Don't know how to calculate? Then visualise...
In the previous post, we have established that I am *not* a Maths guru. In fact, I am not even a Maths amateur. Unfortunately for you and me, Mathematics is everywhere – particularly if you, like me, are inclined to like science. So, what to do!?
Every Mathematical equation or function tells a story. It tells us what something is doing, depending on certain inputs that we may choose. The problem is that, for people like you and me, the story is told in a language that we do not know. That does not mean that the story is not valid, or that is not worth reading, but it does mean that, if we are to understand what it means, we will have to do some work. It is similar to picking up a book in a language you know nothing about. There is a story there, but you, as a non-speaker, cannot decipher it.
Just like ancient languages and representations, we may not know what some (or all) of the symbols mean. This does not mean that they have no meaning. Our job is to understand the story, whether or not we understand all of the symbols that are available to us.
What can we do, then, to understand the story, if we don't know the language? Well, you and I have at least three options:
Give up. But we 'ain't' quitters, so this is not really an option.
Spend time to understand the language of Mathematics, its symbols, and syntax. This is an option for the committed. I believe that, if given enough time and the right resources (and mindset), one could learn Mathematics. But this may incur a huge opportunity cost, meaning that you could use your time learning Mathematics into developing other skills that will benefit you and those around you (more about personal and professional development in a later post).
Translate the story with images. Well, you could ask for someone to translate the story for you – or for you to have a copy of the story with images rather than codified words and symbols. In this case, you wouldn't know word-by-word of what is said, but you would nevertheless understand the story. How is this possible with Mathematics, though? The answer lies in the graphs.
Graphs are the life-saving boat of those with sub-optimal Mathematical understanding – like myself. I do not know what Mathematical symbols mean, nor how they fit together to tell me the story. But, I can understand a graph. So what if we approach every Mathematical problem with a pictorial view of the story that the equations and symbols are telling us? Graphs are just the translator we need to understand the story without actually having to read it word-by-word.
Graphs are the way you and I can understand the story from Mathematical formulas without actually understanding the meaning of the codes that the story is written.
At this point, you might be asking: but if I don't know Mathematics, and what symbols mean, how can I graph anything? I am glad you asked. Thanks to the power of computers, and the ingenuity of kind people around the world, we have online, free softwares that enable us to obtain graphs and properties of our Mathematical formulas with very little hassle. Here I will mention two, one for simple graphs and the other, for more detailed properties of whatever Mathematical function you are dealing with. As this is not mean to be as tutorial of how to use it, I will provide the link to their pages, and you can explore at will. Here, the purpose if to show you that they exist, and that you can make use of them.
The first one is Desmos, which enable you to quickly type in the formula you want to explore, and gives you back the Mathematical story behind it. It is a simple tool, but can take you very far in actually understanding rather complex functions (they work great with functions with x's elevated to the nth power; aka polynomials). The second is the Wolfram Alpha, which allows you to type even more complex equations (e.g., double integrals) and to obtain the properties of those equations back, with little to no hassle. This is obviously for more experienced users, or for users which are faced with the inglorious task of actually dealing with complex Mathematical stories. It does provide you with the graphs, but with a load of other characteristics of your equation (e.g., its derivative, etc).
For those who struggle with Mathematics, like myself, softwares like those are life-saving, as they allow me to gain a good understanding of the behaviour of my Mathematical formulas without actually understanding the meaning of each of the symbols and syntax involved in constructing it. This is obviously not an alternative for understanding Mathematics and its intricacies. But it is a way in which you can start seeing Mathematics in a more friendly way, which may one day stimulate you to spend some time having a go at learning the underlying details and meanings that underpin the world of Mathematics.