Intuitive Mathematics, Part II

In an earlier article, Intuitive Mathematics, Part I, I argued that complex mathematical concepts can be presented in an intuitive way and in doing so, we not only have a better understanding of these concepts, but math in and of itself can become quite fun and in fact a game that anyone can play, including my young five-years-old son. In this second part on “intuitive mathematics” I take the plus and minus concepts described in the earlier article and introduce the concepts of vectors and matrices – to a five-years-old!

First, recall that in my earlier article I claimed that intuition is based on being able to jump back and forth between an abstract and concrete world, inferring “new things” in one world from observations in the other world. In trying to define the abstract vector via a concrete object, I found that an old telescopic radio antenna works best. Since those are hard to find these days, I purchased a bunch of telescopic magnetic pickup tools, instead, as shown in Figure 1.

With our telescopic magnetic pickup tools, which are actually quite fun to play with – and fun is the bonus of intuition, the vectors are defined as shown in Figure 2.

A vector is defined by an origin (or start) and a pointer (or stop). The distance between the pointer and the origin is the length of the vector. 

Second, in the earlier article I described how the number line is a good intuitive explanation of plus and minus, as shown in Figure 3.

With the introduction of vectors, the number line becomes as shown in Figure 4.

The plus numbers are now vectors pointing to the right and the negative numbers are now vectors pointing to the left. Using our concept of vectors, my son and I played the game of pointing the vector in the plus and minus direction, depending on the word I said. While we were playing this fairly silly game of trying to get him to miss-point the vector in the wrong direction (by repeating plus, plus, plus or plus, minus, minus) he all of a sudden stopped and asked:

“Daddy, if plus is to the right and minus is to the left, what is up!?”

He waited a few seconds and then said, “the airplane!” To me, this was a striking example of intuition at work! Even though I have not yet introduced him to the concept of 2D coordinate axes, through his play, he started asking new questions that can lead to new mathematical concepts, which were not initially known to him. This is the power of intuition and why I find intuition sexy. It helps us discover new “things.”

Third, the concept of multiplication and division (scaling) of a vector is the action of extending or shrinking our telescopic tool, as shown in Figure 5.

Fourth, the fun continues when we look at the properties of multiple vectors. The simplest case is when multiple vectors start at the same origin, as shown in Figure 6.

When talking about two vectors, besides length and direction, we can talk about the angles between the vectors, such as the red and green angles from Figure 6. An abstract concept that’s almost always described only in terms of mathematical formulas is the concept of correlation (see Wikipedia or Investopidia). Good luck explaining correlation to a five-year-old using the above definitions. Yet, correlation has a very simple intuitive meaning. It is inversely proportional to the angle between two vectors. The correlation between vectors A and B is lower than the correlation between vectors B and C because the angle between A and B is bigger than the angle between B and C. 

For the advanced reader, I am stretching the definition of correlation a bit. Correlation is indirectly proportional with the angle between two vectors after the constant (also known as DC – from direct current) bias – or projection onto the constant/DC subspace – has been removed.  (Correlation is the cosine of the angle after the DC bias is taken out, but those are non-consequential details.) Of course, this raises questions about why only the DC bias!?  This gets into more advanced topics, which I plan on discussing intuitively, after I introduce vector projections and summation, in a future posting.

Fifth, when talking about two or more vectors, we can introduce the concept of a matrix. From a strictly technical point of view, a collection of multiple vectors is a collection of hyper-vectors (hyper means that they don’t have the same origin), as shown in Figure 7.

However, if we make the requirement that all the vectors share the same origin, then we have a matrix, as shown in Figure 8.

In conclusion, in this article I suggested using a telescopic magnetic tool to concretize the abstract concept of a vector. With the new concretization we introduced the following definitions:

1.    A vector is defined by its origin and pointer. The distance between the pointer and the origin is its length.

2.    Plus numbers are vectors pointing to the right and minus numbers are vectors pointing to the left.

3.    Multiplication makes the vector longer and division makes it smaller (ignoring smaller than one number scaling).

4.    When two vectors share the same origin, the angle between them is inversely proportional with their correlation.

5.    When two or more vectors share the same origin, they are called a matrix.

In my next article I plan to introduce an intuitive explanation of operations that can be performed on vectors and matrices and intuitively define things such as eigen-vectors and eigen-values, matrix multiplications, projections, linear transformations and other.

Keywords: #Mathematics, #IntuitiveMathematics