefficients of this line....
.
It is a special case of an imaginary curve
Imaginary curve
In geometry an imaginary curve is an algebraic curve which contains a finite number of real points.-See also:*Imaginary point*Real point*Imaginary line *Real line*Real curve...
.
It can be proven that there exists no equation of the form
in which a, b and c are all realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
coefficients. However there do exist equations of the form
, but at least one of the coefficients need be complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
.
As follows, it can be proven that, if an equation of the form
in which a, b and c are all real coefficients, exist, the straight line is a real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
, and it shall contain an infinite number of real points.
This property of straight lines in the complex projective plane is a direct consequence of the duality principle
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
in projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
.
In the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
(Argand Plane), we have a term called "imaginary axis".In Argand plane, y-axis is imaginary axis. All numbers in this axis are in form of 0+ib form.
Argument
An argument is the angle or projection of any complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
in the Argand plane on the real axis (x-axis), denoted Arg(z). The argument can be easily found by following procedure:
If a+ib is any complex number foming angle A on real axis then ,
cosA = a/√a^2+b^2 sinA= b/√a^2+b^2 tanA=b/a
arg(z)=A
Properties of argument
- arg(AxB)=arg(A) + arg(B)
- arg(A/B)=arg(A) - arg(B)
- arg(z)=0 if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
z lies in +ve real axis - arg(z)=180 if and only if z lies in -ve real axis
- arg(z)=90 if and only if z lies in +ve imaginary axis
- arg(z)=-90 if and only if z lies in -ve imaginary line
- arg(z) lies in (0,90) in first quadrant, in (90,180) in 2nd quadrant, in(-180,-90) in 3rd quadrant, in(-90,0) in 4th quadrant.
Domain of argument = R
Range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
= (-180,180)
Modulus
Modulus of any complex no. a+ib ismod(z)=√a^2+b^2
In Argand plane, modulus
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
denotes distance between a complex number and the origin (0,0).
Example:
mod(z)=2 denotes locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
of all complex numbers z lying in circle of radius 2 at centre (0,0)
See also
- Imaginary pointImaginary pointA point in the complex projective plane is called an imaginary point if there exists no complex number z such that za, zb and zc are real....
- Real curveReal curveIn geometry a real curve is an algebraic curve that contains an infinite number of real points.-See also:*Imaginary point*Real point*Imaginary line*Real line*Imaginary curve...
- Conic sections
- Complex geometryComplex geometryIn mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
- Imaginary numberImaginary numberAn imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...