What is Math?

If you were to describe "math," what would it be? Numbers? What are numbers?  Somehow people figured out how to use math to describe different shapes, rates of change and even other dimensions. Math is used to describe reality itself. What math is, is a pure form of logic.

There is no such thing as a real life "number." Numbers are man-made symbols that are defined to represent an amount. We assign these values to things to describe how much of it there is. 

Numbers can be used in conjunction with other symbols to create more complex ways of describing how much of something there is. For example, if you have the expression: (3*(2+2)apples)/2,  you are saying that you have half of 3 groups of the quantity of 2 apples plus 2 apples. More simply, you could have just said that you have 6 apples. Both statements mean exactly the same thing. This is where equations come into play.

An equation is a way of showing how expressions are related. 

If you had the equation 2 + 2 = 4, it is as if you are saying "two plus two is the same as four." 

When an equation contains "unset amounts that can vary" (aka variables), the logic gets more complex.

Lets say you have the equation y(apples) = x^2 + 5(apples). This equation simply states that "y amount of apples is five apples more than x amount of x." You could then ask yourself, "what are all the different possible combinations that x and y can be?" You could write down an infinite amount of combinations! 

If you wanted to visualize it, you could grab a piece of paper and draw two lines that are perpendicular. From there, define one line to represent x values and the other to represent y values, with a defined direction along each line to be increasing in value, and the intersection of the lines to be zero for both variables. Now, with each possible x value, move from the intersection in the direction of the x-line the amount that corresponds to the value of x. From there, move in the direction along the y-line that corresponds with what the y value would be for that value of x. At that point, make a dot. 

If you sat there long enough to plot the infinitely many possible dot locations, you would have a continuous, smooth curve that represents the "graph" of this equation. Note that y would just represent a number, while x would represent a number of apples.

With a graph such as this, you could examine it in many different, strange ways. You could come up with an equation that represents the distance between two points. You could use even more complex equations to represent the instantaneous rates of change of a variable with respect to the other, as well as finding the area between the graph and one of the axis lines for some interval along the direction of the axis. You could do even stranger things, like determine the volume of the region that would be occupied if you were to sweep the area bounded by the graph and an axis along some interval along the other axis around the axis.

All this from just a graph showing a relation between some numbers and some numbers of apples.

With mathematics, we can describe how many aspects of the universe and reality are related.