Question

We have used the natural numbers to solve problems. This was the right set of numbers to work with in discrete mathematics because we always dealt with a whole number of things. The natural numbers have been a tool. Let's take a moment now to inspect that tool. What mathematical discoveries can we make about the natural numbers themselves?

This is the main question of number theory: a huge, ancient, complex, and above all, beautiful branch of mathematics. Historically, number theory was known as the Queen of Mathematics and was very much a branch of pure mathematics, studied for its own sake instead of as a means to understanding real world applications. This has changed in recent years however, as applications of number theory have been unearthed. Probably the most well known example of this is RSA cryptography, one of the methods used in encrypt data on the internet. It is number theory that makes this possible.
What sorts of questions belong to the realm of number theory? Here is a motivating example. Recall in our study of induction, we asked:
Which amounts of postage can be made exactly using just
5-cent and 8-cent stamps?
We were able to prove that any amount greater than 27 cents could be made. You might wonder what would happen if we changed the denomination of the stamps. What if we instead had 4- and 9-cent stamps? Would there be some amount after which all amounts would be possible? Well, again, we could replace two 4-cent stamps with a 9-cent stamp, or three 9-cent stamps with seven 4-cent stamps. In each case we can create one more cent of postage. Using this as the inductive case would allow us to prove that any amount of postage greater than 23 cents can be made.
What if we had 2-cent and 4-cent stamps. Here it looks less promising. If we take some number of 2-cent stamps and some number of 4-cent stamps, what can we say about the total? Could it ever be odd? Doesn't look like it.
Why does 5 and 8 work, 4 and 9 work, but 2 and 4 not work? What is it about these numbers? If I gave you a pair of numbers, could you tell me right away if they would work or not? We will answer these questions, and more, after first investigating some simpler properties of numbers themselves.

Answer 1

The natural numbers are fundamental to discrete mathematics, and their study leads us into the domain of number theory. Number theory has practical applications, notably in RSA cryptography, and raises questions about the properties of numbers, such as in the case of determining possible postage amounts with certain stamps. Coprimality between stamp denominations impacts these possibilities, reflecting the universal nature of mathematics.

Step-by-step explanation:

We have used natural numbers to solve problems, finding them to be the ideal set for working with discrete quantities in discrete mathematics. These numbers are integral to the field of number theory, which explores the intrinsic properties of numbers. As we delve into this realm, we realize its significance extends beyond pure math; number theory has found applications in modern technology such as RSA cryptography, a method for encrypting internet data. Number theory poses intriguing questions about numbers, such as determining which sums of postage can be created using stamps of certain denominations. For example, we find that with 5-cent and 8-cent stamps, any postage above 27 cents can be made, while with 4-cent and 9-cent stamps, any amount over 23 cents is possible. However, when examining a combination like 2-cent and 4-cent stamps, we see that it's impossible to create odd amounts of postage since both stamps represent even values. This leads us to discover that coprimality, or the lack of a common factor greater than one, between two numbers is what dictates such possibilities. This is just one illustration of how number theory allows us to investigate deeper mathematical truths, highlighting the universality and consistency of mathematical principles irrespective of time and cultural context—as illustrated by historical accounts from India and the global transmission of mathematical knowledge.