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!

Brilliant!!! This helps so much!!! So much easier than anything else I have found. This is now simple.

Thanks again for showing the simple way of doing this. I look forward to the next video!! Absolutely brilliant.

Once again, excellent! I’ve watched many others, that have not helped nearly as much as these. Thanks for making it so easy to understand. When things get complicated, it is easy to make a mistake, but your method of writing down the structure helps to prevent mistakes.

Thanks a million!!! I’ve watched many videos and read tutorials, but still could not get two in a row correct on Khan Academy. Your explanation was so clear and simple that it finally made sense to me and now I can get all of the problems correct!!! Thank you very much.

It all makes total sense now! Thanks very much for all the derivative videos!!! Very helpful.

MemorizeAndLearn (on four different CCM YouTube videos about Differentiation)

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