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Complex numbers are numbers that have both a real part and an imaginary part, typically expressed in the form \(a + bi\), where:

• \(a\) is the real component.
• \(b\) is the imaginary component, paired with the imaginary unit \(i\), satisfying \(i^2 = -1\).

These complex numbers extend the idea of one-dimensional real numbers to the two-dimensional complex plane by using \(a\) and \(b\) as coordinates. They are crucial in many areas of mathematics and engineering as they provide solutions to equations that no real numbers can solve. In the example of \(-10 + 10i\), \(-10\) is the real part and \(10i\) is the imaginary part.

The magnitude or modulus of a complex number is a measure of its size or distance from the origin in the complex plane. For a complex number \(a + bi\), the magnitude is calculated using the formula: \[r = \sqrt{a^2 + b^2}\]This value provides the length of the vector representing the complex number. For the given complex number \(-10 + 10i\), the magnitude is calculated as:

• \((-10)^2 = 100\)
• \(10^2 = 100\)
• The sum \(100 + 100 = 200\)
• Taking the square root, \(\sqrt{200} = 10\sqrt{2}\)

Thus, the magnitude \(r\) is \(10\sqrt{2}\). This distance plays a central role when converting complex numbers to their trigonometric form.

The argument of a complex number is the angle that the vector representing the complex number makes with the positive x-axis in the complex plane. It is noted as \(\theta\) and can be calculated using the tangent function: \[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]However, the location of the complex number in the plane (quadrant) also affects the calculation. For \(-10 + 10i\), where \(a = -10\) and \(b = 10\), we first calculate:

• \(\tan^{-1}\left(\frac{10}{-10}\right) = \tan^{-1}(-1)\)

Since this complex number is in the second quadrant (negative real, positive imaginary), we add \(\pi\) to correct the angle:

• \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)

This gives us the correct argument for the number.

Polar coordinates express a complex number in terms of its magnitude (radius) and angle (argument). This is useful for simplifying the multiplication and division of complex numbers, among other computations. The trigonometric form of a complex number is given by:\[r(\cos \theta + i \sin \theta)\]where:

• \(r\) is the magnitude.
• \(\theta\) is the argument.

For the complex number \(-10 + 10i\), we calculated the magnitude \(r = 10\sqrt{2}\) and the argument \(\theta = \frac{3\pi}{4}\). Thus, its trigonometric (polar) form is:\[10\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\]This representation helps in visualizing and analyzing complex numbers in two dimensions, allowing for straightforward derivation of properties and behaviors.