As we continue our series on complex numbers, we return to the main series of posts, having concluded what I called a ‘pit stop’ last week. The pit stop introduced us to trigonometry. It should be observed that I have only introduced trigonometric ideas that contribute to our understanding of complex numbers. There is much more that we could have studied. Perhaps at a later date, I will revisit trigonometry and take us beyond to some fascinating results, such as conditional identities and the properties of triangles. Anyway, let us trudge back to our study of complex numbers.
Revisiting the Modulus
When we began our pit stop on trigonometry, we had just introduced the idea of the modulus of a complex number. Hence, we saw that, given the complex number z = a + bi, the modulus is defined as
We saw that this way of defining the modulus is not affected by the rotation of the axes. In other words, this metric is invariant to the rotation of the axes. Of course, if we rotate the axes, the orientation of the complex number with respect to the axes will change. In other words, even though the modulus is invariant, the orientation is not. This is only to be expected since the orientation is determined in relation to the axes and the axes are rotating.
Numbers on the Complex Plane
However, given a complex number z = a + bi, it is clear that this number represents a units in the real direction and b units in the imaginary direction. Hence, the point representing z on the complex plane is fixed. We can, therefore, join this point to the origin with a line that is unique to the point and, hence, this line makes a unique angle with respect to the positive real axis. This is depicted in below.
In the above diagram, we can see that the real and imaginary parts of z can be written as
Introducing the Argument
Hence, the above complex number has an orientation equal to θ with the positive real axis. This angle θ is called the ‘argument’ of the complex number. Now, since the trigonometric ratios are periodic in nature, we can also conclude that
In other words, due to the periodic nature of the trigonometric functions, which is something obtained from repeated rotations about the center, there are infinitely many values of the angle that will land us on the same point on the complex plane. However, there is a unique angle with the smallest magnitude that gives the same orientation. This angle is called ‘the principal value’ of the argument and will take values between –π and +π. Let us see how this works.
Suppose I have a complex number that I have obtained as follows: It has a modulus of 8 and an argument of 22π/3. Since the argument is great than π, I repeatedly subtract 2π to get
Due to this we can conclude that
Since the angle now lies between –π and +π, it is the angle with the smallest magnitude that gives us the same orientation. Hence, while we can say that the same complex number can have arguments of 22π/3, 16π/3, 10π/3, 4π/3 and -2π/3, the principal argument of the complex number is -2π/3. Where would this complex number lie on the complex plane? All we have to do is calculate the real and imaginary parts and plot the point. This is shown below
Till now we have specified complex numbers in terms of their real and imaginary parts as a + bi. This is known as the Cartesian form, since it depends on the system similar to the normal Cartesian coordinates in which we have two axes intersecting at right angles. Of course, since the modulus and argument of a complex number specifies a unique point on the plane, we can represent the complex number in terms of these two numbers. Hence, the above complex number can be written as (8, 22π/2), where the first number specifies the modulus and the second the argument. This is known as the polar form of the complex number. By convention, the modulus is denoted by r, since the complex number having a modulus of r lies on the circumference of a circle of radius r centered at the origin.
Adding and Subtracting with the Polar Form
Now, let us see what happens when we add and subtract complex numbers using polar coordinates. Consider two complex numbers z1 = (2, π/3) and z2 = (3,π/4). Here, the complex numbers have been specified in polar form. We can obtain the following
Now, the terms in red are clunky. And there are no trigonometric identities that would help us reduce these to forms that are neater. So, why bother with the polar form if all it does is give us clunky terms? I’m glad you asked.
Multiplying and Dividing with the Polar Form
The power of the polar form lies in what it does when we multiply and divide complex numbers. Let us consider the same two complex numbers and see what happens when we multiply them.
To get from the third to the fourth lines we used the sum formulas that we obtained in the previous post for the terms in red. Of course, from earlier in the present post we can recognize that the terms in green are simply the complex number z = (6, 7π/12). However, note the following
That is, when multiplying complex numbers, their moduli get multiplied and their arguments get added. That is really convenient and reduces the effort of multiplication considerably. Knowing the two moduli were 2 and 3, we could right away conclude that the modulus of the product is 6. And knowing that the two arguments were π/3 and π/4, we could easily determine that the argument of the product is 7π/12.
The same power is evident when we try to divide two complex numbers. We should be able to infer that, when dividing z1 = (2, π/3) by z2 = (3,π/4), the result should be a complex number with modulus equal to 2/3 and argument equal to π/3 – π/4 = π/12. Mimicking college mathematics textbooks, I leave the proof to the reader!
Looking Ahead
What we have seen is that the polar form of complex numbers significantly simplifies multiplication and division. Addition and subtraction, however, are not really improved at all and may, in many cases, be actually more tedious. However, the polar form has more advantages as we will see in the next post. That will be two weeks from today as I will be taking a break next week for Holy Week and Good Friday.
In our study of complex numbers, we are reaching the end of a pit stop dealing with trigonometry. I ended the previous post by mentioning that the trigonometric functions are periodic in nature. I claimed that this is an important feature in the study of complex numbers. In this post, we will look at what periodicity means and what it reveals concerning the trigonometric functions.
If you read the preceding paragraph carefully, you will probably have realized that my vocabulary has changed. I am no longer speaking of ‘trigonometric ratios’ but of ‘trigonometric functions’. One obvious reason is that, once we have transcended the definitions based on sides of a right angled triangle, there are no sides, the lengths of which can be placed in a ratio. In fact, as we saw in the previous post, we are now defining sine and cosine of an angle in terms of the coordinates of the corresponding point on the unit circle. Moreover, when we say that the values of sines and cosines repeat, we must have a way of quantifying how this repetition occurs. In other words, we want to be able to relate the angle made by the line joining the point under consideration to the origin and the positive x-axis. That is, we want to obtain expressions for the coordinates in terms of the sines and cosines of angles. This means we are treating the sines and cosines of the angles as functions of the angles themselves. Hence the change in vocabulary was needed.
In case you are wondering about the title of this post, it is a feeble riff off the majestic Concluding Unscientific Postscript to Philosophical Fragments by the Danish Christian philosopher Søren Kierkegaard, a work that is quite unwieldy.
Anyway, the focus of this post is on the periodicity of the trigonometric functions and some corollaries that can be obtained from this periodicity. This is precisely where I wanted to reach before returning to the main series of posts on complex numbers. So let us proceed.
Identifying Periodicity
Suppose, we consider a point P on the unit circle corresponding to the angle θ. Then, from what we have learned concerning the unit circle, the coordinates of P will be (cos θ, sin θ). This is shown below.
Now, if we rotate the radius OP counterclockwise by 2π radians, then the new position of the radius will be the same as the original position. In other words, now, the same point P corresponds to a radius making an angle of θ + 2π radians. Since the new position of the radius is the same as the old position, the new coordinates of P, corresponding to the angle θ + 2π radians, will be the same as the old coordinates, corresponding to the angle θ radians. Hence, we can conclude that
Of course, if we rotate OP by another 2π radians, we will be back where we started. This holds true for any integral multiple of 2π radians in the clockwise or counterclockwise direction. Hence, we can conclude that
What this tells us is that the values of the cosine and sine functions will repeat every 2π radians in the clockwise or counterclockwise directions. Now, it is only by convention that an angle is often denoted by the letter θ. Normally, when we plot graphs, we use x for the independent variable and y for the dependent variable. Hence, if we plot the graph of y = sin x, we will get the following
Similarly, if we plot the graph of y = cos x, we will get
In like manner, if we plot the graph of y = tan x, we will get
The regular repetition of the pattern for all three graphs is what makes these functions periodic.
Corollaries of Periodicity
Now, if we are considering rotations of the radius, how do these values relate to each other? That is if we double the angle, how does this affect the values of sine, cosine, and tangent? Before we can answer that question, we need to look at the more general case of adding two angles. So let us deal with that.
Sum and Difference of Angles
The question we are asking is, “How do the sine, cosine, and tangent of A+B relate to the sines, cosines, and tangents of A and B? Here, it is possible for us to derive the following formulas
If you are interested in knowing how these are derived, below are the proofs.
If the above proofs are difficult to follow, please do ask me for an explanation. Of course, having obtained the values of sine and cosine for the sum of angles, we can obtain the values for the difference of angles as below
Double and Half Angle Formulas
Using the expressions for the sum of angles and putting A = B, we can obtain the following expressions for the sine, cosine, and tangent of the double angle 2A. This is shown below
Going in the reverse direction and using the expressions for cos 2A, we can obtain
Triple Angle Formulas
Using some of the above results, we can obtain an expression for the cosine of a triple angle as follows
In a similar way, we can obtain an expression for the sine of a triple angle as shown below
Finally, we can obtain the expression for the tangent of a triple angle as follows
Given the fact that the trigonometric functions are periodic and that there are expressions for the trigonometric functions of sums, differences, multiples and sub-multiples of angles in terms of the trigonometric functions of their component parts, there are an indefinite number of results that can be obtained. However, what we have so far is sufficient for our journey dealing with the study of complex numbers.
Getting Back on Track
However, in case you are wondering why we took this pit stop, recall that we have a geometric interpretation of a complex number on a two dimensional plane in which the horizontal axis is the real axis and the vertical axis the imaginary axis. We had dealt with this in It’s Not Rocket Surgery! and Modulating an Invariant Metric. Recall that, it was immediately after the latter that we began our pit stop.
Recall that we can depict the numbers 3 + 2i, -4 + 4i, and -2 – 3i as shown below.
We had also introduced ourselves to the ‘size’ of a complex number, which is called the modulus. This is nothing but the length of the line joining the origin to the point representing the number. This line, however, has a well defined orientation with respect to the positive real axis. Hence, we can specify the complex number in terms of a distance from the origin (i.e. its modulus) and an orientation with respect to the positive real axis. This orientation has a technical term, of course. However, we will deal with it and some other aspects related to operations on complex numbers in the next post. I know this may bug you, especially if you are the impatient sort. But from you I will accept no argument!
René Descartes’ concept of the Cartesian plane provides valuable insights for trigonometry. (Source: World History Encyclopedia)
As we continue with our pit stop in the series on complex numbers, let us remind ourselves where we have reached. In Trigonometric Identity Crises, we looked at some trigonometric identities. This followed on the heels of Round and Round, in which we saw why a right angle is defined to have 90°. We also defined the radian. Before that, in A Mathematical Partnership, we took a historical tour, starting from India and ending in Rome, during which we understood how the trigonometric ratios are defined and why they have their names. The pit stop began with the post that preceded it, My Trigger, No Metric, Beginnings, in which I explained my long standing fascination with trigonometry, beginning with my explorations with Pythagorean triples.
However, what we have seen so far is that the ratios are defined in terms of the lengths of sides of a right angled triangle. Now, students of Grade 6 (or maybe 7) and above will know that the sum of the angles of a triangle is 180°. This means that, in a right angled triangle, since one angle already is 90°, the other two need to add up to 90°. In other words, as defined, the ratios only work for angles that are acute, that is, strictly between 0° and 90°, not including both of them.
Now, since we can have angles that measure 0°, 90° and everything above 90°, including reflex angles, this means that the definitions of the ratios do not include most of the angles we may encounter. This is a serious drawback.
Descartes to the Rescue
Mathematicians do not like artificial restrictions. Hence, when faced with such an obstacle, they become creative and look for ways to circumvent the obstacle. In many such cases, the innovativeness involved is just mind boggling. Of course, since we are living some centuries after these innovations, we lack the ability to view them as something fresh that contributed to the advancement of mathematical knowledge. Indeed, hardly any teacher spends any time showing the students what the ‘before’ and ‘after’ of these innovations was, thereby impoverishing the students by depriving them of this knowledge.
In the case of extending the definition of the trigonometric ratios, the innovation involved is truly remarkable and underscores something I attempt to communicate to my students often, namely that the different branches of mathematics form one unified body of knowledge in which everything dovetails perfectly with everything else.
So let us see what they did. We begin with the Cartesian plane, named after René Descartes, the French mathematician. It consists of two number lines at right angles (we know what that means!) to each other and intersecting at their respective zero points. Now, let’s consider our right angled triangle. However, we place the vertex of interest at the origin of a two dimensional Cartesian plane, with one of the arms that forms the vertex aligned with the positive x-axis. This is shown below.
Beginning a Redefinition
When we defined the ratios and understood why they have their names, we had considered a unit circle. In keeping with that, the hypotenuse of the triangle (side OA) is going to be 1 unit. Now, we can see that
Now, as indicated in the diagram, the point A has coordinated (x, y). However, the x coordinate of a point is the directed distance of the point from the y-axis. Similarly, the y coordinate of a point is the directed distance of the point from the x-axis. Since the term ‘directed distance’ may be unfamiliar to some, allow me to explain. If we start at the y axis and move toward the right to reach the point, then the distance is considered positive and the x coordinate will be positive. If, however, we more toward the left to reach the point, then the distance is considered negative and the x coordinate will be negative. In a similar way, starting from the x axis, an upward movement is considered positive, while a downward movement is considered negative. Hence, we can conclude that, to the right of the y-axis the x coordinates will be positive while they will be negative to the left of the y-axis. Also, above the x-axis the y coordinates will be positive while they will be negative below the x-axis.
However, with reference to the above diagram, it is clear that OB is the directed distance of A from the y-axis, while BA is the directed distance of A from the x-axis. In other words, the lengths OB and AB, both being positive in the above diagram, give the x and y coordinates of the point A. Hence, we can conclude that
In other words, with respect to the right angled triangle OAB, the cosine and sine of the angle at O are equal to the x and y coordinates of the point A.
Now, let us revisit our Cartesian plane. Without any points on it, it would look like
Here, the Roman numerals (I, II, III, and IV) indicate the quadrant number. The signs below indicate the signs that the x and y coordinates have in that quadrant. Hence, in the second quadrant (i.e. II), the x coordinate is negative while the y coordinate is positive. This follows the directed distance convention we saw earlier – right/up is positive, left/down is negative.
Completing the Redefinition
We can see here that no quadrant has the same combination of signs for both the x and y coordinates. This allows us to extend the definition of the trigonometric ratios as follows. Consider a point (say P) on the circumference of the unit circle. Draw the radius joining P to the center O. Let the angle made by OP with the positive x-axis be θ. Of course, here we have the need for a further convention. Since any point on the circumference can be reached through a clockwise and an anticlockwise rotation, we define the anticlockwise rotation, that sweeps the quadrants in proper order, to be a positive angle measure. Hence, a clockwise rotation would be a negative angle measure.
Anyway, let the angle made by OP with the positive x-axis be θ. Then we define the cosine of θ to be the x coordinate of P and the sine of θ to be the y coordinate of P. We can depict this as follows.
Ratios for Allies
We can see that, since the angle made by the axes with each other is 90° (or π/2 radians), angles that differ by or add to multiples of 90° (or π/2 radians) will have trigonometric ratios that are related to each other. Such pairs of angles, which differ by or add to multiples of 90° (or π/2 radians) are called allied angles. Using these extended definitions, we can obtain the following results
The above table may be a little confusing. For starters, I have used radian measures for the angles. Recall that 180° = π radians. So, let’s see how the table is to be read. Suppose I want to determine sin α, where α = π – θ. The row for sin α is the second row of the table, while π ± θ is the third column. The element in that cell is ∓ sin θ. Since I am interested in π – θ and the minus sign is second in the symbol ±, I take + sin θ, which is second in the term ∓ sin θ. Hence, I can conclude that
How do we obtain these results? If you are interested, you can find visual proofs using a general, not unit, circle here. With our extended definitions of the trigonometric ratios, we can obtain their values for any angle. If you wish to see this for yourself, you can check here, here, and here.
Rotations Without End
Now, of course, we are treating angles as though they were also on somewhat of a number line, just one that wraps around itself. We have adopted the convention that counterclockwise rotations are positive and clockwise rotations negative. Also, we realize that there is nothing that hinders a radius from continuing to rotate beyond the full cycle. So we could have angles that exceed 2π radians (360°) and angles that go below -2π radians (-360°). An angle of 7π radians, for example, would involve three full counterclockwise rotations, each of which accounts for 2π radians, followed by a further rotation by π radians. Similarly, an angle of -810° would involve two clockwise rotations, each of which accounts for -360°, followed by a further rotation of 90°. Of course, with each rotation, the point P will reach identical positions as in the previous rotation. This means that, after every rotation, the values of the trigonometric ratios repeat themselves. This is a feature known as periodicity, which is crucial in the study of complex numbers. Since it is a very important part of the study of trigonometry, we will turn to that in the next post. Till then, keep twirling!
We are in the middle of a pit stop in the middle of our series on complex numbers. The pit stop is intended to introduce the reader to the basics of trigonometry. In the previous post, we saw why we have 360° in a circle and 90° for a right angle. We also defined the radian, which I demonstrated was not an arbitrary unit, unlike the degree. In the post preceding that, we saw how partnership between mathematicians from India, Persia, Greece, and Rome gave us the names of the trigonometric ratios.
At the conclusion of this pit stop, when we return to complex numbers proper, we will see how integral the ideas we uncover in trigonometry are to the study of these numbers. For now, as we begin the next part of our pit stop, let us remind ourselves about the definitions of the trigonometric ratios.
Nomenclature
With respect to the figure above, the angle we are focusing on is ∠ABC, which we are calling θ. The side opposite θ is AC, which has a length equal to b. The side adjacent to θ is BC, which has a length equal to a. Finally, the hypotenuse of the triangle is AB, which has a length equal to c.
Per our definitions of the trigonometric ratios
The three ratios relate as follows
We also define three other ratios as follows
Here, csc stands for cosecant, which is often shortened to cosec. Similarly, sec is short for secant and cot is short for cotangent. With reference to the post made two week back, I leave it to the reader to discover why the secant has its name.
Pythagoras Again
Now, from Pythagoras’ theorem applied to the triangle ABC, we know that
If we divide the entire equation by c2, we will get
Pay close attention to the notation above. What the last equal says is that
Consider the equation
This is the first identity students learn in trigonometry. It is called a Pythagorean identity because it is derived from the Pythagorean theorem. But what do we mean by an identity?
An identity is a mathematical statement that is true for all values of the variables. In the above equation, the angle θ is the variable since we did not decide on its value. The equation
is true for all values of θ and is, therefore, called an identity. An identity, while in this case written in the form of an equation, differs from common equations in that one cannot ‘solve’ an identity to obtain a value of the unknown since it is true for all values.
Now, starting with this identity, let us divide it by cos2θ. We would then get
This last equation is also a Pythagorean identity since it too is derived from the Pythagorean theorem. The reader can begin with the first identity and divide it by sin2θ to obtain the third Pythagorean identity
Proving Identities
Note that the two additional Pythagorean identities did not require any additional trigonometry apart from the definition of the secant, cosecant, and cotangent. This is a big feature in trigonometry, where new identities can be proved based on existing ones. Let us consider some examples.
Example 1
Suppose we were asked to prove that
We could proceed as follows
Here LHS and RHS denote ‘left hand side’ and ‘right hand side’ and indicate the direction in which we are moving to prove the identity.
Example 2
Let’s consider another example. Suppose we were trying to prove that
We could proceed as follows
While this proof is longer than the earlier one, note that in both we use simple algebra, such as was discussed earlier in Primer to Complex Numbers, the post with which I began the series on complex numbers.
Example 3
As a final example, suppose we were asked to prove
We could proceed as follows
Interrelatedness of the Trigonometric Ratios
Note that, along the way, we made use of the Pythagorean identities quite often. This is because the trigonometric ratios simply conceal the Pythagorean theorem in their definitions. In fact, because of this feature, namely that all the ratios are related to each other through the Pythagorean theorem, we can actually show the following
That is, each of the ratios can be expressed completely in terms of any one of the other ratios.
‘Sining’ Off
However, if you have been paying attention, you will realize that we have a problem. We know that no triangle can have an angle equal to 0° or 180°. We also know that a right angled triangle cannot have another right angle. We also know that a right angled triangle cannot have an obtuse angle or a reflex angle. In other words, defining the trigonometric ratios in terms of lengths of sides of a right angled triangle means that most of the angles cannot have trigonometric ratios defined for them. This is a remarkable drawback. However, mathematicians have gotten around this restriction in quite an ingenious manner. We will turn to that in the next post. Till then here are a few identities you can try proving.
We have reached a remarkable junction in our journey through complex numbers. We are in the middle of a pit stop in which we are looking at some trigonometric ideas. And on the occasion of Pi day 2025, I think this post makes a lot of sense. Two weeks back, in My Trigger, No Metric, Beginnings, I introduced the idea of right angled triangles and Pythagorean triples. As a reminder, Pythagorean triples are sets of three integers that satisfy the Pythagorean theorem
In that post, I looked at different ways of generating these triples, an exercise with which I occupied myself shortly after I was introduced to these triples. Last week, in A Mathematical Partnership, we looked at how the common trigonometric ratios are defined. We also understood how the ratios – tangent, cosine, and sine – obtained their names.
It is time now to understand how we humans have decided to measure angles. After all, that is precisely what the trigonometric ratios deal with – the ratios of the lengths of sides of a triangle corresponding to the measure of the angle being considered.
A Degree of Definition
Most of us would have started our journey into angle measurement with the knowledge that a right angle is equal to 90 degrees, where the ‘degree’ is the unit of measuring angles. However, when I have asked students how to define a ‘degree’, many, if not most, of them are flummoxed. They know that 90 of these units give us a right angle. And some students may be able to then say that a degree is one-ninetieth part of a right angle. And while this is perfectly correct, the definitions are circular since the degree is defined in terms of the right angle and the right angle in terms of the degree. I could just as well define the angle below as a right angle and obtain a different measure of a degree than what we conventionally use.
You may object and tell me, “That angle is not a right angle.” But I could very well respond, “How do you know?” Indeed, since the definition of each term – right angle and degree – depended on the other, there is no way of deciding what each measure actually is.
Of course, there is a better way out. Unfortunately, none of my mathematics teachers told me this. And given the flummoxed expressions on the faces of most students when asked to define the degree or the right angle without reference to the other, it seems that this unfortunate practice continues to date.
So let us begin with two intersecting straight lines as shown below
Now, apart from their location and orientation, there is nothing that differentiates the line AB from the line CD. Hence, the angle form by each of these lines at the point of intersection, O, must be identical. In other words, since ∠AOC + ∠BOC forms the straight line AB and, since ∠BOC + ∠BOD forms the straight line CD, the two measures should be equal. So we can write
This is nothing other than saying that vertical angles (or vertically opposite angles) are equal. Now note that ∠BOC is adjacent to both ∠AOC and ∠BOD. These pairs of adjacent angles (∠AOC and ∠BOC or ∠BOC and ∠BOD) have the same angle sum, as we have stated above. However, there is only one configuration in which the adjacent angles will have equal measure. This is shown below
We can now define the four angles formed, all of which will have equal measure, to be right angles. From here, we can define the degree to be one-ninetieth part of the right angle and have no circularity in our definitions. But why do we define a right angle to be 90°? Why not a neat, nice round number like 100°? Of course, there is a unit of angle measure, the gradian, that is defined as one-hundredth part of a right angle. But I bet not too many people know about it. Why, when 100 is a part of the decimal system, being a power of 10, has this unit not caught on? And why do we function with the right angle being defined as 90°? The answer to this is two-pronged.
Celestial Constraints
We live on a strange planet, the only one we know of that can sustain the kind of life that we are accustomed to. Our planet revolves around an unimpressive, medium sized star in a orbit that takes it about 365.25 of its own rotations about its axis. That’s just a convoluted way of saying that one earth year is equal to approximately 365.25 earth days. The earth also has one natural satellite – the moon – which revolves around the earth in an orbit, which when viewed from earth, takes between 29 and 30 earth days to traverse.
The seasons are, for the most part, governed by the solar cycle of the earth. That is, the earth’s position relative to the sun has a greater impact on the seasons than does the moon’s position relative to the earth. However, it is difficult to determine exactly where in the solar cycle we are, since measuring exact times for sunrise and sunset is impractical almost everywhere. This means that determining the exact dates for the equinoxes, where the length of daytime and nighttime are equal, or the solstices, where the lengths of daytime and nighttime are extreme, is practically impossible.
Due to this, most human cultures have found it easier to mark the progress through the year ironically with the lunar cycles. After all, it is quite easy to look into the night sky and realize that the moon is not visible! Hence, most cultures have measures this progress through the year by starting months when the new moon cycle began.
However, as mentioned earlier, the moon takes 29 or 30 days to complete one cycle around the earth, as viewed from earth. This means that we encounter a new moon once every 29 or 30 days. Hence, most cultures have measured the progress through the year in terms of months that are 29 or 30 days long, following the lunar cycles. However, this results in a shortfall of about 11 days when compared with the solar cycle, resulting in a drift. To compensate for this, most cultures have the practice of adding an extra month every 2 or 3 years to get the calendar synchronized with the solar cycle.
So what we have is a solar cycle of approximately 365.25 days and a lunar cycle, twelve of which amount to between 353 and 355 days. The number 360, which is four times 90, falls neatly in this interval between the period of twelve lunar months and one year. The number 400, which is what we would get from the gradian, lies outside this interval and hence would not have been suitable for measuring anything on planet earth.
A Deliberation on Divisors
Of course, the second consideration, beyond what is forced upon us by the weirdness of the interaction between the sun, the earth, and the moon, is that we want a number that has a large number of divisors. This allows for dividing the year into many smaller fractions without having fractional days to account for. It is here that we see another advantage of 360 over 400. Here at their divisors
Apart from 1, which is a divisor of all numbers, and 400 and 360 respectively, since every number is automatically a divisor of itself, 400 has 11 divisors, while 360 has twice as many. In other words, 360 allows us to divide the year in 22 different ways without fractional days, while 400 only gives us 11 options.
Now, if we extricate ourselves from the realms of planetary rotations and revolutions and turn to geometry, we can see that dividing one complete rotation about a point into 360 parts gives us more options than dividing by 400 parts. On account of this and the link with the lunar and solar cycles, the degree was defined to be one-ninetieth part of the right angle.
However, we can see that it is an arbitrary division. Divisibility, which I discussed in Behind the Smoke and Mirrors, plays no role in determining the units of other quantities. Why should it play such a prominent role when it comes to measuring angles? In fact, let us look at how some fundamental units are defined.
Unimaginable Units
The second is defined as 9,192,631,770 cycles of radiation associated with the transition between the two hyperfine levels of the ground state of the cesium-133 atom. If that number is strange, consider that the metre is defined as the distance traveled by light in a vacuum in 1/299,792,458 seconds. The kilogram is defined in an even more roundabout way by first defining Planck’s constant to be 6.62607015 × 10−34 joule second and then using the definitions of the second and the metre and the fact that one joule is equal to one kilogram times metre squared per second squared to arrive at the measure of the kilogram. What this lets us know is that, when it comes to units of measurement, divisibility is the least of our concerns.
What this means is that choosing a unit for angle measure based on divisibility is arbitrary. And since we understand that the twelve lunar cycles and one solar cycle are not sufficient grounds for measuring angles, any number we decide upon, whether 360 or 400, is going to be arbitrary at best. So, how do we decide on a better, more appropriate unit? We have seen earlier, in our series on e and π, that we are often forced to accept constants that are inconvenient. When I say inconvenient, I mean that we would never have thought to consider these numbers as significant. In fact, were these numbers not forced upon us, we would never in our wildest dreams have settled on anything even remotely close to their values.
So, how do we go about defining a unit for measuring angles? The previously mentioned units (second, metre, and kilogram) were all defined in terms of physical processes or quantities. We do not have anything like this for the angle since an angle is an idealized geometric entity. So, we are forced to consider other ways to define a new unit for angle measure.
Reining in Rotation
We start with the observation that an angle simply represents a rotation. So, for example, in the figure below, the angle AOB can be viewed as a rotation of OA in the counterclockwise direction with O being the center of rotation until A coincides with B.
So the measure of the angle can be quantified as the amount by which we must rotate from the starting point until the angle is formed. We also recognize that a logical point would be one complete rotation about O, which would bring OA back to coincide with itself. Once we have done this, the point A would have traced the circumference of a circle with center at O and radius equal to the length OA as shown below
Now, we know that, if the radius of this circle is r, then the circumference is 2πr. We can also understand that, if point A traverses a distance that is twice the distance covered from A to B along the circumference, the angle by which OA would have rotated would be twice the original angle AOB. In other words, the distance covered is proportional to the angle through which OA rotates. We can also see that the distance is proportional to the radius. That is, if the radius is doubled, the distance covered would be double the original distance.
We now look at the formula
Since r represents the effect that the radius has on the circumference, we ask ourselves what the 2π represents. Remember that that distance covered during a rotation is proportional to both the radius and the angle of rotation. Since, r represents the proportionality to the radius, it must be the case that 2π represents the proportionality to the angle of rotation. However, the circumference itself represents a full rotation. Could it be then that 2π is the angle corresponding to the full rotation? What would that mean?
Now, while OA is rotating about O, point A traces arcs of the circle. Let the arc length corresponding to an angle θ be s. Since arc length is proportional to the angle of rotation, we can write
Now, if we keep the angle of rotation fixed at θ, then we can understand that, if the circle was twice as large, that is, if the circle had double the radius, then the arc length would be double what it was. Hence, the constant k must be none other than the radius r. This gives us
Realizing the Radian
Comparing this with the equation C = 2πr we can see that, since C corresponds to the arc length for a full rotation, 2π does indeed correspond to the angle of rotation. Using the equation s = θr, we can see that, if we set θ = 1, we will get s = r. This gives us the definition of the new unit of angle measure, called the radian. The radian is the angle subtended by an arc of a circle that has length equal to the radius of the circle at the center of the circle. In other words, starting from OA, when we have rotated through an angle equal to 1 radian, the arc length involves would be equal to the radius of the circle. This is depicted below
With this definition of the radian, we can conclude the following
One full rotation or the angle about a point is equal to 2π radians.
A half rotation or the angle for a straight line is equal to π radians
A quarter rotation or a right angle is equal to π/2 radians
From this we can also get the following conversion equation
The superscripted ‘c’ is the symbol for the radian, indicating ‘circular measure’ since the radian is the unit of angle measure defined solely based on the measurement of the circle. Of course, since the radian is the standard unit for angle measure, by convention it is often not indicated. Hence, in the equation
it is understood that the left hand side specifies π/3 radians.
Coming Full Circle
What we have seen in this post is that the degree measure is an arbitrary unit. It was adopted on account of two considerations – the orbital period of the earth (i.e. approximately 365.25 days) and the convenience introduced by the fact that 360 is a highly composite number with many distinct divisors. However, we saw that units for other quantities did not pay heed to any such requirements. This allowed us to think of the unit for rotations in terms of the features of the circle, leading us finally to the radian.
However, while almost all the other units have a certain degree (pun intended) of arbitrariness to them, the radian is completely defined in terms of the internal structure of the object it is designed to describe. For example, with the different measurement systems that humans have devised, we can imagine a world in which the standard unit of length is a cubit instead of a metre and the standard unit of mass is the pound instead of the kilogram. The second, based as it is on the period of rotation of the earth and the convenience of dividing successively by 24, 60, and 60, might have a slightly less amount of arbitrariness to it, at least for creatures who inhabit this planet. However, no matter where one may find oneself in the universe, all circles on Euclidean planes will yield the same measure for angles regardless of the name any extraterrestrial species may have given the quantity.
Having defined the radian, we will continue with our pit stop in the next post in which we will consider some trigonometric identities. Until then, hope you have seven turns for the better!
In the previous post, we began our first pit stop during our journey into the wonderful realm of complex numbers. We looked at Pythagorean triples and some ways of generating sequences of triples. I admitted at the end of the post that this was not strictly a study of trigonometry and that we normally begin by defining the ratios since, cosine, and tangent. I promised you that I would tell you how we got these weird names. That is the agenda for this post. We will consider the ratios in the reverse order because the story of sine will take us on two globetrotting journeys that I am sure many, if not most, of you have never undertaken before this.
Triangle Nomenclature
We begin our journey by considering some nomenclature. In what follows, I will assume that the triangle is ABC and that the right angle is at vertex C. Further, the length of the side opposite a given vertex will be denoted by the lower case letter of that vertex. Hence, the length of side BC, which is opposite vertex A, will be denoted as a. The figure below should clarify the nomenclature we will use in the discussions from now on.
Now the trigonometric ratios for angle B are defined as follows
How did we get these names?
Off on a Tangent
For the definition of the tangent, we begin with a unit circle. This is simply a circle with radius equal to 1 unit. Let us consider a unit circle with center at O. A point, P, is chosen on the circumference of the circle and a tangent is drawn at that point. From what we know about circles, this tangent is perpendicular to the radius OP. Now consider another point, Q, on the circumference of the circle, such that the angle made by the radius OQ with the radius OP is θ. Now we extend the radius OQ and the tangent at P so that they intersect at point R. All of this is shown in the figure below.
Now it is clear that ΔOPR is a right angled triangle with the right angle at vertex P. Now, with respect to the angle θ, RP is the opposite side while OP is the adjacent side. Hence, per the definition of the tangent above it should be clear that
However, OP = 1, since we are considering a unit circle, hence we obtain
However, RP is the length of the tangent corresponding to the points P and Q, the radii of which makes the angle θ. Hence, the name of the ratio. The tangent of an angle is the length of the tangent formed as above in a unit circle.
Cosplay Anyone?
Now, that’s all well and good. What about the cosine and sine? We go back to our original right angle triangle. However, now we focus on the vertex A. With respect to angle A, the trigonometric ratios will be as follows:
Note that
However, since C is a right angle, it follows that
In other words, A and B are complementary angles. When we say that
We are saying that the sine of an angle is the cosine of the complementary angle and vice versa. In other words, cosine is simply a shorthand for complementary sine, that is, sine of the complementary angle. But how in the world did the ‘sine’ get its name?
A Journey from Sanskrit to Latin
The ratio of the opposite side to the hypotenuse was used in many parts of the world. However, the name ‘sine’ strangely has its roots in the Indian subcontinent.
With reference to the figure above, the red arc and the green chord can be seen as a bow and string. Hence, Indian mathematicians named the string, that is, the chord, jyā, which is the Sanskrit word for ‘bowstring’. The half-chord was called ardha-jyā, which means ‘half bowstring’. Over time, the mathematicians realized that the ardha-jyā had more common use in the study of the triangles than did the jyā. Hence, they dropped the prefix and began calling the half-chord jyā.
From here we have two stories, equally fascinating, and equally plausible (or implausible).
One pathway indicates that a synonym for jyā in Sanskrit is jīvá. This term was adopted by Arabic scholars, who simply attempted to transliterate it into Arabic script as jība. However, since Arabic is not written with short vowels, this was rendered simply as jb. However, jība is not an Arabic word. In time, jb came to be pronounced as jayb, Arabic for “bosom”. When Latin scholars found these Arabic works, they were reluctant to use the word for “bosom” in a mathematical work. Instead, they chose the word ‘sinus’, the Latin word for the hanging fold of a toga over the breast, from which we get the final form ‘sine’.
The second pathway also observes the synonym jīvá but recognizes that jīvá also means ‘life’ in Sanskrit. From here the proposal is that the transformation happened earlier, when the wordjīvá, now understood to mean ‘life’, was translated as ‘zind’, the Old Persian word for life. From there the name went westward to the Latin world where, unfortunately, the final ‘d’ of the name was misread as a breathing mark. This is understandable since, in Old Persian the letter for ‘d’ is 𐭣 and the breathing mark is 𐭥, making for a very small difference. Hence, the name got transliterated as ‘sine’, with the initial ‘z’ getting replaced with a ‘s’.
Both the above pathways are based on hypotheses concerning when the Sanskrit texts were translated into either Arabic or Old Persian. We do not possess any extant manuscripts that indicate which pathway was taken. It is also possible that both pathways were taken a few centuries apart from each other, leading to a strange convergence when the various corruptions and mistranslations reached the Latin world. However, both pathways indicate that the origin of our name ‘sine’ is from the Sanskrit word jīvá.
Knowledge Transcends Borders
People from the Indian subcontinent may express pride at the fact that the main trigonometric ratio (sine) obtains its name from a Sanskrit word and from the imagination of some native mathematician who saw a segment of a circle as a bow and string. While there is no denying this, the length corresponding to the ardha-jīvá was used in many places, from Egypt to Persia. In other words, the usefulness of the length of the half-chord in relation to the angle formed by the half-chord was recognized by mathematicians the world over. For me, the journey, regardless of the pathway we decide upon, indicates that the world has always been one in which knowledge moved across all artificial national borders. Beginning with a Sanskrit word jīvá we have ended with a Latinized word ‘sine’. And I can only hope that this permeability of knowledge continues despite all the obstacles that are being introduced.
Having seen how the trigonometric ratios received their names, we will continue our pit stop in the next post to understand how we decided to measure angles. Why do we have multiple units – the degree and the radian? Which one do I prefer? It will be pi day and I think the next post will be an appropriate one for the occasion. Stay tuned.
A plot of the primitive Pythagorean triples with the odd leg on the horizontal axis and the even leg on the vertical axis. (Source: Wikipedia)
My mathematical world was thrown wide open when, in the ninth grade, my classmates and I were introduced to the study of trigonometry. Following that, when I studied at IIT – Bombay, I wrote a research paper titled The Synthesis of Coupler Curves and designed a mobile industrial robot, both projects requiring the use of trigonometry in the extreme. Hence, this post and the next couple are ones that I have actually been looking forward to.
Of course, this post forms a part of the ongoing series on complex numbers. However, this post forms the first in a pit stop that I had forewarned the readers about in the first post of the series. Yes, I said ‘first in a pit stop’. You see, unlike in an F-1 race, this pit stop will take some time!
The Theorem of Pythagoras
The journey into trigonometry begins with the ubiquitous theorem of Pythagoras or the Pythagorean theorem. This is not the place to debate where the theorem was first discovered or used or even if Pythagoras himself was an actual historical person or just a mythic figure concocted by the school that bore his name. Pythagoras’ theorem (as we call it in India) or the Pythagorean theorem (as people in North American tend to call it) states that, in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Students are often instructed to memorize
However, without knowing what the symbols a, b, and c denote, the above equation is meaningless. In particular, without recognizing that c represents the length of the hypotenuse, the equation is not even valid! However, in my years of teaching students in high school, I have realized that many students are not even taught what the hypotenuse is!
An Unfortunate Realization
Most students are introduced to right angled triangles with figures like the ones below
In all three triangles, the side AB is the hypotenuse. However, the unfortunate fact is that in the above three figures AB is oblique (that is, neither vertical nor horizontal). And so many students have come to me after middle school with the understanding that the hypotenuse is the oblique side. So consider the figures below.
All I have done is rotate the earlier triangles. The hypotenuse should not change. In all three cases it is still side AB. However, now AB is either horizontal or vertical. And some students have understood that the hypotenuse is oblique. Hence, they conclude that the hypotenuse is AC or BC. The hypotenuse, however, is the side opposite to the right angle. This is something that teachers in the middle school need to ensure students understand. It is not the orientation of a side that makes it the hypotenuse but its relation to the right angle.
Pythagorean Triples
Anyway, returning to the equation for Pythagoras’ theorem, we have
I remember when I was introduced to this theorem back in my middle school days. My teacher introduced us to Pythagorean triples like (3, 4, 5), (5, 12, 13), etc. I remember spending a lot of time trying to find patterns to these triples. “Was there a way of finding new triples?” was the question I asked myself.
My search for patterns revealed that
That is, the square of the smallest number was the sum of the two larger numbers in the triple. Did this always hold true?
I discovered that
with (7, 24, 25) and (9, 40, 41) forming Pythagorean triples. Hence, it seemed that the pattern I had discovered did form Pythagorean triples. Could this be formalized in any way?
Formalizing the Sequence
Scatter plot of the legs (a and b) of Pythagorean triples (Source: Wikipedia)
Later introduction to sequences and series allowed me to recognize that, in all the triples I was dealing with, the smallest number was an odd number. Moreover, the two larger numbers happened to differ by 1, the smaller being even. Knowing that 2n + 1 is always an odd number if n is a natural number, we can do the following
So, the square of the smallest number 2n + 1 can be split into
where the two numbers differ by 1 and the smaller is even. But do these three numbers satisfy Pythagoras’ theorem? Let’s check.
It is clear that
meaning that I had found a way to generate Pythagorean triples. Using this we can generate the triples (11, 60, 61), (13, 84, 85), (15, 112, 113), etc.
Another Sequence of Triples
However, it was clear from the triple (8, 15, 17) that I had not found a way for generating all the Pythagorean triples. Indeed, this triple does not satisfy the original premise for the triples because
Was there a pattern that governed these three numbers? And could that pattern be extended, as before with the first pattern, to generate a new sequence of triples? Trying different patterns, I realized that the square of 8 was equal to twice the sum of 15 and 17. That is,
Was this something important? Using this pattern, I discovered that (12, 35, 37) was also a Pythagorean triple in which
Could this be generalized? Like before, where 2n + 1 is an odd number, 2n will be an even number, which happened to be the case with the smallest numbers in each of the new triples. Further, the two larger numbers differ by 2 and are one less and one more than the square of half the smallest number. So let’s test this. If the smallest number is 2n, then the square of half of 2n would be n2 and the two larger numbers would be n2 – 1 and n2 + 1. When we test these three numbers we get
We can see that we have found a way of generating a new sequence of Pythagorean triples. Using this method, we can generate (16, 63, 65), (20, 99, 101), etc.
Yet Another Sequence
However, the presence of the triple (20, 21, 29) indicates that this second sequence of triples also does not exhaust all the possible Pythagorean triples. Despite this, I tried my hand at many such patterns, just to entertain myself. The first sequence consisted of triples having the form
The second sequence has the form
The third sequence is formed with the triples having the form
from which we can obtain (20, 21, 29), (28, 45, 53), and (36, 77, 85), etc.
Next Steps in the Journey
This was all recreational mathematics in the service of some higher understanding of how these triples, which have a geometric significance could relate to each other. There are, of course, other sequences of triples than the three we have uncovered. For example, the triples (33, 56, 65), (39, 80, 89) and (65, 72, 97) do not fit into the three sequences we have uncovered. I leave the reader to think of other possible ways of characterizing Pythagorean triples. And perhaps also think about the question, “While it is true that each sequence for generating Pythagorean triples generates infinitely many triples, is the number of such sequences also infinite? And can it be proved either way?”
Our journey into complex numbers required a pit stop at which we will learn some trigonometry. However, the journey is way too exciting, as we have hopefully seen in this post. The simple idea of trigonometry based on the right angled triangle, presented an opportunity to look into Pythagorean triples and some patterns with which an infinite number of triples can be generated. This, of course, is not strictly the study of trigonometry, but rather just my idiosyncratic diversion within this pit stop. Most of us know that trigonometry begins with defining ratios like sine, cosine, and tangent. Such strange names, especially the first two. Ever wonder how we got them? Keep your eyes glued for the next post.
Our journey into the wonders of complex numbers continues. In this series I hope to introduce the reader to the reality of complex numbers. In the previous post, we reached the stage where I introduced the square root of -1 as the unit of the imaginary numbers, parallel to the number 1, which is the unit of the real numbers. We looked at how addition, subtraction, and multiplication works with complex numbers. And I promised to look at division in this post.
Division of Irrational Numbers
Now, those who are familiar with mathematics taught till about grade 9 or 10, will know that the set of real numbers includes rational and irrational numbers. Further, a rational number cannot be irrational and vice versa. For example, numbers like
are rational because they can be expressed as the ratio of two integers. However, numbers like
are irrational because they cannot be expressed as the ratio of two integers. Now, when we add or subtract one rational number and one irrational number, the result will be an irrational number. Hence, numbers like
are irrational, with one part being rational and the other part being irrational. Now, if we were asked to divide one such irrational number by another, we would proceed with the method of rationalizing the denominator. So, if we were asked to evaluate
we would begin by multiplying the numerator and the denominator as follows to get
Note that the denominator has now become rational, which would allow us easier ways of manipulating the expression. The factor 5 + √7 is known as the irrational conjugate of the denominator 5 – √7.
Dividing Complex Numbers
So, the irrational conjugate, when multiplied by the irrational factor results in a rational factor. Could it be, then, that a similar ‘complex conjugate’ could transform a complex number in the denominator into a real number? Before we try it on an actual division, let us see if the product of a complex number and its complex conjugate actually gives us a real number.
If we consider 3 + 4i, its complex conjugate (taking a hint from the irrational conjugate) would be 3 – 4i. When we multiply the two we would get
So, our intuition that multiplying a complex number and its complex conjugate would result in a real number was right. And this is because i2 = -1. Let us use this to divide one complex number by another. Let us divide 4 – 5i by 3 + 4i.
Note that the terms in red contain i2 and hence have a sign reversal, as indicated by the terms in green. Of course, the final result has the form a + bi, where a and b are real numbers. Hence, there is a real part, a, and an imaginary part, bi.
A New Perspective on Multiplication and Division
Now, it is all well and good that we managed to divide one complex number by another. However, whatever does the division mean? We know what division with real numbers means. For example 10 ÷ 5 can be viewed as the number of times 5 can fit into 10, giving the answer 2. We can also view it as a reduction of 10 by a factor of 5, giving 2. This allows us to see division and multiplication as, in essence, the same operation. Hence, division by 5 can be considered multiplication by 1/5. So both division and multiplication can be viewed as scaling a given number. Of course, as we have seen, negation, which can be viewed as a multiplication by -1, is nothing other than an anti-clockwise rotation about zero of 180°. And since division and multiplication are inverse operations, we can consider division by -1, which also is equivalent to negation, as a clockwise rotation about zero of 180°. Of course, a 180° rotation clockwise or anti-clockwise results in the same final position. However, we will see that maintaining a difference in direction of rotation makes sense when we consider complex numbers.
Now, while we have given some meaning, in terms of scaling and rotation, to multiplication and division of real numbers, how can this relate to complex numbers? From the previous post, we saw that we can visualize complex numbers on a plane with the horizontal axis depicting the real part of a complex number, while the vertical axis depicts its imaginary part.
The ‘Size’ of Real Numbers
However, viewing multiplication and division as scaling requires us to have some idea about the ‘size’ of the number. I mean it is only meaningful to say that division of 10 by 5 is a scaling of 10 by a factor of 1/5, resulting in 2, if we have some idea that the number 10 has 5 times the ‘size’ of 2.
Of course, using the number line, we understand that the ‘size’ of a real number, also known as its absolute value, is the ‘distance’ of the point representing that number from the zero on the number line. Hence, the ‘size’ of both 4 and -4, shown below, is 4 units because the points representing 4 and -4 are 4 units away from the zero.
A Complex Cul-de-sac
Now we saw that we can represent complex numbers on a two dimensional plane as below
What ‘measure’ can we use to determine the ‘size’ of these numbers? We could simply add the distances moved in the real and imaginary directions. So the ‘size’ of 3 + 2i would be 3 + 2 = 5. Similarly, the ‘size’ of -4 + 4i would be 4 + 4 = 8. However, this measure is dependent on the orientation of the axes. For example, consider the point 4, depicted below
If we now rotate the axis by 30° clockwise, we will get the following
The distance from the imaginary axis will now be the length of the purple line, which happens to be 3.464 (rounded to 3 decimal places). The distance from the real axis will be the length of the green line, which is 0.5. Hence, as indicated, once we have rotated the axes, the new number becomes 3.464 + 2i, as indicated in the diagram. However, this means that the ‘size’ of the number is now 3.464 + 2 = 5.464. This means that a simple rotation of the axes has changed the ‘size’ of the number. We know that this is absurd. Changing the orientation of an object or our orientation with respect to an object should have no effect on the ‘size’ of the object. Hence, the measure involving simply adding the distances along the real and imaginary axes is a mathematically absurd measure.
Vectors Again to the Rescue
We could take a cue from the study of vectors, remembering that it was vectors that first gave us the intuition of adding another axis to depict the imaginary numbers. Now, the ‘size’ of a vector is known as its magnitude and is calculated using Pythagoras’ theorem. So if a vector depicts a move of 4 units to the left and 3 units down, we would first depict it as below
Please do not confuse i, the imaginary unit that is equal to √-1, and î, the unit vector along the x direction. While mathematicians have felt free to adopt letters from other scripts, such as α, β, θ, etc. from the Greek script, we still predominantly use the English alphabet. And there are only 26 symbols to choose from. So some overlap is to be expected.
Anyway, the magnitude of the above vector (i.e. -4î -3ĵ) is determined using Pythagoras’ theorem as follows
The Modulus of a Complex Number
We now use the same method to define the ‘size’ of a complex number, which is called its modulus. Hence, the modulus of the complex number z = a + bi is defined as
Does this hold up for the earlier experiment with rotating the axes? Recall that the original complex number was 4 + 0i. It is trivial to see that its modulus is 4. After rotating, the number became 3.464 + 2i. We calculate its modulus below
The final number, 3.999911999, might seem strange. However, this is the result of the rounding we had done earlier. Remember that the length of the purple line was 3.464 rounded to 3 decimal places. But how did I know if was 3.464? The measuring instruments I have at my disposal are an assortment of ruler, none of which has a least count smaller than 0.5 mm. So, if I used the rulers, I couldn’t have gotten anything better than 3.45 cm. What gives? Am I pulling the wool over your eyes? Absolutely not! You know I would not do that! So, there must be a way in which I have obtained the rounded value 3.464. Yes, indeed, there is. The actual value is 2√3, which the calculator gives as 3.464101615 and Wolfram Alpha gives as 3.46410161513775458705489268301174473388561050762076125611161395890386603381 to 64 decimal places. You can click on ‘More digits’ to get more digits, if that’s something you like to spend your time doing!
Of course, if the actual length of the purple line is 2√3, then the complex number is 2√3 + 2i, which would yield
Hence, with the exact value of the length, we get the same modulus for the complex number even after rotating the axes. Since, this is what we would expect, this definition of the modulus is what is accepted. But how did we obtain that the length was 2√3? In the first post of this series I had forewarned you that we would be making some pit stops along the way. Well, it’s time for one such pit stop, which I will deal with in the next post. Till then, modulate or remodulate yourself!
We are in the middle of a series on complex numbers. I started this series with Primer to Complex Numbers, in which I shared some basic algebra tools that we would need for our exploration of complex numbers, ending in a brief discussion about the importance of the discriminant. In the second post, A New Kind of Number?, I continued with the exploration by looking at what the discriminant tells us about the kind of numbers we are considering. That discussion included the idea that negation could be considered an anti-clockwise rotation about zero of 180° rather than a reflection in zero. Quite obviously, rotation would mean ‘stepping out’ of the constraints of the number line into a new realm. What would that entail?
Operations with the ‘Normal Numbers’
We begin by considering this new kind of number as something that cannot be combined in normal ways with the numbers we normally encounter. For example, we know that the basic mathematical operations can be performed on all our normal numbers. So, for example, we know that 11 and 12 can be added to give 23; that 13 can be subtracted from 10 to give -3; that 3 can be multiplied by 16 to give 48; and that 50 can be divided by 6 to give a quotient of 8 and a remainder of 2.
We do not have to restrict ourselves to integers, though. We know that the following are all valid
We do not even have to restrict ourselves to rational numbers. In fact, the expressions
can be evaluated to any desired degree of accuracy simply by calculating each irrational number and performing the necessary operations after that. This works even for the transcendental numbers, like e and π. For example, while it may be difficult to evaluate by hand, a computer could give us the value of the expressions below to any desired level of accuracy.
In all the above cases, the operations, while in some cases extremely tedious, we possible to carry out. Furthermore, we had a clear idea about what these operations meant since all the numbers could be located on the number line.
However, if we were now proposing that we escape the confines of the number line, what exactly are we attempting to do?
Assistance from Mechanics and Vectors
When students in school learn mechanics, either in the context of physics or mathematics, their study of motion is first confined to motion in a single dimension. A particle is considered to move along one axis, normally not even named since there is only one dimension of motion. However, since most motion in real life is three dimensional, the single dimension proves to be too unrealistic a constraint. Students are then introduced to motion in two dimensions, the most common case considered being that of a projectile, like a ball, being thrown at some angle of incline.
Students are taught a key concept when studying motion in two dimensions – the vertical and horizontal dimensions are orthogonal to each other. The basic meaning of this is that the vertical and horizontal dimensions are at right angles to each other. However, there is a deeper insight to be gained: No horizontal motion will contribute to vertical motion and vice versa. In other words, what happens horizontally does not affect and is not affected by what happens vertically. This allows the students to study the motion in each direction separately before combining the results to describe the motion as a whole.
This forms the basis of the study of vectors too. We conceive of mutually perpendicular axes so that we can isolate what happens along one axis from what happens along other axes. In both cases, what we are saying is that what happens in one dimension or axis is cannot be combined with what happens in another dimension or axis, but must be kept distinct.
We express this technically by saying that vertical and horizontal motion are independent of each other. Similarly, the x and y directions are independent of each other. And this is all because the directions are at right angles to each other.
The Imaginary Unit
We now use this same insight to think of this new kind of number, which when squared can give a negative result. We recognize that all the numbers we normally encounter do not change when multiplied by 1. In other words
We, therefore, consider the number 1 as the ‘unit’ of the normal numbers. Further, when faced with the square root of a negative number, we recognize that we can do something like the following.
In other words, we recognize that any negative number can be written as the product of its absolute value and -1. Now, we proceed to define the square root of -1 as the ‘unit’ of the new kind of number. By convention, we will denote it as i. In other words,
Linking the Normal and New Numbers
Now, we allow ourselves to perform operations with the new numbers as usual, but ensuring that the normal numbers and the new numbers are never combined to as to lose their distinctiveness. So, for example, we can do the following.
But what would happen if we tried to multiply these numbers? We would get the following.
The above rearrangement is legitimate because multiplication is commutative. However, note that we have the square of i at the end. From the definition of i as the square root of -1, we should have
However, note that -1 is a normal number. In other words, by squaring the new number, we get a normal number. But not just any normal number! Rather, we have obtained the negation of the ‘unit’ of the normal numbers. Recall that we had defined negation as an anti-clockwise rotation about zero of 180°. This would indicate that i, the unit of the new numbers, is obtained by an anti-clockwise rotation about zero of 90°, as a result of which squaring would involve two such rotations for a total of 180° as required by the result that
So what we have obtained in that 1, the unit of the normal numbers, and i, the unit of the new numbers, are oriented at right angles to each other! In other words, 1 and i are independent of each other. Since the normal numbers have their own number line, we define another number line for the new numbers that is at right angles to the original number line. Also, since
we realize that zero can be written as a normal number and as a new number. Hence, the two number lines will intersect at zero. We can depict this as below.
Operations in the Complex Plane
Now, I have to stop using ‘normal numbers’ and ‘new numbers’ simply because these are not the accepted conventions for referring to them. However, with respect the the above diagram, the normal numbers, known as ‘real numbers’, are represented along the horizontal or real axis, while the new numbers, known as ‘imaginary numbers’, are represented along the vertical or imaginary axis. The entire plane, consisting of the two axes, is known as the ‘complex plane’, another unfortunate term.
Numbers that are a combination of real and imaginary numbers can be located on the plane as shown below
We can describe the above numbers as follows:
3+2i : Starting from the zero on both axes, move 3 units to the right and 2 units up.
-4+4i : Starting from the zero on both axes, move 4 units to the left and 4 units up.
-2-3i : Starting from the zero on both axes, move 2 units to the left and 3 units down.
Since the real and imaginary axes are perpendicular to each other, they are independent of each other. Hence, we can add and subtract as follows
This is because moving 3 units to the right and 2 units up followed by 2 units to the left and 3 units down, is, in effect, to move 1 unit to the right and 1 unit down. Also, moving 3 units to the right and 2 units up followed by the reverse of (i.e. negation of) two units to the left and 3 units down, is, in effect, to move 5 units to the right and 5 units up.
When it comes to multiplication, however, we encounter a new phenomenon. It is meaningless to talk about multiplying directions. Hence, it is absurd to say that two upward directions multiply to give a leftward direction. However, when it comes to numbers, we saw that negation can be considered as an anti-clockwise rotation about zero of 180°. Further, we saw that i can be considered an anti-clockwise rotation of 90°, allowing for the square of i to be equivalent to negation. This means that it is legitimate to multiply complex numbers and we can do this using FOIL (see Primer to Complex Numbers) as below.
However, since the square of i is -1, we can simplify the above as
The Fork Ahead
So far we have seen that it is possible to conceive of imaginary numbers and real numbers as two quite different species of numbers that are independent of each other. Due to this, they can be added and subtracted in the conventional way with the only restriction being that we keep the real and imaginary parts separate. However, when we conceive of the real and imaginary numbers as independent, we face a corollary that the square of i is equivalent to negation. This gives us the justification for multiplying complex numbers using conventional algebraic rules.
This also explains the title of this post. No, I did not make a mistake. Yes, I have combined two ideas – rocket science and brain surgery – that normally do not go together, much like the real and imaginary numbers. Not everything is smooth, as we will see. However, with the single insight that squaring the imaginary unit is equivalent to negation, we can actually make the two otherwise disparate elements work together.
Of course, there is much more to complex numbers. We still have to give further meaning to multiplication. We also have to see how division would work, not to mention exponentiation. And that is just the start. There is so much that the imaginary numbers open up for us and I hope you are as excited as I am for the journey ahead. In the next post, we will consider how division works with complex numbers. Till then, I won’t say, “Stay real” but “Stay true.”
Last week I began a series on complex numbers with the post Primer to Complex Numbers. At the end of that post, I considered the discriminant in the formula for solving a quadratic equation. We saw that it is easy to comprehend what a zero or positive discriminant would mean simply because the square of zero is zero itself and the square of a positive or a negative number will always give a positive result. However, what do we do with quadratic equations in which the discriminant is negative?
Visualizing Quadratics vis à vis the Discriminant
Here, it would help for us to visualize the functions corresponding to the quadratic equations. In other words, given the equation
we can consider this to be the equation f(x) = 0, where
This is nothing less than finding the points where the graph of y = f(x) cuts the x axis. So, let us consider three different quadratic functions
If we plot the graphs of these functions we will get
We can factorize f(x) and g(x) giving us
Here we can see that the factors x – 1, x – 5, and x – 2 correspond to the solutions of the corresponding equations f(x) = 0 and g(x) = 0. These also are the x-coordinates of the points where the graphs of y = f(x) and y = g(x) cut the x-axis. However, we can see that we cannot factorize h(x). Nor does the graph of y = h(x) cut the x-axis. If we calculated the discriminants of the three quadratics we would obtain
So, when the discriminant is negative, we can see that the graph of the corresponding quadratic function does not cut the x-axis. This is precisely what we mean when we say that the quadratic equation does not have real roots or real solutions.
However, if the quadratic equation does not have real roots, what kind of roots does it have? In other words, what kind of number can we propose which when squared gives us a negative number?
Reimagining Negation
Let us take step back and ask ourselves how we can visualize negating a number. What I was taught, and I think what most students today are taught, is that negating a number involves taking its reflection in the zero point. Hence, the negation of 4 is -4 and the negation of -3 is 3. We can visualize both of these as follows.
However, we can also visualize this as an anticlockwise rotation of 180° as below.
I will later drop the degree unit for measuring angles and replace it with the radian. I will also explain why I am considering an anti-clockwise rotation when we make this change. But for now, it should be clear that an anti-clockwise rotation about zero of 180° on the number line produces the same effect as negating a number.
We can now consider the positive sign to represent no rotation and the negative sign to represent a rotation of 180°. Then the product of two positive numbers will involve no rotation, that is, a positive sign. Similarly, the product of a positive and a negative number will involve no rotation and a rotation of 180°, that is, a negative sign. And the product of two negative numbers will involve two rotations of 180°. This will be a rotation of 360°, which is a complete revolution, or the equivalent of no rotation, yielding, once again, a positive sign.
So what if we considered the square root of a negative number to involve a rotation of 90°? In that case, the product of two such numbers would be two rotations of 90°, which equals an effective rotation of 180°, that is, a negative sign.
The Next Step
What we can see is that, if we are willing to ‘step out’ of the constraints of the number line by introducing the idea of rotations, then we can conceive of a way in which a ‘number’ which when squared gives a negative number. Of course, since we have ‘stepped out’ of the number line to understand these ‘numbers’, it is clear that these ‘numbers’ do not belong on the number line. And since the number line is intended to represent all ‘real numbers’ we are forced to conclude that these new ‘numbers’ are not ‘real numbers’. It is an unfortunate fact of mathematical history that these numbers were given the name ‘imaginary’.
In fact, it was the philosopher and mathematician, René Descartes, who rubbished the idea that there could be numbers whose squares were negative and who then jokingly called them ‘imagined’ numbers, that is, no more than a figment of our imagination. The name stuck. However, while it is great to have some historical humor in the nomenclature, there are many mathematics teachers today who actually think these numbers are entirely made up. As we will see in future posts, this is not the case. So, in the next post, we will look at understanding some arithmetic and algebra related to these numbers. Till then, let your imagination soar!
Acutely Obtuse 
