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The absolute value of a complex number is also known as its modulus. It is a measure of the distance from the origin to the point representing the complex number on the Argand plane. For a complex number in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, the absolute value is calculated using the formula \[\sqrt{a^2 + b^2}\]. This formula comes from the Pythagorean theorem, which helps determine the length of the hypotenuse in a right triangle. For example, to find the absolute value of the complex number \(2 - 6i\), we plug in \(a = 2\) and \(b = -6\), resulting in \[ \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]. This value represents the distance from the origin to the point \( (2, -6) \) on the Argand plane.

The Argand plane is a two-dimensional plane used to represent complex numbers graphically. It is similar to the Cartesian coordinate system, but instead of representing just real numbers, it includes an additional dimension for imaginary numbers. When plotting a complex number on the Argand plane, the horizontal axis (real axis) represents the real part of the complex number, while the vertical axis (imaginary axis) represents the imaginary part. To visualize the complex number \(2 - 6i\), locate \(2\) on the real axis and \(-6\) on the imaginary axis. The point corresponding to this complex number will be at the coordinates \( (2, -6) \). This visual representation makes it easier to understand the relationships between complex numbers and to perform operations like addition, subtraction, and finding the modulus.

A complex number is composed of two distinct parts: the real part and the imaginary part. The real part of a complex number \(a + bi\) is the term \(a\), and the imaginary part is the term \(bi\). The imaginary part includes the imaginary unit \(i\), which is defined as \(\backslashsqrt{-1}\). For example, in the complex number \(2 - 6i\), the real part is \(2\) and the imaginary part is \(-6i\).

Understanding the real and imaginary parts is crucial for performing operations on complex numbers and for plotting them on the Argand plane. When identifying these parts, simply separate the terms involving \(i\) from those without it. This separation helps in graphing the number as well as in calculating additional properties like the absolute value.