# Putting two and two together: mathematical expressions

As somebody who loves words and English literature, I have often been assumed to be a natural enemy of the mathematical mind. And, if we’re being honest, my days of calculus and the hypotenuse are behind me, but, with those qualifications under my belt, I did learn that the worlds of words and numbers are not necessarily as separate as they seem. Quite a few expressions use numbers (*sixes and sevens*, *six of one and half a dozen of the other*, *one of a kind* etc.) but a few are more closely related to mathematics than you’d expect…

### Put two and two together

Let’s start with an easy one. It doesn’t take a mathematical whizz to know that 2 + 2 = 4, and, indeed, that’s the heart of this expression. To *put two and two together* is used to mean ‘draw an obvious conclusion from what is known or evident’. Conversely, if you say that somebody might *put two and two together and make five*, you’re suggesting that they are attempting to draw a plausible conclusion from what is known and evident, but this conclusion is ultimately incorrect. *2 + 2 = 5* was famously used in George Orwell’s *Ninteen Eighty-Four* as an example of a dogma that seems obviously false, but which the totalitarian Party of the novel may require the population to believe: ‘In the end the Party would announce that two and two made five, and you would have to believe it.’

### Fly off at a tangent

I remember a moment of surprise in the middle of one of my mathematics A Level classes. It made a nice change from the almost unbroken moments of bewilderment which characterized the experience. It was when we were looking – as you do – at the equations relevant to what happens when something rotating on an axis suddenly stopped. Well, guess what? It would go off at a tangent.

So, what is a tangent? It’s a straight line that touches a curve at a point, but (when extended) does not cross it at that point. (It’s also apparently ‘the trigonometric function that is equal to the ratio of the sides [other than the hypotenuse] opposite and adjacent to an angle in a right-angled triangle’, but the less said about that the better.)

In common parlance, of course, it simply means ‘a completely different line of thought or action’. While we’re mentioning the *hypotenuse*, you may well recall that it is ‘the longest side of a right-angled triangle, opposite the right angle’, but may not know the word’s origin: it ultimately comes from the Greek verb *hupoteinein*, from *hupo *‘under’ + *teinein* ‘stretch’.

### The lowest common denominator

In a fit of pique, you might have described a person or a group as *the lowest common denominator*. It is often said, in a derogatory way, to mean ‘the level of the least discriminating audience’, for example ‘they were accused of pandering to the lowest common denominator of public taste’. But what actually *is* a denominator?

Cast your mind back (if you will) to the world of fractions – specifically vulgar fractions (those that are expressed by one number over another, rather than decimally). The number above the line is the *numerator* and the number below the line is the *denominator*. In ½, for instance, the numerator is 1 and the denominator is 2. In mathematics, the *lowest common denominator* is ‘the lowest common multiple of the denominators of several vulgar fractions’. For instance, the lowest common denominator of 2/5 and 1/3 is 15, as that is the lowest common multiple of the denominators 5 and 3; the fractions would become 6/15 and 5/15 respectively. It isn’t entirely clear how this sense transferred to the broader, non-mathematical sense.

Can you think of any other expressions that developed from the world of mathematics? Let us know in the comments!

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