# 1    One    1

Interesting numbers --- zero --- one --- complex --- root 2 --- golden ratio --- e --- pi --- googol --- infinity

### Properties of one

One is sometimes called a unit.

One is the basic digit of all number systems. Starting with the number one and the process of addition, you can generate all the rest of the numbers. Many early number systems are unary, where two is shown as two of the one symbol, and three is three one symbols, and so on. Tallying is nothing but a collection of ones, crossed out every five to make it easier to add them up.

Any number multiplied or divided by one stays the same. Any number raised to the power of one also stays the same.

### "As sure as one and one is two"

I have always been irritated by the expresion "As sure as two and two is four". It's not necessarily sure at all. If we take the simpler expression "one and one is two", what else can we make "one and one" be?

 Conventional 1 + 1 = 2 True (sometimes!) Binary 1 + 1 = 10 There is no 2 in binary. Boolean logic 1 AND 1 = 1 This means True and True is True (see below). Roman or Egyptian 1 + 1 = 11 In unary number systems, two is represented by the symbol for one, twice. Greek + = The symbol is iota, which represents ten. or kappa is twenty. Mod 2 1 + 1 = 0 Modular arithmetic is circular. For Mod 2, you divide by 2, and the answer is the remainder. So 25 mod 2 is 1, and 12 mode 2 is 0. Rounding 1 + 1 = 3 This might surprise you! But you can get this effect with rounding. 1.4 + 1.4 = 2.8. But if each number is rounded to the nearest whole number, then you do get 1 + 1 = 3. Try it on the arithmetic calculator set to whole numbers. BODMAS 1 + 1 x 2 = 3not 4 If you replace 1+1 with 2 before doing the multiplication, you will get the incorrect answer of 4. English one as a pronoun "If something happened to one and one didn't know what to do..." You can't replace one and one with two there!

All right, this is a bit of fun! But it does show some important ideas. It's important to realise that rounding can lead to unexpected results as above. It can happen in real life. For example, if you make a report of money, rounded to the nearest pound, then sum the unrounded money and round the result to the nearest pound, you may get a different answer than if you sum the rounded money. Perhaps it might be a good idea to say plus rather than and if that is what you mean, so you avoid confusion with Boolean AND.

Those observant among you reading the page may have noticed that the original statement "As sure as two and two is four" can't be treated in exactly the same way. Boolean logic only works on zero and one, so does binary. There is no two in Roman, Egyptian or Greek number systems. Still, we can say that 2+2=11 (base 3) and 2+2=10 (base 4). Two can be expressed as a binary number, then AND'd with itself, so 2 AND 2 = 2. Modular arithmetic gives us 2+2=0 (mod 4) and 2+2=1 (mod 3). We still have to be careful with BODMAS and rounding. As for an English sentence, how about "Jo came too and two more people said they'd come later." Hm - that's perhaps going too far!

### Boolean logic

Boolean logic is a way of combining True and False values to see if the answer is True or False. It is very important in computing. In Boolean logic, zero represents False and one represents True. Operations are And, Or, Exclusive Or and Not. There are many different symbols for these operations, in Mathematics and in programming languages. I am going to use AND, OR, XOR, NOT.

To some extent, you can understand Boolean logic by thinking about ordinary English. If you say "This is a cat and it is white", then that statement is either true or false. If it's a black cat, it is false. If it is a white dog, then it is false. If it is a grey elephant, then it is obviously false. To be true, then it MUST be both white and a cat. Of the four possibilities of the two components being true or false (FF, TF, FT, TT) only one result is true.

0 AND 0 = 0     0 AND 1 = 0     1 AND 0 = 0     1 AND 1 = 1

There are two different types of Or. In English, you can say "This or that" and you usually mean only one of them. Winnie the Pooh was offered honey or condensed milk, and he was considered very greedy to accept both! This type of Or is Exclusive Or, or XOR. So you can have honey and no condensed milk, or condensed milk and no honey. But you can't have both, and you can't have neither.

0 XOR 0 = 0     0 XOR 1 = 1     1 XOR 0 = 1     1 XOR 1 = 0

In Boolean logic, the more usual Or allows both conditions to be true. If I say "This is a cat or it is white", it's rather a strange thing to say, but logically I mean that it is white, or a cat, or both. The only condition which would make the statement false is if it was neither white nor a cat. The grey elephant would still register as false!

0 OR 0 = 0     0 OR 1 = 1     1 OR 0 = 1     1 OR 1 = 1

The operation Not is different from the rest. Not True is false and not False is True. "This is not an apple" is true if it is a pear, since it would be false to say "This is an apple" if it was a pear. But "This is not an apple" is false if it is an apple, since it would be true to say "This is an apple" if it is a apple.

NOT 0 = 1     NOT 1 = 0

You may feel that this is all totally obvious, and can't see why anyone would bother to write these rules down. But this means that you can write down more complicated conditions and they have a predicatable answer. In particular, what is the opposite of a condition? You may feel that the opposite of one statement AND another statement must be NOT one statement AND NOT another statement, but it isn't. The opposite of "This is a cat and it is white" is not "This is not a cat and it is not white". Remember that it had to be both a cat and white to satify the first condition. The opposite of this is everything except white cats. We will allow black cats and white dogs and grey elephants. But "This is not a cat and it is not white" does not allow black cats or white dogs. The opposite statement that we want is "This is not a cat or it is not white". Now black cats are allowed (since they are not white), so are white dogs (since they are not cats). Grey elephants are there as well, since they are neither. This shows that it is the OR which we need as the opposite of AND, since XOR would get rid of the elephants!

The opposite of    A OR B    is    NOT A   AND  NOT B

The opposite of    A AND B    is    NOT A   OR  NOT B

Boolean logic circuits are used in computers. Computers use binary numbers, which are made of ones and zeroes. If you want to add two numbers in binary, then the result is the two numbers XOR'd toegther, added to the carry, which is the numbers AND'd toegther, shifted left one place.