First, we learn to count … “One, two, three … seven .. ten!” I am a parent, and I have observed the process.

We then learn that numbers represent groups of things and have meaning. For example, we count our fingers and other objects.

We also learn that each number can be *written.* That is, it has a numeral associated with it.

Then we slowly learn that we can perform operations with numbers … add, subtract, multiply and divide them. These are all natural processes, but there are hidden traps and those traps have to do with *consistency*. For example, as long as you are only concerned with adding whole positive numbers, your answer will always be a positive whole number, but when you introduce subtraction (the inverse operation), a remarkable thing happens. Sooner or later you will encounter a calculation like 2 – 5 and realise that the answer is not something that you can count on your fingers. You will need to ‘invent’ negative numbers (and then learn how to add, subtract, multiply and divide them!).

For mathematicians, thinking about types of numbers (and the different operations that can be applied to them) leads to concepts called groups, rings and fields.

There are many kinds of numerical and algebraic systems apart from whole numbers, fractions and decimals. For example, one of the topics below (Determinants) is a number type/system that you may not be familiar with. This topic could be easily be placed under the heading of ‘Algebra,’ because determinants were originally designed to solve simultaneous equations. But, when mathematicians realised that they were also *numbers*, they began enquiring about how they could add, subtract, multiply and divide them and whether the order of operations matters, and whether there are equivalent determinants (and ways of simplifying them), etc. The same questions are asked of any new number system. Complex numbers provide another good example.

Playing with numbers provides a surprisingly rich field of research. I hope some of items that I share here inspire you to read and experiment further!

I have been watching your “Integration technique” videos with high expectation and excitement. They never fail to give me a lot of joy and much pleasure. You must have been one hell of teacher at your time! … just continue making highly enjoyable and intellectually stimulating videos. … Unfortunately, what you learn at the university, you hardly ever see again in your office job. That is why I try to repeat some thing I learned: to enjoy it a bit more, to marvel upon beauty, and to keep my brain sharp. There is where your videos come in perfectly: interesting, amusing and thought provoking problems neatly presented. … You do your best to share your passion, knowledge and experience. The least I can do is to say an honest “Thank you!” As I said, your videos give me much pleasure. You may freely let other people know that. … Rest assured that I shall be watching your videos in the future. With joy and excitement. And I shall comment on them from time to time. You just continue making them.

Zoran V (in private correspondence via YouTube, quoted with permission)

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