Complex Conjugate

 given a complex number x = a+bi, its complex conjugate is x =a−bi.

The conjugate allows us to do some interesting things. The first and probably most important is multiplying a complex number by its conjugate:

Notice that the second expression is a difference of squares:

This means that a complex number multiplied by its conjugate always produces a non-negative real number.

Another property of the conjugate is that it distributes over both complex addition and complex multiplication:

                                                                 

Have to admit I could'nt quite grasp what this last assertion mean. I understand 

distributivity in arihtmetic:

                                                           

Multiplication is distributive with respect to addition; and not the other way around!!

But here, what is going on!?

                                                                

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Also asked Gemini about the absolute value of a complex number:

                                                                       The absolute value of a complex number represents its distance from zero on the complex plane. Here's how to find the absolute value of 1 - 2i:

**Formula:**

The absolute value of a complex number z = a + bi (where a and b are real numbers) is calculated using the following formula:

|z| = sqrt(a^2 + b^2)

**Applying the formula:**

In this case, z = 1 - 2i, where a = 1 and b = -2.

|z| = sqrt(1^2 + (-2)^2)

|z| = sqrt(1 + 4)

|z| = sqrt(5)

Therefore, the absolute value of the complex number 1 - 2i is equal to the square root of 5. 

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