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When we talk about the absolute value of a complex number, we often refer to its magnitude. Complex numbers are of the form \(a + bi\), with \(a\) as the real part and \(b\) as the imaginary coefficient. The absolute value of a complex number tells us how "far" the number is from the origin in the complex plane. To find this distance, we use the formula:

• Absolute value: \(\sqrt{a^2 + b^2}\)

This formula is similar to finding the hypotenuse in a right triangle, where \(a\) and \(b\) are the lengths of the sides. For example, if given the complex number \(2i\), we recognize it as purely imaginary. Here, \(a = 0\) and \(b = 2\), so the absolute value would be \(\sqrt{0^2 + 2^2} = 2\). Calculating this is much like using the Pythagorean theorem.

Purely imaginary numbers are a special type of complex number. These have no real component, meaning their real part \(a\) is zero. They look exactly like \(bi\), where \(b\) is a non-zero number. Understanding these numbers can be very simple:

• If a complex number is \(3i\), its real part is 0 and its imaginary part is 3.
• In the complex plane, these numbers lie on the imaginary axis.

This means when dealing with numbers like \(2i\), the value is entirely along the imaginary axis. The concept is straightforward in that you only consider the coefficient of \(i\), without any real part to worry about. Purely imaginary numbers arise in many mathematical contexts, so understanding them is quite useful.

The modulus of a complex number is simply another term for its absolute value. It's a measure of the size or magnitude of that number, regardless of its position in the complex plane. The formula to determine the modulus is the same as the one used for the absolute value:

• Modulus: \(\sqrt{a^2 + b^2}\)

In practical terms, if you are given a complex number like \(5 + 12i\), its modulus would be calculated as: \(\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).For the purely imaginary number \(2i\), the modulus is \(2\), because \(\sqrt{0^2 + 2^2} = 2\).Remember, the modulus gives you a sense of scale and helps in comparing complex numbers based on their size or distance from the origin.