Let's continue our discussion of common sets of numbers we just discussed a moment ago. The set of all integers, which is notated with a capital Z, and the set of all rational numbers. In other words the set of all fractions, which is notated with a capital Q. So, is there anything beyond the domain of the rational numbers? Anything that's not a fraction that we can think of? Well, one of the most famous numbers of them all is actually not a rational number The number "pi". Pi has some sort of quirky properties, right? We know the decimal expansion begins 3.1415... and so forth. One of the interesting aspects of pi is that decimal expansion goes on forever, so it's infinite and it's also non-repeating, or you could say non-recurring. So, the distribution of the numbers in the decimal expansion of pi are, effectively, random. I can't write pi as the ratio of one whole number (one integer) over another. So if a number can't be written in this fashion, if it can't be expressed in rational form then it's called irrational. So pi is an irrational number. There's a number that's not in Q, let's label that in our Venn diagram. Somewhere out here is the number pi. You might know that the number "e", also sort of a special number in math and science, is 2.7... etc, etc, has also this nice property that its decimal expansion is non-repeating and infinite, so it's likewise an irrational number. So e is also out here. In fact, there is an infinite set of irrational numbers. How can I just argue that there is an infinite set of irrationals? One fact is that if you take the square root of a prime number, you get an irrational result. So it's decimal expansion is non-repeating and infinite. Number two, is that the number of primes is infinite. Now, prime, just to remind you is a whole number greater than or equal to 2 whose only divisors are 1 and itself. So the square root of a prime is irrational and the number of primes is infinite. Now, this is not necessarily obvious, right? We don't know that sort of a-priori. In fact, I'm including a little bonus lecture where I proof a la Euclid that there's an infinite number of primes. If you put these two together, then you can show that there is an infinite set of irrationals. Let's check that out. If there's an infinite number of primes and the square root of any prime is irrational then there's the reason again that there's an infinite number of irrational numbers. A slightly different justification. Let's call this justification number one, that there's an infinite set of irrationals. Justification number two, from sort of a "complexity" perspective, that there's an infinite number of irrationals. What I've said before is that the decimal expansion of an irrational number is non-repeating and infinite. So we can go in reverse here, because if I am just constructing infinite non-repeating decimal expansions I could ask, how many are possible? Now, that's a big counting problem, but as it turns out there's an infinite number of such decimal sequences that I can construct. For instance, let's say I just start by flipping a coin or asking a computer to compute a super-random number and according to the outcomes, these coin flips, or whatever, I fill in some number. How many different such decimal expansions are possible? There is an infinite number of ways I can express a non-repeating infinite decimal expansion. So there are an infinite number of irrational numbers, because there's an infinite number of such decimal expansions I could construct. This leads us to our next set of numbers, and the most important set of numbers of them all, which is the reals. The reals consist of the rational numbers let's just say "plus", kind of informally, the irrational numbers. Take note here. The reals, without a doubt, are the most important and commonly encountered number system in any applied science field. So, when you do calculus, you're working with the reals. When you do linear algebra, differential equations, you're basically working with the reals. Sometimes other sets, but generally speaking it's the reals or a subset of the reals. In proper notation, we would say it's really the union of those two sets. So, now, I can add another concentric circle to my Venn diagram, which is showing the sequence of common sets of numbers. Now I've got the reals as the outermost shell here. If I graph the real line, as it's called, I get a straight line, importantly though, with no gaps. On the other hand, if I plot all the rational numbers on the line, I get lots of gaps. In fact, it's a well-known mathematical result that between any two rational numbers lies an irrational number. The reals then give us kind of this full-fledged closed domain, in the sense there're no gaps. In other words, when I plot the reals, I get, importantly, a continuum of numbers. So, if we plot the reals, we get a continuum. Now, what happens if I sort of track the motion of an object or a particle in space? Let us imagine we see a particle moving and we track it, and it goes along this path here. It's starting in the lower left moving to the right and as it goes, say, time elapses. So, the way I experience the motion of that particle is a little bit debatable, but generally the accepted interpretation is that I experience that motion as a continuous process. So, therefore, it would be of use as mathematicians and scientists to use a model that replicates that continuous behavior. So, in other words, we would probably like to use, generally speaking, a number system that is a continuum. Now, you've heard of the space-time continuum, right? I won't get too new agey here, but we generally experience the real world as a space-time continuum, and the number system we want to use to model that is a continuum. Now, similarly, just to add to that notion, for those of us that have maybe had a little bit of calculus or seen functions and, why not? We can think of a function... Usually in calculus we'd like to study functions that are continuous, right? Those have kind of nice mathematical properties. So, if we take a function that's continuous, maybe as a model for the motion of the particle, and we use it in conjunction with the domain for that function that's also continuous, together, we get a continuous model. And that, generally speaking, is gonna be a good description of real world phenomena.