Betsy's Devine Algebraic Limerick

Or is it Betsy Devine's Algebraic Limerick? Stu put up a post about this, and it's cool that Betsy was able to put an equation into a limerick, but is the equation valid? I checked it out with a Google spreadsheet and it is. But with a little background knowledge it is possible to verify it without having to resort to actual computations.

Okay, maybe more than a little background knowledge. You need to understand the basics of algebra, calculus, trigonometry and logarithms. From algebra, you need to understand the concept of a function. From calculus you need to understand the concept of the derivative and the integral.
Basically the derivative of a function is the slope, which tells you how steep the line is at any given point. It is rise over run. The integral is the area under the curve. The derivative and the integral are like inverse functions. The derivative of the integral of a function is the original function, and likewise the integral of the derivative is also the original function.

A function that produces a graph that is a straight line is going to have a constant slope measured with a single number. The slope of a function that produces a curved line is going to be constantly changing. If the rate of change is constant, then the derivative will be a straight line. A popular example is the relation between acceleration, speed and distance.

Chart of Acceleration, Speed & Distance

If you are accelerating at a constant rate, your speed will be constantly increasing, and the distance you have covered will be getting larger by constantly larger amounts. In the above chart, the acceleration is constant: it is always 1. Let's call it one 1 foot per second per second. Speed starts at zero and increases at this rate. After 3 seconds, we are going 3 feet per second. Distance likewise starts at zero, and it builds slowly, which is what we expect as we are not going very fast. But it keeps going up, and as we go faster, we start covering more distance.

After 3 seconds we have covered 4.5 feet. You will notice that the area under the red line is a triangle, and the measure of that area is 3 x 3 / 2, or 4.5. The integral is the area under the curve, the area under the speed line is 4.5, so the value of the integral (the distance) at this point is also 4.5. So distance is the integral of speed, and speed is the derivative of distance. Likewise notice that speed is a straight line. It is at an angle, and that angle never changes. The slope of the speed line is a constant 1, so the derivative of speed is the acceleration.

For more fun with derivatives and integrals this page has an interactive graph you can play with.

From trigonometry you need the concept of sine, cosine and the unit circle. This page has a good explanation.

Logarithms are a trick that was invented a long time ago to let people multiply two numbers by adding. This is how slide rules worked, if you have ever seen a slide rule. The answer generally wasn't as precise as you would get from a calculator, but it was close enough for the real world. I mean, that's how we got to the moon.

So if you've got all that, we can look at Betsy's Limerick.

The left side of the equation is composed of two parts:

1. The integral of Z squared,
2. The cosine of a constant.

First Expression

We know it's a separate expression because of the "dz", which is just what math geeks tack on the end of integrals. Derivatives are the inverse of integrals, so Z squared is the derivative of the integral of Z squared. Taking the derivative of a simple expression like Z squared is relatively easy: you multiply the expression by the exponent and reduce the exponent by one. For example the derivative of Z squared is 2 times Z. Working backwards, the integral of Z squared must have Z cubed. But taking the derivative of Z cubed would give you 3 times Z squared. To get rid of the factor 3, we will need to divide by 3, so the integral of Z squared becomes one third times Z cubed:

(1/3) * Z^3

Now we evaluate that expression for the beginning value (1) and the ending value (the cube root of 3) and find the difference.

When Z has a value of one, the value of the integral is (1/3) times one cubed, or (1/3) times one, which is simply 1/3.

When Z has a value of the cube root of three, the value of the Z cubed is 3 (the cube of a cube root of a number is simply the number), and the value of the integral is 1/3 of that ((1/3) x 3), or 1.

To find the difference we simply subtract the first value from the second value and we have 2/3.

Second Expression

Looking inside the parentheses, 3/9 reduces to 1/3 (3/9 = 1/3), so we get PI/3. Since they are using PI, it is safe to assume they are using radians instead of degrees. The circumference of a circle is 2 times PI times the radius. For these kind of problems we assume the radius is one, so the circumference is simply 2 * PI, so PI is one half of a circle or 180 degrees. 1/3 of 180 degrees is 60 degrees. If we draw an equilateral triangle with sides all of length 1 (the same as the radius of our circle), with the base at the bottom, coinciding with the X axis, and the left end point coinciding with the origin, then the X coordinate of the apex will be 1/2, and since the Cosine of an angle is the same as the X coordinate, the value of this expression is 1/2. This page has an interactive unit circle that illustrates all this.

Left Side

The value of the first expression (the integral) is 2/3, the value of the second expression (Cosine) is 1/2. Multiplied (another math geek convention) together we get 1/3.

Right Side

The Natural Log (ln) of a number is the value that when used as an exponent to the special number e, results in the original number. Well, we already have e to a power here: the power is 1/3. The N-th root of a number can also be expressed as that number being raised to the power of one over N, or 1/N. Here we are taking the cube root of e, which is the same as raising e to the power of 1/3, so 1/3 is the value of this expression.

Summary

Since both sides of the equation evaluate to 1/3, the equation, and the limerick are true. I wrote this for two reasons. One, I wanted to see if I could evaluate the equation, and two, I wanted to see if I could explain it.

Update October 2016 replaced missing image. Limerick text available here. Harbor News link to story about 'how we got to the moon' is dead.

Update December 2017 added link to spreadsheet used to generate chart.