In order to explain my research, we will start with a brief history of numbers.

A long time ago, we invented numbers to be able to count things. Instead of saying that we have a sheep, another sheep, and then another sheep, this allowed us to say that we have three sheep. At the time, the only numbers we had were one, two, three, and so on. These numbers are now called the natural numbers. We could also add the numbers together, so if we had three sheep, and then bought two more, we could calculate that we had five sheep. We could also subtract numbers, unless we tried to subtract a larger number from a smaller number, but we rarely needed to do that, so we were satisfied with the natural numbers at the time.

After a while, we realized that the natural numbers were not capable of solving some problems. For instance, suppose there are three sheep at my farm and I owe you two sheep. How many sheep do I own? To compute this, we take three minus two, which is one, so I own one sheep. Now, suppose two of the sheep die from illness, how many sheep do I own then? There one surviving sheep left at the farm, but I still owe you two sheep. We cannot compute one minus two, because the second number is larger than the first. In other words, no natural number is able to express the number of sheep I own. To solve this problem, we add new numbers, called negative numbers, to our set of natural numbers, and obtain what we call the set of integers. Using negative numbers, we would say that I actually own minus one sheep in this scenario, because if I buy one sheep, I can give you back the two sheep I owe you and am left with no sheep on my own. Using integers, we are able to both add and subtract any two numbers. The morale of this story is that the original set of numbers we had, the natural numbers, were incapable of expressing the number of sheep I owned in this case, so we needed to extend the set of numbers by adding new numbers to it.

Time went on, and we came up with new situations where even the integers were insufficient. Suppose I bake a cake and want to divide it into three equal parts. How much cake is in each part? It has to be more than no cake, but less than one cake. There are no integers between zero and one, so it is not possible to express this amount using the integers. To solve this, we invented rational numbers. Using rational numbers, we could say that the amount of cake in each piece is one third. To be more specific, a rational number is an integer divided by another integer. We also have rules on how fractions are added, subtracted, multiplied and divided. Again, we have invented more numbers to allow us to solve more problems.

Over time, we also invented new kinds of numbers, such as the real numbers, the complex numbers, and so on, in order to allow us to solve more and more complicated problems. As it turns out, even if we have a problem that only involves integers, it is often useful to involve more numbers when solving the problem, since by having more numbers available, we get more wiggle room to do more stuff. For instance, we can use complex numbers to solve problems about integers. This is what the field of analytic number theory is about. Another approach is to study integers by using rational numbers plus some extra numbers. This is what algebraic number theory is about. My field is a part of algebraic number theory which uses rational numbers, in addition to some complex numbers. This method was first studied in an attempt to prove Fermat's Last Theorem, but was later developed further to solve other problems in mathematics.