Equation of the Day #3: Triangular Numbers

I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing!

The triangular numbers are the numbers of objects one can use to form a triangle.

Anyone up for billiards? Or bowling? (Image: Wikimedia Commons)

Pretty straightforward, right? To get the number, we just add up the total number of things, which is equal to adding up the number of objects in each row. For a triangle with n rows, this is equivalent to

\displaystyle T_n = 1+2+3+ \ldots + n = \sum_{k=0}^n k

This means that the triangular numbers are just sums from 1 to some number n . This gives us a good definition, but is rather impractical for a quick calculation. How do we get a nice, shorthand formula? Well, let’s first add sequential triangular numbers together. If we add the first two triangular numbers together, we get 1 + 3 = 4 . The next two triangular numbers are 3 + 6 = 9. The next pair is 6 + 10 = 16 . Do you see the pattern? These sums are all square numbers. We can see this visually using our triangles of objects.

(Image: Wikimedia Commons)

You can do this for any two sequential triangular numbers. This gives us the formula

T_n + T_{n-1} = n^2

We also know that two sequential triangular numbers differ by a new row, or n . Using this information, we get that

\begin{aligned} n^2 &= T_n + (T_n - n) \\ 2 T_n & = n^2 + n = n(n+1) \\ T_n &= \frac{n(n+1)}{2} = \begin{pmatrix} n+1 \\ 2 \end{pmatrix} \end{aligned}

Now we finally have an equation to quickly calculate any triangular number. The far right of the final line is known as a binomial coefficient, read “n plus one choose two.” It is defined as the number of ways to pick two objects out of a group of n + 1 objects.

For example, what is the 100^{\rm th} triangular number? Well, we just plug in n = 100 .

T_{100} = \dfrac{(100)(101)}{2} = \dfrac{10100}{2} = 5050

We just summed up all the numbers from 1 to 100 without breaking a sweat. You may be thinking, “Well, that’s cool and all, but are there any applications of this?” Well, yes, there are. The triangular numbers give us a way of figuring out how many elements are in each row of the periodic table. Each row is determined by what is called the principal quantum number, which is called n . This number can be any integer from 1 to \infty . The energy corresponding to n has n angular momentum values (l = 0, 1,\ldots, n-1) which the electron can possess, each of which has a total of 2l + 1 orbitals for an electron to inhabit, and two electrons can inhabit a given orbital. Summing up all the places an electron can be in for a given n involves summing up all these possible orbitals, which takes on the form of a triangular number.

\begin{aligned} \sum_{k=0}^{n-1}(2l +1) &= 2 \sum_{k=0}^{n-1} k + n \\ &= 2T_{n-1} + n \\ &= n(n-1) + n \\ &= n^2 \end{aligned}

The end result of this calculation is that there are n^2 orbitals for a given n , and two electrons can occupy each orbital; this leads to each row of the periodic table having 2\lceil (n+1)/2 \rceil^2 elements in the n^{\rm th} row, where \lceil x \rceil is the ceiling function. (This complication is due to the Aufbau principle, which dictates how energy levels fill up.) They also crop up in quantum mechanics again in the quantization of angular momentum for a spherically symmetric potential (a potential that is determined only by the distance between two objects). The total angular momentum for such a particle is given by

L^2 = \hbar^2 l(l+1) = 2 \hbar^2 T_l

What I find fascinating is that this connection is almost never mentioned in physics courses on quantum mechanics, and I find that kind of sad. The mathematical significance of the triangular numbers in quantum mechanics is, at the very least, cute, and I wish it would just be mentioned in passing for those of us who enjoy these little hidden mathematical gems.

There are more cool properties of triangular numbers, which I encourage you to read about, and other so-called “figurate numbers,” like hexagonal numbers, tetrahedral numbers, pyramidal numbers, and so on, which have really cool properties as well.


Equation of the Day #2: Cardioid

I made the above figure in Inkscape. A cardioid is the envelope formed by a set of circles whose centers lie on a circle and which pass through one common point in space. This image shows the circle on which the centers of the circles in the above image lie. A cardioid is also the path traced by a point on a circle which is rolling along the surface of another circle when both circles have the same radius (here is a cool animation of that).

What is the cardioid’s significance? Well, it looks like a heart, which is kind of cool. It’s also the (2D) pickup pattern of certain microphones (I have a USB cardioid microphone). If a sound is produced at a given point in space, the pickup pattern shows an equal intensity curve. So, if I place a microphone at the intersection point of all those circles, the outside boundary is where a speaker producing, say, a 440 Hz tone would have to be to be heard at a given intensity. So, the best place to put it would be on the side where the curve is most round (the bottom in this picture) without being too far away from the microphone.

Another interesting fact about the cardioid is that it is the reflection of a parabola through the unit circle (r = 1) . Here’s what I mean; in polar coordinates, the equation of the above cardioid is given by

r = 2a(1-\sin\theta)

where a is a scaling factor, and theta is the angle relative to the positive x-axis. The origin is at the intersection of the circles. The equation of a parabola opening upwards and whose focus is at the origin in polar coordinates is just

r = [2a(1-\sin\theta)]^{-1}

which is an inversion of the cardioid equation through r = 1 , or the unit circle.


Equation of the Day #1: Complex Numbers

Most of us are used to the real numbers. Real numbers consist of the whole numbers (0, 1, 2, 3, 4, \ldots) , the negative numbers (-1, -2, -3, \ldots), the rational numbers (1/2, 2/3, 3/4, 44/7, \ldots) , and the irrational numbers (numbers that cannot be represented by fractions of integers, such as the golden ratio, \sqrt{2} , or \tau ). All of these can be written in decimal format, even though they may have infinite decimal places. But, when we use this number system, there are some numbers we can’t write. For instance, what is the square root of -1 ? In math class, you may have been told that you can’t take the square root of a negative number. That’s only half true, as you can’t take the square root of a negative number and write it as a real number. This is because the square root is not part of the set of real numbers.

This is where the complex numbers come in. Suppose I define a new number, let’s call it i , where

i^2 = -1.

We’ve now “invented” a value for the square root of -1 . Now, what are its properties? If I take i^3 , I get -i , since i^3 = i\cdot i^2 . If I take i^4 , then I get i^2\cdot i^2 = +1 . If I multiply this by i again, I get i . So the powers of i are cyclic through i, -1, -i, and 1 .

This is interesting, but what is the magnitude of i , i.e. how far is i from zero? Well, the way we take the absolute value in the real number system is by squaring the number and taking the positive square root. This won’t work for i , though, because we just get back i . Let’s redefine the absolute value by taking what’s called the complex conjugate of i and multiplying the two together, then taking the positive square root. The complex conjugate of i is obtained by negating the imaginary part of i . Since i is purely imaginary (there are no real numbers that make up i ), the complex conjugate is -i . Multiply them together, and you get that -i\cdot i = -1\cdot i^2 = 1 , and the positive square root of 1 is simply 1 . Therefore, the number i has a magnitude of 1 . It is for this reason that i is known as the imaginary unit!

Now that we have defined this new unit, i , we can now create a new set of numbers called the complex numbers, which take the form z = a + bi , where a and b are real numbers. We can now take the square root of any real number, e.g. the square root of -4 can be written as \pm 2i , and we can make complex numbers with real and imaginary parts, like 3 + 4i .

How do we plot complex numbers? Well, complex numbers have a real part and an imaginary part, so the best way to do this is to create a graph where the abscissa (x-value) is the real part of the number and the ordinate (y-axis) is the imaginary part. This is known as the complex plane. For instance, 3 + 4i would have its coordinate be (3,4) in this coordinate system.

What is the magnitude of this complex number? Well, it would be the square root of itself multiplied by its complex conjugate, or the square root of (3 + 4i)(3 - 4i) = 9 + 12i - 12i +16 = 25 . The positive square root of 25 is 5 , so the magnitude of 3 + 4i is 5 .

We can think of points on the complex plane being represented by a vector which points from the origin to the point in question. The magnitude of this vector is given by the absolute value of the point, which we can denote as r . The x-value of this vector is given by the magnitude multiplied by the cosine of the angle made by the vector with the positive part of the real axis. This angle we can denote as \phi . The y-value of the vector is going to be the imaginary unit, i , multiplied by the magnitude of the vector times the sine of the angle \phi . So, we get that our complex number, z , can be written as z = r(\cos\phi + i\sin\phi) . The Swiss mathematician Leonhard Euler discovered a special identity relating to this equation, known now as Euler’s Formula, that reads as follows:

e^{i\phi} = \cos \phi + i\sin\phi

Where e is the base of the natural logarithm. So, we can then write our complex number as z = re^{i\phi} . What is the significance of this? Well, for one, you can derive one of the most beautiful equations in mathematics, known as Euler’s Identity:

e^{i\tau} = 1 + 0

This equation contains the most important constants in mathematics: e , Euler’s number, the base of the natural logarithm; i , the imaginary unit which I’ve spent this whole time blabbing about; \tau , the irrational ratio of a circle’s circumference to its radius, which appears all over the place in trigonometry; 1 , the real unit and multiplicative identity; and 0 , the additive identity.

So, what bearing does this have in real life? A lot. Imaginary and complex numbers are used in solving many differential equations that model real physical situations, such as waves propagating through a medium, wave functions in quantum mechanics, electromagnetic phenomena, and fractals, which in and of themselves have a wide range of real life application.