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. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis. The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates – the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams.
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These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. All complex numbers, which are represented in the form of a + bi are shown on the complex plane. It is also called as z- plane. It is used to represent the complex numbers established by the help of the real and the orthogonal – imaginary axis. Here if we represent the complex number along the real axis, we can represent the imaginary part along the y- axis. The plane so formed is called the Complex Number plane. We use the concept of the complex plane which allows a geometric interpretation of complex numbers. Now let us look at the complex numbers. The numbers which are written in the form of ( a + ib), where we have a and b as the Real Numbers and I = root (-1) is the representation of the complex number. The Set of all complex numbers are represented by C. For any complex number, z = a + ib, we have a as the real part of z, which is written as Re( z) and b is the imaginary part of z, written as Im( z). (4 + 3i), here 4 is real part of the complex number and 3 is the imaginary part of the complex number.
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We say that the number is a pure real number, when we have the Im (z) = 0, such as 4 , 7, root(3) are all the pure real numbers. On the other hand, we have the pure Imaginary Number, if the real part of the complex number i.e. a = 0. Thus, we write Re (z) = 0 2i, root (7) I, -3i/2 are all pure imaginary numbers. Modulus of the complex number z is represented as |z| = root ( a^2 + b^2). Suppose we have a complex number z = 3 + 4 I, then the modulus of z = root ( 9 + 16) = root (25) = 5.
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