Geometry/the philosophy of number/lines
Logically, a number is an extension amidst space. For example, a mark on a blank sheet of paper. Let's say I draw a short line on the paper. Then I draw another, connected with the first line. I then draw another connected line, then more indefinitely.
The first line is represented by the number 1, etc. We all realize that the more quantity of lines there are, the longer the "number". It is obvious the bigger the quantity, the longer the number is. We can not get around this number logic. Also interesting, if a quantity is smaller then 1, it is a decimal. The smaller the "decimal" number is the longer it becomes, such as .0234523487983 .
The number 1(one) is "1". Any number greater or smaller then "1" is longer or more complex. For example, ".1", ".9", ".2222", or 10, 100 etc.. "1"(one) is just a mark, it is simplicity. I guess "1" was designated as a source number, a beginning to the order of quantity(positive, decimal, zero and negative numbers).
When one represent a stick with a tally mark, 2 sticks with 2 tally marks and so on, the tally marks can grow unwieldily, infinitely long. The tally mark system is simple, but cumbersome with large quantities. The number system is an efficient and effective way to represent quantity.
It is a paradox that the number 1 is the smallest unit/part yet it can also represent the whole such as "1" whole pie!
A straight line is a simplification/representation of extension out to space. Geometry is about inner and outer regions of forms, between space and matter. If we multiply the existence of a straight line to infinity, then it will take over/fill up space. If space grows at minimum the same rate(given they both expand at the same time) of straight line creations, it cannot be taken over by the straight line.
What is the future of numbers/ math? Will we simplify/improve numbers in some way? What is the latest in math research?
Thanks for reading!
I don't know of anyone who can say what the future holds, but I can say what has happened in the past. When numbers were first developed, they were positive integers.
When addition and subtraction was introduced, it was seen that if you had 5 widgets and 5 were taken away, there would be 0 widgets left, so 0 was added.
As math began going for itself and as physicists began working out with one direction being positive, we had to have negatives as well. As the division of two integers arose where the division wasn't even, fractions came up. These were expressed in decimal notation and soon it was discovered that there were numbers that weren't from the division of two integers.
When square roots were computed, the concept of imaginary numbers arose. That is, the square root of -1 is i. If we then add then, we suddenly get a complex number -1+i. If complex numbers are graphed with positive real numbers at 0 degrees and imaginary numbers at 90 degrees, negative real numbers at 180 degrees, and negative imaginary numbers at 270 degrees, multiplication can be seen to follow trig. That as, if we express a complex number (x,y) as (r,theta), where r is the radius and theta the angle, then the product of two complex numbers (r1,t1) and (r2,t2) is (r1*r2,t1+t2).
Using this notation, complex numbers have been found to form a complete set under all mathematical operations. That is, it is known that all addition, subtraction, multiplication, and division can be done in the real numbers, but it take complex numbers to allow for square roots to be computed on negative numbers.
It has been known, using negative numbers, that the square root of 4 is +2 and -2. It has also been found that the square root of -9 is 3i, for 3i * 3i = 9i^2, and i^2 is defined to be -1, so the answer is -9. Not that 3i has an angle of 90 degrees where as -9 has an angle of 180 degrees. Using this, the cube root of 8 would be 2 with the angle being 0, 120, or 240. This is because when multiplied by 3, all angles result in an angle equivalent to some integer multiple of 360 degrees.
It has been shown that there is no operation that can be performed on a complex number that won't result in a complex number.
In case you thought this was all, there is also the more abstract concept of higher lever mathematics on number spaces, such as linear algebra, quantam mechanics, matrices, numberical analysis, and more. In any of these fields, they have found questions that haven't been answered by the rules of mathematics that are known.
What I have seen is that no matter what field is gone into, there is still more that people need to find out that what mankind has already found out.
Is this what you're after?