 So, remember the complex number x plus iy corresponds to the ordered pair xy and the ordered pair xy corresponds to a point in the plane. The distance of this point to the origin is square root x squared plus y squared, which motivates the following definition of the modulus of a complex number. Now, we should read this symbol as the modulus of z. However, because it looks like it, we often just call it the absolute value of z. Now, if a is a real number, this has a meaning as the absolute value of a. If z is a complex number, this has a meaning as the modulus of z. But remember, real numbers are complex numbers, so we should ask if z is a real number, is the modulus of z the same as the absolute value of z? The answer is left for the viewer to determine. The modulus of a complex number satisfies a number of properties, the two most important of which are the following. The modulus of a product is the product of the moduli. And the modulus of a sum is less than or equal to the sum of the individual moduli. The second property is usually referred to as the triangle inequality, because if you interpret the moduli as a length, this expresses the relationship between the sides of a triangle. Now, because it's a theorem, both of these have proofs. Yes, it turns out the first one is much easier to prove. The triangle inequality requires us to develop a little bit more mathematics. The useful strategy for learning mathematics? Always ask, how does this relate to things we've already done? Earlier, we talked about the conjugate of a complex number. So we might ask, how do conjugate and modulus interact? Let's find out. So let's start with a complex number. Then by definition, our modulus is the square root of x squared plus y squared. But now let's consider the conjugate of that complex number. The modulus of the conjugate will be, and remember, equals means replaceable. We have this claim that the modulus of z is square root of x squared plus y squared. We'll hear square root of x squared plus y squared. And so we can replace it with the modulus of z. And so this proves that for all complex numbers z, the modulus of z is the same as the modulus of the conjugate of z. Now remember that the modulus exists because we wanted a complex product to be real. In particular, the product of a number and its conjugate will be a real number. And if we look at that, we might notice that x squared plus y squared is part of the modulus formula. And that means that the square of the modulus will be... So over on the right hand side, we have the square of the modulus. On the left hand side, we have the product of the number and its conjugate, which gives us an alternative form for the modulus formula. For all complex numbers z, the square of the modulus is the product of the number. And it's conjugate.