In the various math classes I have taken, we have studied a wide variety of topics ranging from logic to tessellations. We talk about truth statements, we talk about patterns, we talk about shapes, and rotating shapes. We talk about distance, we talk about length, we talk about area, volume, statistics, calculus, infinity, graph theory, sets, etc. The list goes on. It may seem as though all of these subjects do not belong in one larger category. However, they share a common theme: the role of our number system. Each topic corresponds to our number system in some way.
The nature of our number system plays a vital role in what determines what mathematics is in several ways. First, numbers can represent a ratio. When a young child learns to count, they generally start with one. Then we teach them to say two, then three, then four, and so on. The difference between two consecutive integers is 1. For this reason, we can use a counting system to distribute a set amount of items to a group; we have a system with a 1:1 ratio. This ratio not only allows us to count or distribute goods, but to calculate distance. We know that 5,280 feet is equivalent to one mile. In addition, our number system allows us to play with ratios other than 1:1. For example, instead of saying I am going to travel 15, 280 feet, we can say I am going to travel 10 miles. Now the ratio is 5,280:1. Whatever we do to the one side of the ratio, we need to do to the other. The nature of our number system is what allows us to represent large amounts of something in smaller terms and then do operations to both sides. It also allows us to generalize amounts. I can say that for every x number of people I will give two candies. So if there are 10x groups of people I will give 10(2) or 20 candies. In this way, we discovered algebra.
Secondly, our number system does not change; it is reliable. It is vitally important that if 1+1=2 today, then 1+1=2 tomorrow. Because of this, we are able to establish truths about the area of a triangle, about the nature of multiplication, and about statements in general. This is where logic comes into play and this is why it is a part of our high school math curriculum. Consider the following statement: If you do your homework, then your grades will improve. We know this statement is false if we do our homework and our grades don’t improve. We call this concept math because it is based on claims we assume to be true, like numbers. As long as we accept their truth, we can do proofs and make more discoveries about our number system.
In addition to our number system not changing, it is also build-able and follows a pattern. We can think of a pattern or sequence of numbers on a graph but we can also think of them geometrically. Consider the sequence 1, 4, 9, 16,… On the graph this is represented by by y=x^2 when x<=1.We can also think of this as a pattern geometrically as the set of squares, the first being a 2×2, the second a 3×3, etc. When we take a simple pattern and make it a subset of another pattern, then we get a larger pattern which can also be presented by a pattern of shapes, or a tessellation. In this way, a pattern of numbers can represent more than one thing. They start with a small building block (most often one or zero) and either continue to grow or repeat.This is a fundamental idea in graph theory and sets.
One other important feature of our number system is that it follows rules. In 630 AD, Brahmagupta finally defined zero as the result of subtracting a number from itself. From there, he came up with properties of our number system which today are formally known as the Additive Inverse Property, the Additive Identity Property, the Multiplicative Inverse Property, and the Multiplicative Identity. He also discovered that a negative times a negative equals a positive and a positive times a negative equals a negative. Some of these ideas are hard to grasp (specially negative times a negative), but they are what make our number system unique. Brahmagupta called negative numbers a “debt” or something one doesn’t have. However, this discovery allows us to think about patterns involving negative numbers and also helps us to solve for positive quantities.
Finally, it is important that our number system does not only end with our imagination. It goes further; it reaches infinity. Without the idea of infinity, we would not have calculus. For example, when we take the volume of sphere using calculus, we need to break up a segment into the smallest sections we can imagine, then break those parts in half, and those parts in half until we get infinitely small sections. Although this is hard to imagine, we can perform these types of operations because our number system is testable beyond our limited domain and because our number system is reliable.
In conclusion, the nature of mathematics is based on the characteristics of our number system. It is reliable, provable, and is applicable beyond our physical domain.