Creating New Mathematics
Let's create a mathematical object that's an ordered list of two integers..
(a,b)
where a and b are integers. Order is important, so (a,b) and (b,a) are different objects.
Great, but our objects are not much use unless we can do something with them.
So let's define a multiply operation as follows..
(a,b)*(c,d)=(a*c,b*d)
Here's an example: (3,4)*(2,5)=(3*2,4*5)=(6,20)
This is interesting (1,1)*(a,b)=(a,b)
So the object (1,1) plays the same role with our new objects that the number 1 plays with the integers. So (1,1) is our "number 1". Mathematicians call this the identity element and many groups of mathematical objects have an identity element because it's a really useful thing.
So now our objects can be multiplied together. Notice that when we multiply two objects we always get an object of the same type. Mathematicians say that our collection of objects is "closed under multiplication". This simply means that no matter how frantically we multiply objects together we'll never create a different type of object. Of course, not all collections of objects need to be closed under an operation. But ours is.
What about addition? Our definition of the addition operation may look a bit complex..
(a,b)+(c,d)=(a*d+b*c,b*d)
Here's an example: (3,4)+(2,5)=(3*5+4*2,4*5)=(23,20)
Again, when we add two of our objects we always get an object of the same type. So our collection of objects is also closed under addition.
Are our objects of any use? You bet they are..
Our object (a,b) is usually written a/b and it's called a fraction!
I cheated a bit. I made things look easier than they really are. Inventing the new object (a,b) is easy, the harder part is defining operations such as "multiply" and "add".
These operations were defined to make the objects useful, and to have some basic properties expected of all multiply and add operations. So I did not really invent new mathematics from scratch.
But still, this gives you the flavor of what it's like to create new mathematics. Never be afraid to create your own.
Oh, and you just learned fractions really fast. A nice bonus!
Content written and posted by Ken Abbott abbottsystems@gmail.com