We find it handy to count in the decimal system using 10 numbers from 0,9, known as base 10, before we have to put two together to make 10 to keep counting further — 10 fingers and toes and all. But it turns out you can represent all numbers equally using just two digits, known as base 2, a 0 and a 1. This is called the binary system and is credited to Gottfried Wilhelm Liebniz in the 1600s.

The binary system is handy because 1s and 0s can be represented by simply by on/off and reproduced in as simple means as pebbles in trays , the sign of a magnetic field eg positive/negative, or a gate as open or closed. This made them the choice for designing computers and is how digital information is stored and transmitted today.

When we write a decimal number we use positional notation with each successive position representing 10 to the next power. So 739 is understood as (7 x 10^2) + (3 x 10^1) + (9 x 10^0). 10 to the power 2 is 10 x 10, so is the number of 100s in the number. Anything to the power 0 is 1.

The same is true in binary, so 1101 is understood in decimal as:

1101 = (1 x 2^3) + (1 x 2^2) + (0 x 2^1) + (1 x 2^0)

1101 = (1 x 8) + (1 x 4) + (0 x 2) + (1 x 1)
     = 8 + 4 + 0 + 1
     = 13

Each 0 or 1 in a binary number is known as a bit — named by Claude Shannon as short for binary digit — and 8 bits is known as a byte.

Translating to decimal looks fiddly, but computers don't have to do that, they can just add, subtract or multiply the binary numbers directly. Amazing to think that the device you may be reading this on now is just incredibly efficient at manipulating 1s and 0s.

For a readable and visual introduction to the history and operation of computers — from binary, logic gates, transistors, circuits, and Moore's law through to software and AI — you could do a lot worse than my Dad's book The Computing Universe 😀

Binary

You have to drop off 2 things after school, get to the shops and back home — which route should you take? You drive an Amazon delivery van and you have 145 parcels to deliver across London — what’s the most efficient route?

It’s pretty clear that for situations like the first one, with fewer stops, assuming you know travel distances or travel times, you could figure out the best route by trying out the different combinations until you hit on the shortest. The trouble is as you keep adding extra stops it doesn’t get just a little bit harder — instead the difficulty continues to increase along with the time it will take to find an answer.

Figuring out the shortest route to visit all the stops and return back home is known as the travelling salesman problem. It’s a problem formulated over 150 years ago that still has relevance and interest today whether it’s for delivering parcels, stocking shelves or soldering transistors.

The travelling salesman problem

You may remember this little trick from school. If you’ve not seen it before, give it a quick try 🙂

The 9 times table on your fingers

This is the fascinating observation that it’s not straightforward to say how long a coastline is. If you were to measure the coastline of a country by using a ruler on a globe, you would come out with a vastly different number than if you were to pace around the edge. The closer you look, the more wiggles and squiggliness you come across and instead of converging on a more accurate length, the coastline just keeps getting longer. The smaller your ruler, the longer it gets.

This was originally spotted, incredibly, in the 1950s, by an Englishman, Lewis Richardson, when trying to check a theory he had that the likelihood of war between countries depended on the length of their shared borders. Remarkably, he found that the quoted lengths of borders varied significantly. While measuring on maps at different scales he saw that the smaller scale map he used, or the smaller the width of his calipers he was measuring with, the length systematically increased. When looking at coastlines, instead of borders, some countries had wigglier coasts and so the length increased at a faster rate with the scale — for instance, Norway’s coastline, with it’s crinkly fjords, increases faster than Britain’s, which in turn increases faster than South Africa’s, as he zoomed in. The rate of this increase later became known as its fractal dimension.

Long after Richardson’s research, Benoit Mandelbrot published a paper How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension that discussed how the wiggliness of something like a coastline at one scale can be repeated at smaller and smaller scales. The work led to the later term fractals. Many other things exhibit fractal-like behaviour such as river networks, borders, brains, frequencies, lightning or even the stock market.

More mindbending mapping, and the difference between Great Britain, the UK and the British Isles.

There’s a super section on this in Scale, by Geoffrey West.

The coastline paradox

In my everyday life I’d have been hard pushed to put my finger on the difference between accuracy and precision but, in engineering at least, you learn that there is a clear difference between the two.

Precision describes how variable an outcome is, or how close repeated measurements or actions are to each other.

In contrast, something may be very precise but not be accurate to your intent or to the true value of a thing. That’s accuracy — how close you are to the true or intended value. If you’re hitting near your intended target then you’re accurate.

Hmm…one of those things where a sketchplanation works a whole lot better than words.

For the more spiritually inclined: Everything is aiming.

Or for the more statistically inclined: Measures of central tendency (mean, median, mode), and Common distributions (normal, skewed, pareto).

That accuracy is true to intention and precision true to itself I learned from Simon Winchester in Exactly: How precision engineers created the Modern World.

Accuracy and precision

Idempotence is essentially an operation that, no matter how many times you do it, you’ll still get the same result, at least without doing other operations in between. A classic example would be view_your_bank_balance being idempotent, and withdraw_1000 not being idempotent. It’s a property that’s often handy as you can retry the operation without worrying about unintended effects.

Idempotence is a concept from mathematics and computer science but can be applied more generally. For example, simple on and off buttons are idempotent, a property that can be very useful in emergencies.

Idempotence

I like this radius. It explains some of why asteroids are lumpy and bumpy, kind of like potatoes, and yet planets and moons are smooth round balls.

From the potato radius size, about 2–300km, the mass of the rock itself gets large enough that it is pulled in on its own gravity and consequently starts to round itself off.

I learned about this from Brian Cox’s show Forces of Nature.

Not sure about asteroids, meteors and meteorites?

The potato radius

Loosely based on the game show Let’s Make a Deal, hosted by Monty Hall, what’s known as the Monty Hall problem is a simply constructed puzzle. Imagine in a game show contestants saw 3 doors, and behind one of which is a prize, and behind the other two doors, goats. The contestant chooses a door, but instead of opening it, Monty opens one of the remaining doors that reveals a goat. He gives you a choice: Do you want to change your door or stick with the one you picked? The problem is: which is the better choice?

If you’re interested it’s pretty easy to find out the real answer. Confirming to yourself why is another thing.

The Monty Hall problem

It’s a funny thing, when you think about it, that if you owned a stock in the market and the value went up 50%, and then went down 50%, that it wouldn’t end up at what it started.

Another way to look at it, is if you owned something worth 100 and it lost 50%, it’d be worth 50. To get back to being worth 100 you’d then need a 100% gain. 

Asymmetry of returns

The time to a target depends on the ratio of distance to the target over it’s size. So clicking on a small target close by might take a similar time to clicking on a large target further away. It’s a simple model and there are a few little games out there to bring it home in practice.

I’m reminded however, of an observation from a mentor of mine, Darrell Mann, that just because people call it a law doesn’t mean you have to treat it like one. The example of Apple’s standard program icons at the base of the screen that expand when you mouse near them, are a simple technique that help to overcome the limitations of Fitt’s law. There are plenty more. Don’t be sucked in to laziness by a name.

Fitts’ Law

Very easy to forget the difference between the mean, the median and the mode.

Median: the middle value of a list in ranked order.

Measures of central tendency: mean, median, mode

Normal (the weight of apples, measurement errors, IQ…), skewed (Number of children in a household, age at retirement…) and Pareto (80/20 rules, long-tails, size of meteorites).

Common distributions: normal, skewed, pareto

OK, there are probably many candidates, but it is remarkable to me how such esoteric and difficult to define numbers - derived independently - have such a simple elegant relationship.

Sketchplanations tagged with mathematics