The Nature of Mathematics | The Role of Our Number System

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.

Grecian Influence in the History of Math

One of the greatest tools we have today in mathematics is the ability to use previously proven ideas such as theorems in order to keep discovering more mathematics. This is exemplified in the mathematical findings done by the Greeks. They used the ideas of others as a stepping stone to develop their own, and these ideas were later used by other mathematicians such as Euclid and myself, specifically their discover of the deductive proof method. Although the Greeks adapted many of their ideas from the Romans and Egyptians, they were also able to develop some of their own.

One of their most well known developments was in the field of Geometry. To begin, in roughly 600 BC Thales developed what is known as Thales Theorem which states that if one side of a triangle is the diameter of a circle, then that triangle is a right triangle. Thales also developed the Intercept Theorem which describes the similar areas created when parallel lines intersect to intersecting lines.

During this time, the most well-known Greek mathematician, Pythagoras, also made substantial contributions to Geometry in Greece.  Unknown to many, he was the first to think of math as a system. He realized that geometry could represent numbers. His most well known work is the Pythagorean Theorem which he developed using a rope. Although Thales and Pythagoras both in some way discovered geometry and made contributions to the field, Pythagoras is more well known.

Later in the fifth century BC, a third important contributor to geometry, Hippocrates of Chios, surfaces. He worked on one of the three famous problems: constructing a square with an area equivalent to that of a given circle. His book entitled The Elements was written in 440 BC and highlighted the basics of geometry. Chios’ and The Elements were later used by Euclid.

Also in the fifth century BC, Zeno of Elea used a version of the famous Tortoise and Hare story to come to develop the idea of infinity. He considered the following situation: If the tortoise starts the race ahead of Achilles, and the Achilles always catches up to the tortoise, how can the Achilles beat the tortoise?  In theory, Achilles should have gained a little more distance between himself and the tortoise during the race until he was eventually within an infinitely small distance away from Achilles, but he never passed Achilles.

zeno_paradox

This paradox was called the Dichotomy Paradox and it was used by Zeno to introduce the concept of infinity.

Similarly, Democritus is thought to have come up with the idea of breaking a whole into smaller parts and again, dipping into the theory (at this point) of infinity during the late fifth and early fourth century BC. Instead of looking at the Dichotomy Paradox, he worked with volume and divided matter into cross-sections.

Around 360 BC, Eudoxus of Cnidus, student of Plato whose focus was largely logic, developed a solution to find the volume of a prism by breaking it up into smaller sections. This may have also led to the idea of proportions he developed and irrational numbers.

Later in the third century BC, Euclid used many of the ideas stated previously such as the Pythagorean Theorem, number theory, ratios of divided sections, and plane geometry in the various books he wrote. Euclid formalized and did further research in these areas which were first developed by the Greeks. Although the Greeks laid much of the groundwork, Euclid is the most well-known the work done in them. For example, we have Euclidean geometry and Non-Euclidean geometry.

Even though the Greeks made many contributions to Euclid’s work, their greatest contribution to mathematics was developing a new method of proof. Previously, the Egyptians had used inductive reasoning and recurring patterns in developing truths in mathematics. The Greeks used rules of logic to prove and to disprove statements. This discovery is tremendous in the field of mathematics because it allows us to prove things that are not seemingly possible or imaginable.

One of our human limitations is time. If we solely rely on finding patterns and physically counting or measuring something in order to make a conclusion, we will run out of time. When we use proofs, we are able to use previously proven truths in order to prove more true things. We can generalize and be 100% sure in our findings without having to check the physical measure. This also allows us to prove things which are beyond our physical domain such as Euler’s famous equation

 

euler-identity.jpg.png

He did not prove this by guess and check, but proved it by using a series of true things.

The deductive proof method lets us consider what happens when a process occurs an infinite amount of times, better known as taking the limit of something. We don’t know what the limit of an expression is as it approaches infinity because it would take us to long to get there. But we can use proofs to find out.

In conclusion, the Greeks played a large role in the development of mathematics by discovering key groundwork ideas used by Euclid and other mathematicians. One of their greatest contributions was the deductive proof method which allows us to prove things beyond our physical domain.

 

 

 

What is math?

When I think of one of the underlying themes in math, I think of patterns. Even in the earliest forms of math when people unknowingly used it to decide how much food to prepare, they were forming a ratio of a certain amount of food per person and multiplying that amount by how many people there were. This is a pattern that begins with one unit and continues to the next. This is just one of the types of patterns in mathematics.

I used to love math because it meant purely experiencing the joy of problem solving. This is probably why I like algebra. I enjoy taking a lot of information that sometimes seems unrelated to each, breaking it down into small pieces, and building it back up into one clean answer. It doesn’t matter how I got the the clean answer, but just that I got there. Math is problem solving. There are always multiple ways to find the solution, but a solution always exists even if it is the empty solution.

Another reason I enjoy math is because I think it suits my learning style. With any new information I take in or learn, it is really important for me to first look at the big picture. From there I dive deeper into the topic looking at what things are related and connected. Math is like this in a sense that it takes big ideas and problems and breaks them down. It seeks to solve these problems by finding a pattern starting with the smallest details, and growing from there.

These small details often form a pattern because of the logic or building blocks they are built on. Math is empirical; it is not founded on opinion or values. We can ask “what if?” and “what if not?” and are able to make solid consistent conclusions every time. The nature of the rules of logic is what allow us to find patterns in mathematics.