Why do we count starting from zero?

https://cseducators.stackexchange.com/questions/5023/why-do-we-count-starting-from-zero

by inostia • 8 years ago

1 1 8 years ago

cseducators.stackexchange.com

ddingus• 8 years ago

Address lines. So much boils down to address lines, and how they are binary.

Address 0000 is the lowest one, first one, initial address, right?

Well, either we start, given 4 address lines:

0000 = 0

, or

0000 = 1.

Which is it?

Either we have to talk to people about zero, address lines, various indexing and pointer math

, or

We have to start all those conversations with some reason why 0000 = 1.

The former seems to make the most basic sense, and is rooted deep in the hardware.

I have found explaining address lines, and the powers of two very useful. This is how I was introduced to all of this back in the 80's, when confronting computers, memory, hex, pages, and all sorts of basic things.

It made immediate sense. Later, moving to other higher order, and or more complex environments, there are sometimes details needed, but those basics really do not change much, leaving the real problem connecting whatever higher order tools one might be using with the lower level nature of the computer itself.

When I've gone through this, I typically use a 16 bit environment, and 8 bit CPU example. It's 6502, 6809, Z-80 like. Does not matter much to the newbies. We are never going to go there, unless they have an interest.

But, that tiny space, and encouragement to just memorize the first 16 powers of two make for a great foundation!

They can easily do hex and binary with that info. ASCII, and other core things, put into tabular listings of various kinds, all sort out pretty easily. There are lots of examples, like 8 bit color, 24 bit color, those weird "color numbers that have letters in them." Just one of many simple things people can be shown, or talked to about and have it all make some sort of sense.

Why is $000000 black? And why is $FFFFFF white? And just why are there colors from 0 to 255, or levels of red, green, blue?

Decimal conversions are not that difficult either. Just add up all the ones and you are done. Division works the other way. Some people won't need or want those, so no worries. Skip them. They can return later.

Time spent on all that stuff is kind of fun time, and it pays off big. All one needs is a white board, the calculators found in whatever teaching environment is available, and or their computers.

Invite questions. Many people will have seen stuff, and if they ask anything at all, that's your opportunity to connect the dots in a way that will be useful to them and endure easily.

While there, and say it's a school setting, or the people are headed somewhere it may be needed, we've somehow introduced this "metric" or "SI Unit Standard" for Kilobyte as being 1000. I won't argue it, just as I would not for Imperial vs Standard International units, as found in the USA. Not a good hill to die on.

The better thing is to just recognize the units and know what they mean. There are just units, they get used, and the most important thing is to know what those units actually mean. Preferences, politics on them are fun, but secondary.

Kilobyte - 1024, because of address lines. The other unit can make sense for disks, packets, and other things not necessarily tied so fundamentally to the powers of two. It's just enough that they get the differences.

Pages? While those vary considerably, the fundamental idea is each of those powers of two can be a page. On old computers, pages were 256 bytes, 0 - 255 inclusive, for example. They were 8 bits, and that's 8 address lines worth, which is...

Pages in modern machines and languages vary widely, but 4K seems to come up a lot. Again, not too important, just link that concept to the address lines, chunks of memory, etc...

And then finally some math. Bytes, words, longs, and so forth. Nibbles too. 4 bits, 8 bits, 16 bits, 32 bits.

When they do the conversions, a whole lot will make sense, and the most basic thing, say 4 bits being 0 - 15, will make sense, as does the "page", and the number 16 being a power of two, and how all that adds together to make most efficient use of address lines. The colors are one super easy way to talk about this. And they can all bring up some program and type them in, get some sense of things that way.

Truth is, with address lines, there will be X number of them, and they have two states. (technical issues, like floating states aside) Why do anything other than use them? It's a waste to do otherwise, and again technical things aside.

Having taken newbies (and I've personally done middle-school and high school student sessions on request many times) through this stuff, they come away with a very basic understanding about the nature of the computer.

If that's all they get, it's fine. Specific numbers, kilobyte, megabyte, megabit, all have some basic meaning, and the unit issue stands a fair chance of being understood too.

Other questions like, "How come it's always 8GB, or 32Kb?", or "How come we never see a 29GB USB drive?" Have easy and obvious, if not always technically correct, answers they will find make sense.

That all leads straight to, "Count starting from 0" having good reasons, and it can make some basic sense. That's all people need. Should they get to a point where it's a real debate, it's a very nice problem for them to have, if you ask me.

Those who advance and want to do programming will build on these ideas and have little trouble with counting from zero, and when and why it is done, or not done.

Didn't put this on Stack Exchange because I'm not on a machine with those credentials, so here it is for anyone who might need it as an introduction for someone new. It works well --even for very non-technical people. And BTW, the non-technical ones will often just get it, then accept it with little more than, "so that is why computer numbers are weird!

Great, that's the point. Just enough to build on.

And again, way back in the day, all that made so darn much sense to my peer group. We grokked it, and literally got better at understanding a whole bunch of things without even feeling it very much.

Finally, a CPU 8 bits, registers, and all that stuff may not be appropriate. Depending on the audience, address lines, and the on-off state is enough. Use houses, blinkenlights, whatever they can identify with and just link it to how the computer can access things. Talk through 4 bits worth, and they will get it.

Doing that is enough for understanding and for those interested to make good progress with somewhat less stumbling.

Why do we count starting from zero?

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