How To Find Argument Of Complex Number
Complex planes play an important role in Mathematics. It is also known as the z-plane, which consists of two perpendicular lines called axes. The horizontal line represents real numbers and is called the real axis. The vertical line, on the other hand, represents imaginary numbers and is called the imaginary axis. The complex plane is used to represent the geometric interpretation of complex numbers. This plane is similar to the Cartesian plane which has the real and imaginary parts of a complex number along with the X and Y axes. There are two concepts related to complex numbers. They have magnitudes and arguments. In this article, we will learn in detail about the definition, properties, and reciprocal formulas of complex numbers.
What are complex numbers?
Contents
A complex number is a number written as a + ib, where “a” is a real number and “b” is an imaginary number. Complex numbers containing the symbol “i” satisfy the condition i2 = −1. Complex numbers can be thought of as an extension of a one-way number line. In the complex plane, a complex number denoted a + bi is represented as a point (a, b). It should be noted that a complex number with no real part, such as – i, -5i, etc., is said to be completely imaginary. Also, a complex number with an imaginary part of zero is called a real number.
Arguments of Complex Number Definition
The opposite of a complex number is defined as the angle of inclination from the real axis in the direction of the complex number represented on the complex plane. It is denoted by “θ” or “φ”. It is measured in standard units called “radians”.In this graph, the complex number is denoted by the point P. The OP length is called the magnitude or modulus of a number, while the angle at which the OP is inclined relative to the positive real axis is given as the argument of the point. P.
Complex number formula arguments
In polar form, a complex number is represented by the equation r (cos θ + i sin θ), where θ is the argument. The argument function is denoted by arg(z), where z is a complex number, i.e. z = x + iy. The complex argument calculation can be done using the following formula:arg(z) = arg(x + iy) = tan-1(y / x)Therefore, the argument θ is represented as: Read more: How to Skip Christmas | Top Q&Aθ = tan-1 (y / x)
Attributes of arguments of complex numbers
Let’s discuss a few properties shared by the arguments of complex numbers. Assuming that z is a non-zero complex number and n is an integer, thenarg (zWOMEN) = n args (z)Assuming, z1 and z2 are two complex numbers, the following isoforms are:
- arg(z1/z2) = arg(z1) – arg(z2)
- arg(z1 z2) = arg(z1) + arg(z2)
How to find arguments of complex numbers?
Example
Question:Find the argument of the complex number 2 + 2√3i.Solution: Let z = 2 + 2√3i. Here, the real part, x = 2 Read more: how to build a wooden driveway gate Imaginary part, y = 2√3 We know that the formula for finding the argument of a complex number isarg(z) = tan-1(y /x) arg(z) = tan-1 (2√3 / 2) arg(z) = tan-1 (√3) arg(z) = tan-1 (tan π / 3) arg(z) = π / 3 Therefore, the argument of the complex number is π / 3 radians. Register with BYJU’S – Learning App and download the app to watch interactive videos. Read more: How to get the brad pitt . function
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