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

This is a fun thought-experiment highlighting the challenges of communicating over an unreliable channel — where your message may or may not get through, and where you won’t know if it has.

The situation is that two armies, either side of a hostile valley, are trying to coordinate an attack, and they want to attack at the same time to be sure to succeed. The trouble is, to coordinate they have to send brave messengers through the hostile valley to the other camp and the messengers may or may not be captured along the way.

To coordinate the attack you have to know that your message got through which you’d know if you received a confirmation. But how would the general sending the confirmation know that their confirmation got through unless they received a confirmation of the confirmation. And on…

The two generals problem

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

Translation: The energy efficiency of computers doubles roughly every 18 months.

In a new twist beyond Moore’s Law, which shows signs of reaching it’s limits, there’s Koomey’s Law. In his own words, the idea is:

“…at a fixed computing load, the amount of battery you need will fall by a factor of two every year and a half.”

Why does this matter? It means we can fit more more computing power into smaller and smaller devices. Which is much of the reason the phone in your pocket is probably more powerful than the EDVAC there in the chart.

Koomey’s Law

This chart is from the original he published - just the five data points. Pretty simple huh? The “observation, originally made in 1965, that the number of transistors on an integrated circuit would continue to double every year. Although this increase has slowed to a doubling in eighteen to twenty four months, Moore’s law has held true for nearly fifty years and is the foundation for the astounding Information Technology revolution we see around us.”

For those interested in Moore’s law, remarkably, I think the best information is my Dad’s book, The Computing Universe, which is also the source of the quote.

Moore’s Law

Big data + Machine learning + Cloud = Informed, intelligent decisions. As a very minor example of this I have been enjoying the Dark Sky app.

Sketchplanations tagged with computing