Tag: Calculus

Categories
Assessments Conceptual Understanding Teaching Ideas

When Imaginary gets Real

One of the many interesting aspects of imaginary numbers is that we can use them to find out “real” facts (facts about real numbers).

Perhaps the most used examples are the derivation of the trigonometric identities for \(sin 2\theta\) and \(cos 2\theta\)*.

This post offers something more exciting and less-known: the radical forms of \(cos \frac{\pi}{12}\) and  \(sin \frac{\pi}{12}\).

Consider \(\frac{\frac{1}{2}+\frac{\sqrt{3}}{2}i}{\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i}\)

Algebraically, simplify it and write it in the form of \(a+bi\).

\[\frac{\sqrt6+\sqrt2}{4}+\frac{\sqrt6-\sqrt2}{4} i\]

We know the answer. But as a wise man once said, mathematics is all about seeing the same thing from different angles (or something like this).  So let us, calculate the same thing in some other way.

We write \(\frac{1}{2}+\frac{\sqrt{3}}{2}i\) and \(\frac {\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\) in the form \(r e^{i\alpha}\). And then evaluate the original fraction , that is going to be:

\[\frac{e^{i \frac{\pi}{3}}}{e^{i \frac{\pi}{4}}}\]

That can be simplified as \[e^{i \frac{\pi}{12}}\] That, can be written as \[cos \frac{\pi}{12}+i  sin\frac{\pi}{12}\] Compare this result with the one we had in the form of \(a+bi\) above, where \(a\) and \(b\) were in a radical form. They are the same things, represented in two different forms. Thus, their real parts should be equal with each other, and also their imaginary part should be equal with each other. So, we have: 
\[cos\frac{\pi}{12} = \frac{\sqrt6+\sqrt2}{4}\]
\[sin\frac{\pi}{12} = \frac{\sqrt6-\sqrt2}{4}\]

 

Design a problem to find \(cos \frac{\pi}{8}\) and \(sin \frac{\pi}{8}\) in the radical form. 

* We know that (a post will be written for this): 

\[(cos \alpha + i sin \alpha)^2=cos (2\alpha) + i sin(2\alpha)\]

Remember what the wise man once said and find \((cos\alpha+i sin\alpha)^{2}\) by actually finding the power. Then we have:

\[(cos \alpha + i sin \alpha)^2=(cos^2\alpha -sin^2\alpha)+ i (2 sin\alpha cos\alpha)\] The right sides of the above two equality are the same represented in two different forms (both are equal to the same left side). So the real parts of the two should be equal, and the imaginary parts of the two should be also equal.

\[cos 2\alpha=cos^2 \alpha -sin^2 \alpha\]\[sin 2\alpha=2 sin \alpha cos\alpha\]

 

 

Design a problem to find \(cos 3\alpha\) and \(sin 3\alpha\) .  

Categories
Assessments Conceptual Understanding Teaching Ideas

i Cycle

Where is \(i^{127}\)? 

If you have read i Cycle, the Sum of Powers, you have already experienced the powers of i, and their hidden cycle. This is a much basic problem and in fact, it is a prerequisite for understanding i Cycle, the Sum of Powers and everything else about complex numbers (okay, it is a bit exaggeration to say so, but you got the point, it is very important).

What is the problem here? 

Notice that I could ask the “same” problem just using a different power. For example, I could ask: Where is \(i^{39}\)?  You see, the problem here is not about finding this or that power of \(i\), it is about finding the pattern governing these powers. 

The fundamental power in this case is two, as it is the power used in the definition of \(i\): \[i^2=-1\].

Furthermore, So if our question was Where is  \(i^2\), we knew the answer is C. From, this starting point, we can see \(i^3=i^2*i=-i\) is the point D, \(i^4=i^2*i^2=1\) is on the point A, \(i^5=i^4*i=i\) is on the point B, and \(i^6=i^4*i^2=-1\) is on the point C, that happens to be the point we started our 90 degrees anticlockwise journey. 

It doesn’t matter where we start our rotation. But, maybe, the point A is the most natural starting point. Starting from A, the powers are: 

\[i^0, i^1, i^2, i^3, i^4, i^5, …\]

They look different from each other, but we know that there are only four fundamentally different number, as the above sequence is in fact the following sequence: 

 \[1, i, -1, -i, 1, i, -1, -i, …\]

So, to find the answer to our original question, we only need to put aside the fours that goes into 127 and see what remains. 

 

 

 

Starting from the point A, has another interesting advantage. It gives meaning the the very definition of \(i\). Think of \(i\) as \(i \times 1\). You might say, okay so what, isn’t it obvious that \(i \times 1=i\). Yes, but, notice that the left side is the process of multiplication and the right side is the result of the process. In this case, the process has an important interpretation, that is rotating 90 degrees anticlockwise. That is to say, when we multiply the point A (that happens to be \((1,0)\)) by \(i\) it goes to the point B, and likewise, when we multiply the point B by \(i\), it goes to the point C. In other words, \(i\times i\times 1=-1\). 

Voila: \[i^2=-1\]    

Categories
Assessments Conceptual Understanding Teaching Ideas

i Cycle, the Sum of Powers

\[1+i+i^2+i^3+…+i^{57}\] Do I need to say what the question is?

“Find the sum!” is the immediate question that comes to mind when you start mathematics. When you gain more experience in mathematics, you learn to ask a deeper question:

What is the pattern?

You know, 57 is irrelevant. The only important thing here is the general question of the finding the sum without actually doing all the additions. However, to get to the point of finding the sum without actually doing all the additions, we should experiment with actual additions first! For example, let us find the sum of the first seven terms.   

Please find it before continuing reading. 

Did you notice that to have 7 terms, the last power in the sum should be 6 and not 7, and that means we should calculate \[1+i+i^2+i^3+i^4+i^5+i^6\] and not \[1+i+i^2+i^3+i^4+i^5+i^6+i^7\]. 

This was not a trick. In fact, thinking in terms of the number of terms is the key to the pattern that we are looking for. 

What is the sum of the first two terms? 

What is the sum of the first three terms? 

What is the sum of the first four terms?

What is the sum of the first five terms? The first six? The first seven? 

Let us also include “the sum of the first term”. It is silly, but you see why it is useful. Here are the results starting from “the sum of the first term”, following by “the sum of the first two terms”, then, the sum of the first three term”, and so on. 

\[1,1+i, i, 0, 1, 1+i, i, 0, …\]

See, the sum of the first four terms is 0, the sum of the first eight terms is 0, and so on.  And then, after each 0, we have 1, 1+i, and i. Now if we want to know what is \(1+i+i^2+i^3+…+i^{57}\) we only need to know how many fours goes into 58 (that is the number of terms). This number of fours only matters to us, as long as it helps us to find how many terms remain. In this case, no terms remains. So, the sum is 0. 

\[1+\frac{1}{i}+\frac{1}{i^2}+\frac{1}{i^3}+…+\frac{1}{i^{2019}}\]

Do I need to say what the question is?

Categories
Teaching Ideas

Proof-Generated Definitions!

Okay, for a long time, I didn’t know what to blog about. Now, I have decided to write about my teaching ideas that take ages to turn into a piece of research. That is why I have started with this strange title for my first blog. The title comes from the exact same phrase coined by Lakatos in his book Proofs and Refutations.  Here is how I used the idea recently.

Last week, I was going to teach the definition of a convergent sequence. Previously, we had played a lot with sequences geometrically: how they are represented on a number line and how they are graphed as a function on the coordinate plane. We hadn’t done any calculations; just we drew a lot of figures to see how different sequences (i.e., convergent, divergent, bounded, increasing, decreasing) behave graphically.  The question I (as the lecturer) had to answer was which “theorem” could justify and motivate the definition of a convergent sequence. Indeed, at that stage of students’ knowledge, there was perhaps only one choice; but, surprisingly that choice seemed to be a perfect fit:  if a sequence converges, then its limit is unique. There was only one small problem, students have no reason to think otherwise: how on earth a convergent sequence could have two limits? Thus, I needed to raise the question and also play devil’s advocate. Here is how the plan goes in the class.

I: How do we know that this sequence has only one limit (I drew the following figure)?

Students: What? Obviously, it has only one; see, it is getting closer and closer to the red line.

I: But, what if there is another line very close to the red line that the sequence is getting close to it too (I didn’t draw that other line on the graph).

Students: It (the sequence) is going to pass that line; see (one of the students drew the following (blue) line):

I: What if the line I thought of is closer to the red line.

Students: After a few other terms, it is going to pass that line as well.

Here, suddenly one of the students said: “What if the sequence also passes the red line?” As a result, I changed my role from someone who was “against” uniqueness to someone who was defending it.

I: But, it can’t go that far from the red line, can it?

Then, I cheated and I told the rest of the story! After all, it was the first time that I tried to generate the definition of convergnce of a sequence via a theorem in the class. I more motivated the definition and less generated it, though, we were very close to hit the target and generated a number of key ideas that were used in the definition. Next time that I teach sequences, I’ll try to go all the way and report it in another post.

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