 I want to offer another example that you might not be familiar with, and this is called the ring of Gaussian integers. It's commonly denoted as Z bracket I, where this I is understood to be the to be a complex unit that is a square root of negative one. So the ring of Gaussian integers, that's what they're named after Gauss, of course, the ring of Gaussian integers Z sub I, this would be all the integer combinations of an integer with a purely imaginary number. So that is to say all the numbers of the form a plus bi, where a and b are both integers. This thing kind of resembles the complex numbers, right? Because the complex numbers as a set can be described as all of the numbers of the form x plus yi, where x and y are arbitrary real numbers. So the complex numbers are forms by taking linear combinations of real numbers and purely imaginary numbers, where those coefficients of the real part or the purely imaginary part could be any real number. The Gaussian integers are by analog here, but now our combinations is the real part is an integer, and the imaginary part is also an integer. It turns out that this is an integral domain. For the same reason that the integers are an integral domain, the Gaussian integers form a subfield of C. And so if there was a product of two Gaussian integers that equal to zero, then we'd have a product of two complex numbers which are equal to zero, which can't happen because the complex numbers form a field. The ring, or I should say the domain of Gaussian integers is a very important ring in the study of number theory, and particularly in algebraic number theory. I want to mention that the Gaussian integers, they're an integral domain, just like the integers, but they're not a field. There do exist Gaussian integers that don't have multiplicative inverses. In fact, the number of units inside the Gaussian ring is actually really, really small. The integers only have two units plus or minus one. Gaussian integers actually have four units, plus or minus one, and plus or minus i. We know those to be units. One and negative one are their own inverses, multiplicatively speaking. And then of course, i times negative i is equal to one. So i and negative i are inverses of each other. And these are the only ones. How does one prove that? Because after all in the complex numbers, everything is a unit other than zero. But if you have to use integers as your coefficients, why that one? Why can't you have more? And so I want to introduce a technique that we're going to return to later on in this series. And it's the idea of using a modulus, something that kind of mimics an absolute value. So when it comes to complex numbers, we can introduce this norm, sometimes called that too. And as such, they often use the Greek letter nu. So norm. So we introduce a norm on the complex numbers. So the norm of a complex number z, we define to be the absolute value of z squared, commonly referred to as the modulus. It's a generalization absolute value, right? So this is a Gaussian integer a plus bi. In particular, this is equal to the complex number times by its complex conjugate. So when you have a complex number here, its norm is the square root of a squared plus b squared. So you square it together, it's, well, excuse me, you square the real part, imaginary part, add it together, take the square root, that's its norm. But this sum of squares a squared plus b squared, this is equivalent to the product a plus bi times its complex conjugate. And no stipulation on the numbers a and b needs to be here. This is just how the number i works. And so in particular, if you take the norm of a Gaussian integer, this will give you a sum of squares of integer. So in particular, it'll be a non-negative integer. I guess it could be zero, of course, it could be a natural number if you have, if z is zero itself. But if z is a nonzero Gaussian number, Gaussian number, then its modulus will be something, it'll be a positive integer. So in particular, nu of z is always greater than equal to zero. Now what's interesting about this norm, aka this modulus, is that it's multiplicative, that if I take two different, or it could be even the same, but if I take two complex numbers c and w, the norm of z times w is the norm of z times the norm of w. This is exactly how absolute value behaves on real numbers. The absolute value of a b is the absolute value of a times the absolute value of b. That's a very important observation there. This norm very much mimics that. And so I also want to point out to you that if z is a unit, if z is a Gaussian number, a Gaussian integer with a multiplicative inverse, because of this factorization, the norm of z times the norm of z inverse, this is the norm of z z inverse, which z z inverse is one, the norm of one is itself one. So if you have a multiplicative inverse, then the product of the two norms is equal to one. But the norm has to be an integer by our previous observations. So how many integer equations are there to the equation a square plus b square equals one? So in number theory, we refer to this as a data fontine equation. We look for only integer solutions, integer choices for a, integer choices for b. And the only possible way is one of these squares is one, one of these squares is zero, which leads to four possibilities where you get one, negative one, i, and negative i.