Mathematics – The Melancholy Roman

24 February 201921 April 2019

The Mrs Warren quotation

Felicity Kendal, in a still from the 2010 production of Mrs Warren’s Profession

There’s a fairly famous quote by George Bernard Shaw that I came across recently. It is:

The people who get on in this world are the people who get up and look for the circumstances they want, and if they cannot find them, make them.

In the recent occurrence, the person quoting it was obviously approving and gave the source as “Mrs Warren.”

The quote is actually from Shaw’s 1893 play Mrs Warren’s Profession and is said by the character of Vivie, not Mrs Warren herself. Most of the rest of the scene following the quote is a repudiation of the sentiment expressed.

Let’s unpack it a little more:

The quote is famous and I first heard it during a speech at my high school graduation. I assume the speaker had gone through a book of quotations and picked it as fitting the message he wanted to convey. These days you would search for quotations online.

Sometimes a slightly longer version is given:

People are always blaming their circumstances for what they are. I don’t believe in circumstances. The people who get on in this world are the people who get up and look for the circumstances they want, and if they cannot find them, make them.

It was years after my high school graduation that I was in a theatre in London’s West End, watching a production of the play, and was astonished to hear the quote word for word. I hadn’t realised till then that it was by Shaw, hadn’t realised that it was from a play, and definitely hadn’t realised that it was expressing a character’s view and wasn’t necessarily Shaw’s opinion at all.

As it happens, what the character Vivie says in full is:

Everybody has some choice, mother. The poorest girl alive may not be able to choose between being Queen of England or Principal of Newnham; but she can choose between ragpicking and flowerselling, according to her taste. People are always blaming their circumstances for what they are. I don’t believe in circumstances. The people who get on in this world are the people who get up and look for the circumstances they want, and, if they can’t find them, make them.

If you read or better still, watch the play, by this part of the scene you will have already learnt details about the character. She is a young woman and has grown up wealthy. She is also highly intelligent, having achieved equivalent grades to the third wrangler at Cambridge. That’s an obscure reference these days but it refers to the third highest scorer in the undergraduate maths degree at Cambridge. More than 125 years after the play was written it remains a benchmark for formidable intellectual achievement. Philippa Fawcett had achieved a higher mark than the first wrangler in 1890 and this is presumably where Shaw got the idea for the reference.

But like I said, the rest of the scene argues against what the character has just stated. The title of the play is “Mrs Warren’s Profession” and the reason for this title is the hidden scandal of how Mrs Warren makes a living: she is a former prostitute and now a brothel owner. The wealthy and middle-aged Mrs Warren was only able to climb out of poverty, to find or make the circumstances she wanted, by turning to this disreputable trade.

It’s not exactly the stuff of stirring quotations.

Of course, Shaw also explored wealth and class in his more famous play Pygmalion, which was adapted into the even more famous and popular musical My Fair Lady, the musical rather missing the point of the play. But in both plays, Shaw comes from a recognisably leftist position that the world is unfair to the poor and disadvantaged. In other words, his own opinion is presumably the opposite of the famous quotation.

One can easily reject Shaw’s views and to a certain extent one absolutely should. He was an anti-Semite and held other unpleasant beliefs. You can also argue quite convincingly that opportunities for women have improved somewhat since 1893. Nonetheless, given the context of the play, it really isn’t the best inspirational quotation.

The Melancholy Roman

7 March 201810 November 2018

The Binomial Theorem

This picture has absolutely nothing to do with this blog post.

Or: practicing using LaTeX in WordPress.

We’ll start with factorial notation. In mathematics, $n!$ reads “ $n$ factorial” and signifies $n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1$ . For example,

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.$

Now, imagine that we have $n$ objects and we want to select $k$ of them (assuming that $k \leq n$ ). How many different ways could we do this?

Assuming we can’t count the same object twice, we have $n$ choices for the first object, $n-1$ choices for the second object, $n-2$ choices for the third object and so on. In fact we have

$\underbrace{ n \times (n-1) \times \ldots \times (n-k+1)}_{k\ \text{terms}} \ \ \text{choices}.$

We can put this much more simply as

$\frac{n!}{(n-k)!}.$

If you can’t quite see this, consider that

$n! = n \times (n-1) \times \ldots \times (n-k+1) \times (n-k)!$

and then look at the expansion below and see how the terms cancel out:

$\frac{n!}{(n-k)!} = \frac{ n \times (n-1) \times \ldots \times (n-k+1) \times \text{\sout{$(n-k)!$}} }{ \text{\sout{$(n-k)!$}} }.$

So, to be clear, if we have $n$ objects and we are choosing any $k$ of them,

$\frac{n!}{(n-k)!}$

gives the total number of possible permutations.

But what if we don’t care about permutations? What if, selecting $k$ different objects, we don’t care what order they’re in? The above formula will give many duplications. For example, if we’re picking three different letters from $A$ , $B$ , $C$ , $D$ , $E$ , $F$ and we choose $A$ , $B$ and $E$ , there are six different ways of doing this: $ABE$ , $AEB$ , $EAB$ , $EBA$ , $BEA$ , $BAE$ . But they’re all equivalent, just different permutations of the same three letters.

How many such permutations are there? If you’re arranging $k$ different objects, there are $k$ choices for the first item, $k-1$ choices for the second item, $k-2$ for the third and so on. In fact there are $k \times (k-1) \times (k-2) \times \ldots \times 1 = k!$ possible permutations.

Now, since

$\frac{n!}{(n-k)!}$

counts every possible permutation, and since $k$ objects must have $k!$ possible permutations, we simply divide by $k!$ to give the number of ways of choosing $k$ objects from $n$ if we don’t care about the order. This gives us

$\frac{n!}{k!(n-k)!}.$

Next we introduce some notation. The symbol

$\binom{n}{k}$

is pronounced “ $n$ choose $k$ ” and means that given $n$ objects, we want to choose $k$ of them. This notation is used when we don’t care about permutations and as you may have guessed, we define

$\binom{n}{k} = \frac{n!}{k!(n-k)!}.$

To demonstrate how useful this is we’ll give an example. Imagine we have $5$ coins on a table and we want to pick up any $2$ of them. How many ways can we do this?

A bit of mental visualisation suggests that the answer is $10$ but we can apply the formula:

$\binom{5}{2} = \frac{ 5! }{ 2! (5-2)! } = \frac{ 5 \times 4 \times 3 \times 2 \times 1 } { (2 \times 1 ) \times (3 \times 2 \times 1) } = 10.$

Clearly more complicated examples can be beyond the power of mental visualisation and then the formula comes into its own.

So much for the introduction. We can now get to the nitty gritty. Let $a, b \in {\mathbb{C}}$ . Then

$(a+b)^0 = 0 \ \ \text{ (of course*)}$

and

$(a+b)^1 = a+b,$

and

$\begin{align*} (a+b)^2 &= (a+b)(a+b)\\ &= a^2 + 2ab + b^2, \end{align*}$

and indeed

$\begin{align*} (a+b)^3 &= (a+b)(a+b)(a+b)\\ &= a^3 + 3a^2b + 3ab^2 + b^3. \end{align*}$

These calculations are fairly easy to do by hand but as $n$ gets bigger the algebra for calculating $(a+b)^n$ rapidly becomes tedious. We know that the terms will be $a^n$ , $a^{n-1}b$ , $a^{n-2}b^2$ and so on, but what are the coefficients?

Every term in the multiplied out expansion is the product of an $a$ or a $b$ in each $(a+b)$ term. And this means we can easily work them out.

For instance, let us take the $a^3b^7$ term in the expansion of $(a+b)^{10}$ . We choose the $a$ term from three lots of $(a+b)$ and the $b$ term from seven lots of $(a+b)$ . So how many times would $a^3b^7$ occur if the entire product was multiplied out? The answer is, as many times as there are ways of choosing $3$ of the bracketed terms to give us the $a$ or $7$ of them to give us the $b$ . That is, $\binom{10}{3} = \binom{10}{7} = 120$ .

Generalising, if we multiply out $(a+b)^n$ , each $a^{n-k}b^k$ term occurs $\binom{n}{n-k}$ times. Since $\binom{n}{n-k} = \binom{n}{k}$ and the latter is simpler to write, we say that the coefficient for $a^{n-k}b^k$ is $\binom{n}{k}$ .

And now we can state the compact version of the Binomial Theorem. By convention, we define $0! =1$ . For $a, b \in {\mathbb{C}}$ and $n \in {\mathbb{N}}$ , the Binomial Theorem says that

$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k.$

As ever, we can write the expanded version, which begins

$(a+b)^n = \binom{n}{0}a^nb^0 + \binom{n}{1}a^{n-1}b^1 + \ldots$

Postscript*

As generations of mathematics students have reasoned, if (for example)

$1^0 = 5^0 = 496^0 = x^0 = p^0 = 1,$

that is, if anything to the power of zero is one; while (for example)

$0^1 = 0^{5} = 0^{496} = 0^x = 0^p = 0,$

that is, zero to the power of anything is zero; then what is the solution to

$0^0?$

The answer is that we generally say it is undefined. However, in the context of the Binomial Theorem, by convention we define $0^0 = 1$ in order to make the equation work.

Second Postscript

The caption under the image at the top of this blog post says it has absolutely nothing to do with the rest of the post. That is correct. It is actually detail of a miniature painting called Carretó fantasma by Salvador Dalí, and is displayed in the Dalí Museum in Figueres, Spain.

The Melancholy Roman

24 August 201725 April 2019

The Taj Mahal

The Taj Mahal is the most beautiful building ever constructed. This is my opinion and you are welcome to disagree. But while it’s true to say that I’ve never physically set eyes on some other famous buildings like St Basil’s Cathedral in Moscow or Fallingwater in Pennsylvania, I nonetheless think my opinion is pretty easy to defend.

The story of the Taj is relatively well known. It was built on the orders of the Mughal emperor Shah Jahan in honour of his wife Mumtaz Mahal, who died in childbirth. Her tomb is at the centre of the monument. Shah Jahan’s is there as well but it’s off to one side.

The very image of the Taj is iconic but that image made me curious. We all know what it looks like but I wondered, what do you see if you turn your back to it and look the other way?

I satisfied my curiosity when I visited India several years ago. Below is the picture I took:

The view in the opposite direction: The great gate, or darwaza-i rauza.

It’s worth noting that in addition to the great gate there are a further two buildings in the complex that aren’t the Taj Mahal, that are nonetheless impressive in their own right. One is a small mosque to one side of the mausoleum. The other is a building quite similar to the mosque but on the other side of the mausoleum, built purely to maintain symmetry.

It’s well known that Shah Jahan was deposed by his son Aurangzeb. There is a story that Shah Jahan planned to build a second Taj Mahal in contrasting black marble on the other side of the Yamuna river. His overthrow by his son stopped him doing this.

I don’t believe the story. It’s an attractive myth in that it neatly explains why the emperor’s tomb is off-centre in contrast with the otherwise perfect symmetry of the central mausoleum. It also offers the tantalising suggestion that if only Shah Jahan had stayed in power, the Taj Mahal would be even more spectacular than it is.

But the reality? The offsetting of the second tomb was the normal custom.

Moreover, a black Taj just wouldn’t look as good. The white Makrana marble of the Taj has an unusual translucence, which tour guides demonstrate to visitors by literally shining torches through the stonework. I think this translucence contributes to the slight ethereality that the Taj possesses. Black marble just wouldn’t look the same.

Still not convinced? Then think on the availability of black marble in 17th century India. Surviving Mughal monuments tend to be of white marble or local sandstone. There’s a reason for that.

The Melancholy Roman