Hi, I'm a Bullis student tutor, and in this video I'm going to be talking about number theory. I'll be giving a brief introduction to what this is. Number theory is also called higher arithmetic, and it is essentially a study of mathematical interactions and number types. So we have a lot of different types of numbers.
We've got odds, evens, squares, integers, rational numbers, all sorts of types of numbers. And what the branch of mathematics known as number theory does is take those types of numbers and ask questions about their relationships. And they use formal mathematical proofs to answer these questions.
So let's give an example of one of these questions. About 2,000 years ago, a guy called Euclid looked at the group of numbers known as primes. And we should be familiar with what primes are.
They're numbers that are only divisible by one and themselves. So some examples are 2, 3, 5, 7, and the list goes on. And Euclid, being a number theorist, looked at this group of numbers and asked a question.
He said, how many primes are there? And his guess was that they were infinite primes. that no matter how far you went in numbers, no matter how high the numbers went, there would always be new primes.
And so then he took that hypothesis and he proved it using a formal proof. And he used proof by contradiction, which is a type of formal mathematical proof in which he made the assumption that there were only a finite number of primes, and then showed why that assumption was wrong. So we can see that at its base, number theory is really all about asking questions about numbers.
And for a long time, people thought that number theory couldn't be applied to the real world, that it was really just for the people who loved math and wanted to ask questions about math. But now as we enter the 20th century, 21st century, we realize that Number theory has a lot of real-world applications. One of the major tenets of number theory is the fundamental theorem of arithmetic that basically says, at its essence, that all numbers can be factored into a unique set of primes. And we can see that using a quick example.
If we take the number 12, we can break that up into 6 and 2, we know. And we see that 2 is a prime number. And then we can break 6 into 3 and 2. Those are its factors. And we see that 3 and 2 are also prime numbers. So the number 12 has a unique prime factorization of 3, 2, and 2. Those are its prime factors.
And no other number will have those prime factors. So beginning in the 20th century, and especially now in the 21st century, number theory, and specifically the fundamental theorem of arithmetic, is being used for encryption schemes for large-scale business networks. A lot of times now a business's computer network will be encrypted using prime factorizations.
The key to their network will be a factorization of a huge number. And so that unique prime factorization for that huge number is the key to the business's network. So we see that number theory is about asking questions about math and using formal proofs to answer them.
And we know that in the real world, you can apply these concepts to solve technological problems. This has been a Bullis student tutor video. If you liked the video or found it helpful, please make sure to subscribe, like, and check out our other videos on our YouTube page, Bullis Student Tutors. Thank you.