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Complex numbers are numbers that have both a real part and an imaginary part, usually written in the form \(a + ib\). The magnitude of a complex number gives us the "distance" of the complex number from the origin on a complex plane. It's similar to the Pythagorean theorem and helps us understand how "big" a complex number truly is.

To find the magnitude of a complex number \(a + ib\), we use the formula:

• \(|a + ib| = \sqrt{a^2 + b^2}\)

This formula calculates the square root of the sum of the squares of the real part \(a\) and the imaginary part \(b\).

For example, if you had a complex number \((x-1) + i(y-3)\), its magnitude would be calculated as:

• \(|(x-1) + i(y-3)| = \sqrt{(x-1)^2 + (y-3)^2}\)

Finding the magnitude is useful in both mathematics and engineering, as it gives us a straightforward measurement of the "size" of any complex number, no matter its position on a complex plane.

Subtracting complex numbers involves separating their real and imaginary parts and then subtracting them separately. Start with two complex numbers, like \(z_1 = x_1 + i y_1\) and \(z_2 = x_2 + i y_2\).

To subtract \(z_2\) from \(z_1\), write:

• \((x_1 + i y_1) - (x_2 + i y_2) = (x_1 - x_2) + i(y_1 - y_2)\)

By separating the real and imaginary parts, we keep the operations simple and organized. For our specific expression, we start with \(z = x + iy\), and subtract \(1 + 3i\).

Thus, the subtraction becomes:

• \((x + iy) - 1 - 3i = (x - 1) + i(y - 3)\)

This subtraction results in a new complex number with adjusted real and imaginary parts.

Complex numbers are formed by two parts: the real and the imaginary. These two parts let us express and understand complex numbers in a clear, structured way. Consider a complex number \(z = x + iy\):

• The real part is \(x\).
• The imaginary part is \(y\).

Both parts are essential in performing operations with complex numbers, like addition, subtraction, and finding magnitude.

When performing subtraction such as \(z - (1 + 3i)\), always remember to separately handle these parts:

• Real Part: Adjusted by subtracting the real component (like subtracting \(1\) in \(x - 1\)).
• Imaginary Part: Adjusted by subtracting the imaginary component (like subtracting \(3\) in \(y - 3\)).

When working with complex equations, clearly identifying these parts is key to simplifying and solving them. It ensures each number part receives the right operation for correct solutions.