How 1000100111 may be 227, if you’re not careful

Since the beginning of human existence – probably before that – people have had to count. “How many fish did you catch today?” you could be asked, and unless you could count, you wouldn’t be able to boast about the fact that you caught more than anyone else. So counting is necessary for people to survive.

Now, picture this scenario:

A long time ago (but not in a galaxy far, far away), you get a job for one day. You have to count all the sheep in a farmer’s flock and tell him how many there are, since he keeps falling asleep whenever he tries to count them. So you make a mark – a straight line (|) – on a nearby rock for each one of the sheep. Soon enough, you’ve finished, and the farmer asks you how many sheep he has. But you have no idea – all you can show him is a lot of lines.

This illustrates the problem with counting at face value, or in base 1. So, clearly, humans would need a different system of counting if they were going to count the sheep in a field. We eventually came up with the commonly-used system of base 10, in which each digit is worth ten times the one to its right. This system is so deeply embedded in our brain that we don’t even have to think about it – we look at 8352 and don’t have to think “That’s 8 times 1000, plus 3 times 100, plus 5 times 10, plus 2.” That’s what it is, and we’re so used to seeing these numbers that we compare them, add them, subtract them – in short, juggle them – on a daily basis.

But we could use any base we liked. We’ve only chosen base 10 because we had ten fingers – if we come across alien life with six fingers (or similar appendages), they would probably use base 6. To understand bases, it’s necessary to split up a number. Let’s take 315 (this is still in base 10):

315

= 3*100 + 1*10 + 5*1

= 3*102 + 1*101+ 5*100

Do you notice that everything is multiplied by a power of 10? Changing the base simply involves changing this number to something else – say, 8.

315 =  4*82  +  7*81 + 3*80 = 4738 – the subscript means base 8.

In each place, you can’t have a number greater than the base number itself. This means that, paradoxically, there is no 8 in base 8. They don’t exist – just as 10 doesn’t exist in base 10. 10 is simply a combination of ‘1’ in the x1 place and nothing in the x0 place.

The above rule is because if you have (in base 10, say) ten or more units in place xn, it’ll be ‘carried over’ to place xn+1 – because, as we saw, there’s no symbol representing 10 (it’s just a combination of symbols).

Computers work in the simplest base (though not for humans) – base 2, or binary. In binary, either something is there, or it isn’t – 1 and 0 are the only digits. However, this means that number lengths are longer (a four-digit number in base 10 would take between nine and twelve digits to represent in binary). Basically, there’s a tradeoff – the more symbols you have to remember, the fewer digits you need for numbers.

Binary addition, in particular, is simple – it’s basically XOR addition, in which 0 + 0 = 0, 1 + 0 = 1, but 1 + 1 = 0 (that doesn’t mean that you forget to carry the extra one, though!)

Base 16 is also used, with A, B, C, D, E and F representing the numbers ten to fifteen. So 16435934 in base 10 spells FACADE in hexadecimal.

One last thing: sometimes, base 10 is represented as ‘dec’ before the number; similarly, base 8 (the octal system) is represented as ‘oct’. So, try converting ‘dec 25’ to oct and see how one celebration becomes another!

2 thoughts on “How 1000100111 may be 227, if you’re not careful

  1. raja

    Extremely interesting take on the genesis of numbers and the methodology for expressing them. Good discussion also on different bases, especially why we use base 10 for human counting and base 2 for computer-based counting.

    One question though. In your opening sentence, you say “Since the beginning of human existence – probably before that – people have had to count”. Am a bit puzzled. How could people have been counting even before the beginning of human existence?

    Reply

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