"I Am the Very Model of a Modern Major-General", Part I

A Math Commentary

My school has an “Arts Cycle”. Each quarter there’s a theme for music, poetry, painting, hymns, scripture, folks songs, and “memory work”; every three years, we cycle through it again. This quarter it’s Opera, Lewis Carroll, Impressionists, “My Hope Is Built on Nothing Less”, Isaiah 53, Yankee Doodle, and the Nicene Creed. Part of Opera is Mozart’s Queen of the Night Aria (Konigin der Nacht, as performed by Erika Miklosa and Florence Jenkins), but another part is Gilbert and Sullivan, which is quite relevant to math for the 7th graders. Some of them are memorizing the Major-General's Song from Pirates of Penzance, one stanza of which is

I am the very model of a modern Major-General,

I've information vegetable, animal, and mineral,

I know the kings of England, and I quote the fights historical

From Marathon to Waterloo, in order categorical;

I'm very well acquainted, too, with matters mathematical,

I understand equations, both the simple and quadratical,

About binomial theorems I'm teeming with a lotta news,

With many cheerful facts about the square of the hypotenuse.¹

So what does that mean? If you want to be a major-general in the U.S. Army, you need to know these things. Keep reading. For it is, it is, a glorious thing, to be a major-general.²

We covered the hypotenuse last month, and are doing algebra this month, so I used the song as a peg from which to hang some review and some new material.

The equation “-3 + x = 4” is simple, while “x² = 9” is quadratical. The equation “(x+y)² = 8” is a binomial quadratic equation because it has the two variables x and y (“bi-” as in “bicycle”). We have come to the Distributive Property in class, and it (or common sense) says that we can expand that binomial equation thus:

\(\begin{eqnarray} (x+y)^2& = & (x+y)\cdot (x+y) \\ && \nonumber\\ & = & x (x+y) + y (x+y) \\ && \nonumber\\ & = & x x+xy + yx+yy \\ &&\nonumber\\ & = & x^2+2xy +y^2\\ \end{eqnarray}\)

Figure 1 shows it in a picture. The square is (x+y) on each side, and we can divide it into four pieces, of area x², y² and two pieces of size xy each.

Figure 1: The quadratic binomial, geometrically

There are three different ways to do the quadratic binomial. We could put in a squared minus sign instead:

\(\begin{eqnarray} (x-y)^2& = & (x-y)\cdot (x-y) \\ && \nonumber\\ & = & x (x-y) - y (x-y) \\ && \nonumber\\ & = & x x-xy - yx+yy \\ &&\nonumber\\ & = & x^2 -2xy +y^2\\ \end{eqnarray}\)

Or, we could mix plus and minus signs:³

\(\begin{eqnarray} (x+y) \cdot (x-y)& = & x (x-y) + y (x-y) \\ && \nonumber\\ & = & x x -xy + yx -yy \\ &&\nonumber\\ & = & x^2 -y^2\\ \end{eqnarray}\)

Similarly, we could expand a trinomial equation (x+y+3)². Or, we could expand a cubic (power 3) or higher binomial equation. I won't show you the steps for how to do it, just one result for power 5 to show you how it gets more complicated. Note how everyterm ends up with a total of five things times each other (x⁵ is xxxxx, x³ y² is xxxyy), which is no accident.

\(\begin{eqnarray} (x+y)^5& = & (x+y) (x+y) (x+y) (x+y) (x+y) \\ &&\nonumber\\ & = & x^5 + 5x^4y + 10x^3y^2 + 10 x^2y^3 + 5 x y^4 + y^5 \\ \end{eqnarray}\)

There isn’t a good way to show the expansion to the power 5 in a picture, but Figure 2 has nice pictures from Wikipedia for n = 1, 2, 3, and 4 (Figure 1 was for n = 2).

Figure 2: A Geometric View of the Binomial Theorem for n = 1,2,3,4

The Binomial Theorem gives a formula for expanding (x+y)ⁿ for any value of n you might choose— a real mess, if n = 11, for example. The Theorem is a complicated formula in combinatorics (Pascal’s Triangle is part of it, but just a part of it) which is usually taught in high school rather than middle school, so I won’t try to explain it here.⁴

But I’ve exceeded Substack’s “email length limit”, so let’s stop here for now, and I’ll continue later with Part II. It’s Easter weekend, so maybe, gentle reader, you are looking for something to read and I should strike while you are desperate enough to pick up something with equations in it. Also, Substack permits any length, but I’ll take it as a warning that maybe Part I is enough for one sitting. Substack lacks something real magazines have: editors. One raison d’etre d’editors is to tell the writer when to stop and let his reader take a break. So I’m using the crude AI editor of a length warning. As Luther said when Cordatus inquired of him, “Reverend Father, tell me in a brief way how to preach.”

First, you must learn to go up to the pulpit.

Second, you must know that you should stay there for a time.

Third, you must learn to get down again.

---

If you liked this Substack, you’d like "I Am the Very Model of a Modern Major-General," Part I” and “The Two-Envelope Paradox: Stated but Not Resolved” and “Why 0.9999 . . . Equals 1.0000 . . .”.

---

1. Other vocations have similarly long lists of requirements. An entire PhD sequence could be constructed around Josh Tyra’s 2014 “I Am the Very Model of a Biblical Philologist,” which is in my opinion superior to the original Major-General.

2. The song to which I allude, a reprise of the Pirate King’s Song, is meant ironically. Alas, to be a true pirate king nowadays, Cornwall real estate is far too expensive and you have to go live in Somalia, whereas most major-generals live in northern Virginia. (A major-general needn’t live in the District, which would be like living in Somalia, but with better museums.) Glory isn’t everything.

3. I really should include pictures of squares and rectangles for (x-y) (x-y) and (x+y)(x-y), but I haven’t done that yet. I’ll revise this essay eventually. Till then “it is left to the reader as an exercise,” irritating tho that phrase may be.

4. Of course, forbidden fruit is sweetest, so here is the statement of the Binomial Theorem:

where

\({\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}, \)

which is equivalent, when you recall that x! is x-factorial, which is

x! = x(x-1)(x-2) …  (3)(2)(1)

to

\(\begin{eqnarray} \binom nk& =& \frac{n!}{(n-k)! k!}\\ & & \\ & =& \frac{n!}{(n-k)!} \cdot \frac{1}{ k!}\\ & & \\ & = & \frac{n\times(n-1)\times\cdots\times1}{(n-k)\times (n-k-1) \cdots\times1} \cdot \frac{1}{k\times(k-1)\times\cdots\times1}\\ & & \\ & = & \left(n\times(n-1)\times\cdots\times(n-k+1) \right) \cdot \frac{(n-k)\times (n-k-1) \cdots\times1}{(n-k)\times (n-k-1) \cdots\times1} \cdot \frac{1}{k\times(k-1)\times\cdots\times1}\\ & & \\ & = & n\times(n-1)\times\cdots\times(n-k+1) \cdot \frac{1}{k\times(k-1)\times\cdots\times1}\\ & & \\ & = & \frac{n\times(n-1)\times\cdots\times(n-k+1)}{k\times(k-1)\times\cdots\times1}\\ \end{eqnarray}\)

So often it turns out that forbidden fruit isn’t all that sweet…