Unlearning Mathematics
Why?
Because some of the greatest concepts in the math are taught in a wrong way and it is very difficult to move those concepts into their deserving place in our mind, once they were taught in that wrong way.
Some examples
- i is an imaginary number which denotes the square root of -1
- sin and cos are ratios of a side to hypotenuse in a right-angle triangle
- π is the ratio of circumference to diameter of a circle
- a matrix is a grid of numbers
- a curve shape represents a function of variables
- a vector is an arrow-like value with direction and magnitude
- e is the sum of some infinite series
- numbers are different from the arithmetic operations
- multiplication is a repeated addition
- exponentiation is a repeated multiplication
Hmm..
Where do I start? It may not be the same experience for everyone, but atleast I was taught in that way (almost) in my school, which I'm still trying to unlearn and replace them with a more natural way of understanding these great concepts. These interpretations are not wrong, but they lead you in a wrong direction about whole mathematics ad cause difficulties in understanding the real beauty and nature of math.
I think there should be more natural ways to think about these things. The problem is, there are several ways to look at each of these things. They are actually a combination of all these views. Some of these views or interpretations do not exist by themselves. That is, they exist mutually with others. They are higly dependent on each other, and they do not exist in the absence of the others. In other words, these multiple seemingly different things are actually different views or perceptions of the some core thing. And that core thing doesn't have it's own description except through these views.
The imaginary number
i is more about rotation by 90°. You can denote it by a square matrix[0,-1; 1,0]. When you do such rotation twice (multiplication), you get 180° rotation, which is -1. Square root of i is a 45° rotation which is an equal mix of 1 and i , normalized with a scale factor of 1/√2. Also if you plug in i * π into the Taylor's expansion for e^x, with matrix multiplications, it converges to [-1,0; 0,-1] - another way of arriving at the Euler's identity.
Sine and Cosine
Sin and Cos are probably most misplaced things. They were taught more from a rotation/angle perspective. But their more natural place is in rate of change. Cos indicates rate of change in Sin and vice versa, except that Sin gives a negative rate of change in Cos. The i in matrix form is a special case of rotation matrix built using sin and cos. Also, the very essence of existence of the universe is in the vibrations/oscillations which can be seen as rotations in higher dimensions. The reduction of these rotations into lower dimensions gives rise to waves, oscillations, sin and cos.
Matrix and Tensor
Matrix as a grid of numbers is a bad way to look at this beautiful concept. While i is about rotation, matrix is about deformation of the space. It's a measure of transformation. In a tensor form, it can also represent the concept of number itself, while adding dimensionality to it. Numbers can be seen as operators just like how a matrix is a transformation operator.
Numbers and multiplication
That brings me to multiplication of numbers. In the multiplication of a * b, it is better to see one of these numbers as a transformation operator for the space in which the other number lives as a quantity. For example, 5 * 4 can be seen as expanding or stretching out the domain of 4 such that new unit in the expanded space is 5 times the old unit. The result 20 is the mapping of the expanded quantity of 4 to its old quantity.
Functions and curves
Also a curve such a parabola has a shape only relative to the space. You can make it a straight line by stretching an axis in a quadratic way, similar to a log scale. So the shape of a curve (or a function) is relative to the axial state. This is how polynomial regression works in statistics or machine learning.