 I'm Zor. Welcome to Unizor education. This would be a very simple and short presentation of problem number two in the rational numbers theory. The problem was stated quite simple, proved that the product of two periodic decimal numbers is periodic decimal number. And it's actually a joke. It's not really a problem. And here is why. You see, problem one, I had to really go through all the details when I was trying to prove that the sum of two periodic decimal numbers is periodic decimal number. Why? Because the result of this was used in the lecture where I was proving that every rational number can be represented as a decimal fraction with a period. So I couldn't really cut corners. I really had to go through all the details of that particular problem and think about all the different variations of the period, equal period, not equal period of these two different decimal numbers, etc. I could not use the fact that every decimal periodic fraction is actually a rational number or every rational number is a decimal periodic fraction. But now, since everything has already been proven, the problem number one actually was one of the foundations for a bigger proof that every rational number can be represented as a decimal fraction with a period and vice versa. Now I can use all this. But if I can use this fact that every rational number can be represented as a decimal with a period and vice versa, then just use this fact for this particular problem. And there is no problem because obviously if you have a product of two different periodic decimal fractions, it's the product of two rational numbers because every decimal periodic number is actually a rational number. Now the product of two rational numbers is a rational number because that's how rational numbers were introduced in the very first case from the very beginning. So here is basically the problem. X is a periodic decimal, periodic decimal, Y is periodic decimal. How can they prove that their product is periodic decimal? Very simply. X is rational and so is Y. So the product X times Y is also rational. And since every rational number is represented as periodic decimal, X times Y is a periodic decimal. So we were using here both direct and the converse theory. Direct one being that every periodic decimal number is rational and the converse one when every rational is actually a periodic decimal. So basically that's why it's periodic decimal. It's very important by the way to understand a very important distinction between the first problem when I was trying to prove that the sum of two periodic decimal numbers is periodic and the second problem when I'm talking about the product of two periodic numbers. Because in the first case I couldn't really use all these theorems about every periodic is a rational and every rational is a decimal periodic number. Because that was a consideration which I actually had used during this proof. So logically I couldn't really use it because it would be kind of an infinite logical loop. But now since everything is proven to the product I can use it without any problems and everything is okay. And obviously division if you wish it's also exactly the same thing as long as your denominator is not equal to zero. So that's it. I couldn't really even say this is a real problem. But what's important about this is to understand when I can use and when I cannot use previously proven facts. So for the sum I cannot use this type of equivalence between duration numbers and decimal fractions in the period. But for the product I can already use it. I couldn't use it in the first case because that was my foundation on which the whole proof was built. Now everything is okay so I can use it. Alright short but you know hope you enjoyed it. Thank you very much.