Tricks Of The Trade In Undergraduate Mathematics- Part 1
The foundations of mathematics are inherently philosophical so if we are to fully grasp the subject, we have to grasp its foundations.
Now by foundations I do not mean going all out on a crash course in the philosophy of mathematics(it would be nice if you did though). What I mean by grasping the foundations is an understanding of the basic ideas that stem from the paradigm shift from computational mathematics to abstract mathematical reasoning.
Thinking About Questions
To think mathematically about a problem like say 2x+1=3 seems rather trivial and boring but let’s explore two different approaches:
- Take 1 to the right hand side and change its sign to get 2x = 3-1=2 which reduces to 2x=2. Divide both sides by 2 to get 2x/2=2/2 so x=1.
Okay “=” means the quantities on the left and right side of the equality are equal and my aim is to isolate x on one side. Hmm… why not add -1 to both sides of the equality , things would still stay balanced after all, so 2x+1+(-1)=3+(-1) and hence 2x+0=2. Since 2x+0=2x the last time I checked, we have that 2x=2. Now, the next question is what number when multiplied by 2 gives 2? Any thoughts? It’s the multiplicative identity 1, yay!
Sure the differences between the two solutions seem rather subtle and sure you do not have to explain how x=1 is the solution to 2x+1=3 each time you see it but that’s not my point. My interest goes beyond the question at hand and I’m instead trying to apply the properties of certain mathematical operations and objects. Consider the problem:
Find the range of the real-valued function f(x)=(arcsinx)²+(arccosx)².
A problem like this requires an understanding of what the range of a function is as well as what sort of mathematical objects arcsinx and arccosx are. From High School you were taught that sinx and cosx are numbers between -1 and 1 (this is the range) with their inputs consisting of angles which which range over the real numbers.
Notice that our problem is asking for the inverse of x=siny where y = arcsinx so x is bounded by 1 and arcsinx is actually an angle with multiple branches. Okay , but how does that help us ? Well we now know what we are dealing with so we can now play around with the problem. For some reason I am compelled to explore sin(arcsinx+arccosx) and cos(arcsinx+arccosx).Why ? Well how does a painter know where to lay his brush? Inspiration, experience and sheer luck are the hallmarks of creativity and that in my view is what makes mathematics an art. We proceed below:
Using sin(A+B)=sinAcosB+sinBcosA and cos(A+B)=cosAcosB-sinAsinB as well as sin(arcsinx)=x (the same is true for cosine) and sin²x+cos²x=1 gives us
Thus we have that sin(arcsinx+arccosx)=1 and cos(arcsinx+arccosx)=0 which without loss of generality had pi/2 as it’s solution.
Hence arcsinx+arccosx=pi/2 and so arcsinx=pi/2 — arccosx.
f(x)=(arcsinx)²+(pi/2 — arcsinx)²
= 2(arcsinx — pi/2)² +(pi)²/8.
Thus range of f , = [(pi)²/8,infinity).
The bulk of this article focuses on thinking about problems rather than just the solutions as it could result in more insight to solve other problems. As Grant Sanderson, the founder of the Mathematics YouTube channel 3blue1brown would say just play around with the idea with no particular purpose in mind, and that my friends is what mathematical thinking involves, looking beyond the seemingly obvious. Happy maths!