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Complex Plane, Modulus, Argument, and Graphical Representation of a Complex Number Hello, my name is Mohammad Ali Nayeem, a student at DIU studying Software Engineering. In Math 102, we delve into various concepts related to complex numbers, including the complex plane, modulus, argument, and their graphical representation. Complex Plane The complex plane is a two-dimensional plane used to visually represent complex numbers. It consists of a horizontal axis (the real axis) and a vertical axis (the imaginary axis). Any complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi can be plotted on this plane, where 𝑎 a is the real part and 𝑏 b is the imaginary part. Modulus The modulus (or absolute value) of a complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi is the distance of the point representing 𝑧 z from the origin (0, 0) in the complex plane. It is denoted by ∣ 𝑧 ∣ ∣z∣ and calculated using the formula: ∣ 𝑧 ∣ = 𝑎 2 + 𝑏 2 ∣z∣= a 2 +b 2 The modulus provides a measure of the magnitude of the complex number. Argument The argument of a complex number is the angle 𝜃 θ between the positive real axis and the line segment connecting the origin to the point representing the complex number. It is denoted by arg ⁡ ( 𝑧 ) arg(z) and is usually measured in radians. The argument can be found using the formula: 𝜃 = tan ⁡ − 1 ( 𝑏 𝑎 ) θ=tan −1 ( a b ) The argument indicates the direction of the complex number in the complex plane. Graphical Representation To graphically represent a complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi, plot the point ( 𝑎 , 𝑏 ) (a,b) on the complex plane. The modulus is the length of the line from the origin to this point, and the argument is the angle this line makes with the positive real axis. For example, the complex number 3 + 4 𝑖 3+4i can be plotted as the point (3, 4). Its modulus is: ∣ 3 + 4 𝑖 ∣ = 3 2 + 4 2 = 5 ∣3+4i∣= 3 2 +4 2 =5 Its argument is: 𝜃 = tan ⁡ − 1 ( 4 3 ) ≈ 0.93 radians θ=tan −1 ( 3 4 )≈0.93 radians Understanding these concepts helps in visualizing and manipulating complex numbers, which are fundamental in many areas of engineering and applied mathematics. Best regards, Mohammad Ali Nayeem DIU Software Engineering Student

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