What is the polar form of a complex number Z=3+4i?

Let’s start with a definition of a complex number. A complex number is one that can be written in the form a + ib, where a and b are real numbers and i is a symbol known as the imaginary unit, and that can be solved by the equation i2 = -1. René Descartes dubbed “i” as an imaginary number since no actual number can satisfy this equation. The real portion of the complex number a + ib is called the real part, while the imaginary part is called the imaginary part. The set of complex numbers can be represented by the symbol C. Complex numbers are recognized in the mathematical sciences as just as “real” as real numbers, despite the historical term “imaginary,” and are essential in many parts of the discipline.

When all three roots are real numbers, the solution in radicals of a typical cubic equation contains the square roots of negative numbers, a condition that cannot be corrected by factoring helped by the rational root test if the cubic is irreducible. In around 1545, Italian mathematician Gerolamo Cardano came up with the concept of complex numbers, but his knowledge was basic.

Work on the topic of generic polynomials eventually led to the fundamental theorem of algebra, which states that every polynomial equation of degree one or higher has a solution with complex numbers. As a result, complex numbers constitute an algebraically closed field in which each polynomial problem has a root.

The creation of complex numbers was aided by a number of mathematicians. Rafael Bombelli, an Italian mathematician, devised the methods for adding, subtracting, multiplying, and extracting the roots of complex numbers. The Irish mathematician William Rowan Hamilton created a more abstract framework for the complex numbers, which he extended to the idea of quaternions.

Apart from rectangular form, the polar form of a complex number is another approach to express a complex number. We usually write complex numbers as z = x+iy, where “i” stand for imaginary number. Complex numbers are expressed in polar form as a combination of modulus and argument. The absolute value of a complex number is also known as the modulus. In the coordinate system, polar coordinates of real and imaginary integers are used to express this polar form.

Equation of polar form

The equation of polar form of a complex number z = x+iy is z=r(cosθ + isinθ)

Where,

r=|z|=√(x2+y2)

x=r cosθ

y=r sinθ

θ=tan-1(y/x) for x>0

θ=tan-1(y/x)+π or

θ=tan-1(y/x)+180° for x<0 .

So, polar for of Z=3+4i will be

Here, x=3 and y=4

r=|z|=√(x2+y2) = √(32+42)=5

We know that,

x=r cosθ

y=r sinθ

θ=tan-1(y/x) for x>0

θ=tan-1(y/x)+π or

θ=tan-1(y/x)+180° for x<0

Therefore, in this case, x>0

We can write θ=tan-1(y/x)= tan-1(4/3)=53.130

Therefore, Polar form of Z=3+4i will be z = 5 (cos(53.13°) + i sin(53.13°))

Here, you must understand the application of imaginary number

Real numbers (or even rational numbers) are acceptable descriptions of data for most human tasks. When counting stones, fractions like 2/3 and 1/8 are useless, but when comparing the sizes of different collections of stones, they are crucial. Negative values like -3 and -5 have no relevance for measuring an object’s mass, but they’re crucial when keeping track of monetary debits and credits. Similarly, imaginary numbers are used in a wide range of disciplines and fields, including signal processing, control theory, electromagnetic, quantum physics, vibration analysis, cartography, and many more.

Here are some examples:

· In control theory, systems typically go to the frequency domain of Laplace transformation from the time domain. In the complex plane, the spolesandzeros system is then analyzed. The complex plane is used by the root locus, Nyquist plot, and Nichols plot techniques.

· Complex numbers for periodically different signals are utilized for in signal analysis and other disciplines for easier explanation. In the case of real functions, frequently in terms of sines and cosines, which reflect genuine physical values, matching complex functions are regarded as the true pieces. The absolute |z| value of the corresponding z in respect of a sine wave of a particular frequency is the amplitude and arg(z) is the phase.

· In the mathematical formulation of quantum mechanics, the complex number field is significant, in which complex Hilbert spaces give the environment for such a formulation, which is handy and probably most conventional. Complex numbers are used by the basic basis formulas of quantum physics, the Schrödinger equation and the Heisenberg matrix mechanics.

· Computer Systems arithmetic and logic is a valuable reference to a basic field of computer science and engineering. The purpose is to illustrate the concept behind the algorithmes rather than to address any direct applications by presenting algorithms for the conduct of operations like addition, subtraction, multiplication and division in digital computer systems. Alternative approaches are studied and the basic elements and reasons for theories and instances are explained.

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