Mathematics, the old "queen of sciences", kind of runs on accuracy. It's also eminently logical and not really that hard. So permit me to channel John Allen Paulos for a moment...

It was with a mix of horror and irony that I read David M Peterson's post on O'Reilly's XML blog today: "AWS Drops SQS Pricing By 10,000%?" (the irony comes in the first sentence of the post.)

Oh, dear. Let's see. Suppose the original cost was $1. So 10,000% of that is $100. So reducing the cost by 10,000% would mean that Amazon were now paying $99 for every $1 they were previously charging. Doesn't sounds like a particularly profitable business model. Probably not worth relying on that service to be around very long. Wait ... you mean, that's not what they're doing? Oh dear, it's Marketing Maths at work, again. :-(

So why not be accurate? They've reduced the price for 10,000 requests from USD 1 to USD 0.01. That's a 99% reduction. The new value is now 1% of the old value. Easy.

Ratios aren't symmetrical: reducing by half and then increasing by half leaves you with less than the original amount. That's mathematics. Trying to change it because you prefer symmetry gets you a choice of free admittance to the Flat Earth Society AGM or a poster about the geocentric model of the universe. What it doesn't get you is a passing grade.

Besides, the symmetry already exists here: we use the same starting point (the original quantity) always. If something increases from $1 to $2, you don't say it increased by 50%. Similarly, it's illogical to say that something decreasing from $1 to $0.50 has been reduced by 100%. However, this seems to be a common marketing blunder amongst people wanting to show how extreme something is.

Yeah, David M Peterson doesn't deserve this grief. He having a Zippy the Pinhead ("somebody pinch me!") moment and just made a quick note on a blog. I'll admit that I'm taking a slightly cheap shot just to have a rant. Still, it's the third time in the past day or so that I've seen this nonsense and this was the only case with a URL attached. And it's my blog. Just because your government uses this kind of arithmetic to create fiscal policy, doesn't mean you're allowed to use it in computer science.

Oh, and, in passing... Amazon Web Service's price change is a pretty sweet deal.

Topics: mathematics, venting

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Having started with some thought experiments in the original article, today I want to start to make some concrete progress.

We're starting in the middle of a large map here. Where we are now is the place of knowing about numbers and how we use them in every day life. We can add, multiply, divide, maybe do some trigonometry. In each direction are slight variations on this theme. Removing properties, as we'll do today, or adding in extra features. Some of these directions are no doubt familiar to people, others will be new.

For anybody wanting to follow this series, without having to read any of my other writing, this Atom feed or this web link will contain just the articles in this feed (the Atom feed contains the full text of each article).

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Topics: mathematics/series: deriving

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Another multi-part series of posts (along with storytelling) I want to try here is laying out my thinking about how we can try to understand (and teach?) formal mathematics. Nothing too revolutionary. Everything I'm going to write about has been done before in various places. I'd like to have a shot at putting my spin on the exercise.

My idea is to motivate everything. Start from what we commonly accept as "true" today — and by "we" here, I mean, say, your basic early high-school student. What do we understand about numbers right now? How can we motivate the leaps that mathematicians make to get to abstract things like sets, rings, fields, formal logic, analytic number theory, ... the list is endless.

Given my finite time here on earth, I'm not going to try and cover everything (surprise!). However, I will claim that a lot of what is taught in senior school classes, even in university courses in Pure Mathematics, really isn't that far away from what "everybody" knows. It's partly a matter of notation, but also, if you learn by building mental models for yourself, a matter of seeing the motivation behind particular idea so that you can put it in perspective.

Let's begin with some thought experiments and on Tuesday I'll start working towards answers.

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Topics: mathematics/series: deriving

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Quoting from a Reuters piece, the Ig Nobel prize for mathematics was awarded to a couple of Australians: "Nic Svenson and Piers Barnes of the Australian Commonwealth Scientific and Research Organization, for calculating the number of shots a photographer must take to almost ensure that nobody in a group photo will have their eyes closed."

The point of these prizes is to make you laugh and then make you think. They nailed it on this one.

Some details of their work is available from an interview with the always excellent ABC Science Show earlier this year the same writeup, but with a graph and a little more symbology is here.)

(Original seen in the Nikon Flickr group.)

Topics: mathematics, photography

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A week ago I posted an innocent looking geometry puzzle. Today, I give the solution.

This problem turned out to be a little harder then I remembered (at least, the solution turned out to be longer than I hoped when I wrote it out). I have another couple of problems to post at some point; they should be easier than this one.

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Topics: mathematics/puzzles

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For a change in pace, an old geometry problem I came across again recently.

Suppose you are given two circles in a plane. Their centres are marked. You want to find a point P such that when you draw the four tangent lines from P to the two circles (two to each circle), the angles between the pair of tangent lines for each circle are equal. The following diagram may help here. Find the point P such that the two marked angles are equal.

If you know where the centres of the two circles are (and I'll give you that), you can solve this with only a straight-edge (not a measuring ruler; no measuring required). So you might have lost half of your straight-edge + compass construction kit since high school or whenever you last did these, but you don't need the compass here anyway (I gave you the circles, you don't need to draw them).

[Update: This is rubbish. You do need a compass to solve this problem. At least in the solution I have in mind. There is a need to construct a perpendicular line at one point and that cannot be done with a straight edge alone.]

1. For bonus points, find all the possible points P.

2. For even more bonus points, introduce a third circle into the picture (not on a the straight line connecting the centres of the first two circles). Now there is a unique point P that satisfies the condition (the angles between all three tangent pairs must be equal). How can you find it?

Topics: mathematics/puzzles

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