The modulus of a complex number z gives its distance from the origin (0, 0) in the complex plane.
Here's a breakdown:
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Definition: If z is a complex number of the form z = a + bi, where 'a' is the real part (Re(z)) and 'b' is the imaginary part (Im(z)), then the modulus of z, denoted as |z|, is calculated as:
|z| = √(a² + b²) = √((Re(z))² + (Im(z))²)
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Geometric Interpretation: Imagine plotting the complex number 'z' as a point (a, b) on a 2D plane (the complex plane). The x-axis represents the real part ('a'), and the y-axis represents the imaginary part ('b'). The modulus |z| is then the length of the line segment connecting the origin (0, 0) to the point (a, b). This length is always a non-negative real number.
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Intuition: The modulus can be thought of as the "magnitude" or "absolute value" of the complex number. It tells you how "far away" the complex number is from zero.
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Example: Let z = 3 + 4i.
Then |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
This means the point (3, 4) in the complex plane is 5 units away from the origin.
In summary, the modulus of a complex number z gives its magnitude or absolute value, representing its distance from the origin in the complex plane, and is calculated as the square root of the sum of the squares of its real and imaginary parts.
