What is Number Theory?
Number
theory is simply the study of the properties of numbers. The topics in number
theory range from a variety of fairly simple to extremely complex. (Most people
can understand and follow along with topics in the middle of this range.) The
simplest properties of numbers are still a topic of research in number theory
today. A person can start becoming an amateur number theorist by simply picking
a number and finding all of the ways that number can be mathematically
manipulated, and paying attention to and recording any patterns that arise. While
doing this you can do a little research to find other ways to manipulate the numbers
you chose, and through the induction/deduction process of logic find your own
theories. You might not find anything groundbreaking, but you will certainly
find fun ways to learn and enjoy math on your own. I found an interesting “shortcut
for multiplying by the number nine by simply making a chart of all of the
multiples of nine and analyzed it until I found a pattern. I am not sure if
someone else has already found this or not, but it was a very fun “trick” to
use. You can find a full explanation with examples here Multiplying
by 9,99,999...etc
In high
school, you were taught some of the basic properties of multiplication. Such as: the distributive property, and the associative property. You were most likely
shown a formula to explain these similar to
a(b+c) = ab+bc, Which describes the distributive property.
a(b+c) = ab+bc, Which describes the distributive property.
Well where did we get this? It is simply a result of number theory, or
another way to define it is the logic of numbers. It is difficult to tell who
to actually give credit to for discovering this property; it is one of the most
widely used properties in mathematics. It
was, however; discovered by analyzing the properties and patterns found when
multiplying and dividing numbers. The use of the letters, or variables, comes
from what is called abstraction. Abstraction is simply taking something out of
its original context and making a general form of it. This is not a great definition,
but hopefully you see that when we find a pattern that is very useful, we need
to make a method to apply it to all numbers.
Now some may understand the
distributive property better if it is shown with actual numbers. Like this
2(3+4) = 2x3 + 2x4 but without abstraction some people would look
at the example using numbers and think it only applies to those numbers. Effective abstraction leads into using proofs
to check your pattern to see if it applies to all numbers or just a certain set
of numbers of numbers with specific properties.
I will leave that discussion for another post. Hopefully, this will give
you a basic understanding of what number theory is and inspire you to learn
more.
Yes a very nice article indeed. For many, Maths is generally a difficult topic. And the way it is taught in schools/colleges is mechanical and uninteresting(not anybody's fault, we have to live with that and move to next class). But the idea is that it is interest which matters in Maths. I have seen that good Maths people actually see numbers as different personalities or with great emotions(they have so much love for numbers). This may be the key to success. Unless you love something, you cannot see patterns(it may be maths,a chess position,quantum mechanics...). Some research should be done , about how to induce this love for numbers in students.Of course there could be some brain chemistry behind love for numbers or any other field(that's why we have so many experts in different fields). Try to love any field (fun comes first and then love) and you start seeing patterns.
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