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Expressing -22i in Modulus-Amplitude Form: A Step-by-Step Guide
Expressing -22i in Modulus-Amplitude Form: A Step-by-Step Guide
In the realm of complex numbers, expressing a number in modulus-amplitude form (also known as polar form) is a fundamental skill. This article will guide you through the process of converting the complex number -22i into its modulus-amplitude form, illustrating how to plot it on the complex plane and calculate its modulus and argument.
Understanding the Complex Plane and Modulus-Amplitude Form
The complex plane is an essential tool in visualizing complex numbers. A complex number can be represented as a point (a, b) in the plane, where a is the real part and b is the imaginary part. For the complex number -22i, the real part is 0 and the imaginary part is -22, placing it on the negative y-axis.
Identifying the Modulus (or Magnitude)
The modulus of a complex number z a bi is given by:
[|z| sqrt{a^2 b^2}]For z -22i:
[|z| sqrt{0^2 (-22)^2} sqrt{484} 22sqrt{1} 22]The modulus of -22i is 22, as it is the distance of the point from the origin in the complex plane.
Calculating the Argument (or Amplitude)
The argument of a complex number is the angle that the line joining the point to the origin makes with the positive real axis, measured in radians. For the complex number -22i, which lies on the negative y-axis, the angle is:
[arg(z) pi - tan^{-1}left(frac{b}{a}right)]Since the real part (a) is 0 and the imaginary part (b) is -22:
[arg(z) pi - tan^{-1}(0)]However, we can also write:
[arg(z) pi frac{pi}{4} frac{3pi}{4}]The angle 3π/4 corresponds to the point -22i on the negative y-axis, making an angle of 3π/4 radians with the positive real axis.
Expressing -22i in Polar Form
Using the modulus and argument, we can express -22i in polar form:
[z 22 text{cis}left(frac{3pi}{4}right)]Alternatively, in exponential form:
[z 22 expleft(frac{3pi i}{4}right)]Conclusion
Understanding and expressing complex numbers in modulus-amplitude form not only enhances our comprehension of complex arithmetic but also provides a practical tool for solving advanced problems in mathematics, physics, and engineering. The complex number -22i, when expressed in this form, is 22|cis (3π/4)|, or 22 exp (3πi/4), reflecting its distance from the origin and its orientation in the complex plane.