Tricks Of The Trade in Undergraduate Mathematics-Understanding Problems

Last time we explored the importance of thinking about questions by virtue of two examples; one basic and the other requiring a bit more work. Now in this article we focus on understanding the key players in a mathematical problem. Objects and the operators that act on them.

Objects

What is an object? According to the Oxford Advanced Learners Dictionary, an object is a noun, noun phrase or pronoun that refers to a person or thing that is affected by the action of the verb, or that the action is done to or for. For our purposes however, it would be innocuous to consider an object to be something an operator(in the mathematical sense) acts on.

What comes to your mind when you see 6+4 written anywhere? A task to be done whose answer is 10(procedural perspective) or just another way of writing the number 10(object perspective)? Most people see it as the former due to the fact that we learnt how to add in precisely this way. So which perspective is the correct or right perspective to have? Well, both of them. In pre-school, the main focus was for pupils to learn how to count and counting can be annualized using the “+” operator , hence performing the operation of the operator on numbers results in another number.

A new way of thinking about 6+4 is as an equivalent way of writing the number 10. There are infinitely many ways of writing the number 10 of which just a few are:

3+7, 20/2, 5×2, 1.5+8.5, 1.6+8.4, 11-1, e.t.c.

These are rather boring examples, some unfamiliar ones are:

e^ln10, integral of x from 0 to 2sqrt(6), 20sin²x+20cos²x+10,e^ipi and many more.

This might just seem to be a nice parlour trick to impress the ladies(oh the irony!) but understanding that an object can be represented in many forms is a useful tool in your mathematical problem solving kit.

Observe that I mentioned earlier that integral of x from 0 to 2sqrt(6) is a number (10 precisely) but what about the indefinite integral, integral of x. Well let’s see:

Integral of x = x²/2 + c where c is any real number so for example x²/2, x²/2 -4, x²/2 +6.7 and so on are solutions.

So the integral of x is actually not one, not two, but a collection of infinitely many functions.

Conclusion

So we’ve explored what objects are in mathematics an how they can take different forms , we will explore operators in a future article.

Reference

How to study as a mathematics major by Lara Alcock.