Tag Archives: \ell_{infty}

Getting into Norms: a technical postscript

For mathematicians’ eyes only. The post doesn’t require much theoretical knowledge to understand, but I haven’t given many definitions.

In the last two posts I’ve been talking about an example I made with four-dimensional vectors a, b, c such that \|a\|_1 > \|b\|_1 > \|c\|_1, \|b\|_{2} > \|c\|_{2} > \|a\|_{2} and \|c\|_{\infty}>\|a\|_{\infty}>\|b\|_{\infty}. Finding it was more difficult than I at first expected, so I thought I would write the investigation up, which happily gives me an excuse to introduce a useful inequality.

My first thoughts were to choose something like c=(10,0,0,0) and a=(6,6,0,0) or a=(4,4,4,0), and then I’d got stuck choosing b. So I decided to try to prove the opposite.

First of all, it’s not possible to create such an example in two dimensions. Continue reading

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Getting into norms

This blog is named “Out of the Norm”, and the logo is the list of axioms that a norm satisfies (with some extra graffiti), but there’s a fair chance that it’s just mathematical gobbledegook to you. So what is a norm? At its heart, a norm is essentially a measure of distance, though not necessarily a spatial distance. Here are a couple of examples to give you some informal sense of what a norm is, and how they might be used.

Distance in space

If I asked you to go find some buried treasure 3 units of distance East, and 4 units North, how far would you have to travel? That is, from the origin (0, 0) to the point (3, 4) on the graph or ‘map’ below?

It’s not a trick question: the distance as the crow flies, by Pythagoras’ theorem, is simply \sqrt{3^2 + 4^2}=5.

But what if it were a trick question? Let’s say you were actually a taxi driver in some Manhattan-like city with square blocks (each a single unit in size), and were taking a passenger three blocks East and four blocks North (perhaps to the buried treasure, if you insist). Then the shortest route would be seven units long (7=3+4; there are many different shortest routes).

Two of the shortest routes

So now you’ve seen two different notions of distance in 2-dimensional space: the first very familiar; the second probably less so. Just to reinforce the point, you can do the same thing in 3-dimensional space. The distance as the crow flies directly to a point 5 units above the unnecessary treasure, that is, the point (3, 4, 5), is approximately seven units: \sqrt{3^2+4^2+5^2}=\sqrt{50}. However, our taxi cab driver has to drive around the buildings and then take some sort of car-lift, and travels 3+4+5=12 units.

Diagram not at all to scale.

You can extend this to as many spatial dimensions as you want, though as far as I know it’s impossible for humans to imagine a diagram with the point (3, 4, 5, 2) hovering somewhere. But that doesn’t mean the notion is useless.

Guess the age of the lecturers

The next example has been used by one of my former colleagues to introduce norms to his students. Firstly, he gets the students to each write down a guess of the ages of all the lecturers in the maths department, and writes the guesses up on the board. Then he reveals the true ages of the lecturers, and asks the students how they should decide who has won. Here’s some completely made-up numbers:

Let’s say there are four lecturers: a PhD student, a lecturer, a senior lecturer, and a professor, who are 25, 35, 45 and 60 years old respectively. We’ll write this as the vector (25, 35, 45, 60). Now the three students make their guesses: Adam guesses in order 27, 36, 55 and 66: again, write this as a vector (27, 36, 55, 66). Beth guesses (26, 35, 53, 51), completely misjudging the age order of the older pair. And, because there’s always one in every class, Charlie guesses (27.2, 34, 56.4, 62.4), which reminds us that we needn’t work only with whole numbers.


The students decide to judge the results Continue reading

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