Number Theory

"Mathematics is the study of numbers!"

A very prompt, but slightly misled response.

Numbers are a large part of the field of mathematics, but they most definitely do not define the entire field.

"Ok, well I totally understand the number part, all that adding and subtracting stuff."

But let us consider numbers and mathematics. If anyone is to understand mathematics, most would assume they understand its relationship to numbers. But numbers may surprise you.

Number Theory

Number theory is the branch of math that studies numbers. Questions that number theory raises include:

• Where do numbers come from?
• What is a number?
• How are numbers used?

The origin of numbers can be answered by considering numbers as discovered or created, or a combination of both. Certainly amounts of things have always existed, but the tools to understand them, numbers, were created by man. Numbers by themselves are very abstract. We can say, "5." But what does 5 mean? 5 of what? Immediately we try to think of 5 objects, and it becomes apparent we are impatient and cannot sit and ponder on 5 itself for very long. But let's think about it.

What is a number? Numbers by themselves are quite abstract, as they refer to nothing but a quantity. It is odd to try to think only about the quantity and not the object, even when the objects can be abstract as well (ie. colors). Our straightforward, mathematical numbers have become quite mysterious! What in fact are we using these abstract characters for?

Humans grasp numbers at an early age through learning how to count. As Keith Devlin makes the distinction in his book The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), counting and knowing how many are two different things. The latter develops as we learn the language of operation mathematics, things such as adding, subtracting, multiplying, and dividing. Here we learn of the symbols.

Before, when we were trying to only think of what five is on its own, the symbol of the number, 5, most likely popped into our heads to fill the lack of objects. However, there is a distinction between the number and the numeral. These numerals, or symbols, developed over thousands of years, from clay tokens to the Arabic numerals we have today. The symbols are arbitrary, but they allow us to manipulate numbers and thus open up the world of math. The symbols are what give us the capabilities to do perform operations of math, without them, it would be very difficult to see relationships between numbers!

To know that one of the major components of mathematics has such ambiguous qualities should be alarming to hear, because doesn't such a systematic and direct field need stronger foundation? It turns out mathematics relies on such uncertainty. It just goes to show that math is not all straightforward and calculating, but a mystery that develops and grows with creativity!

References
Devlin, Keith. The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs). New York: Avalon Publishing Group, Inc., 2005. Print.