# BITS & BYTES

This popular occasional series returns with an easy to follow look at binary operations.

Volume 1

Number 11

January 1984

# The inside story of binary operations...

By MIKE BIBBY

IN previous articles, we've seen that binary numbers can be added and subtracted just as our more familiar decimal numbers are. And, of course, we can multiply and divide them.

There are, however, other ways of combining two binary numbers that are extremely useful in dealing with computers. They're also easy to use, so let's have a look at them!

Firstly, we'll see how we can NOT a binary number - simple, one-bit numbers first. By the way, we're going to be dealing exclusively with binary numbers this month, so we can drop the % sign.

The rules for doing a NOT are simple:

If the bit is 1 then it becomes 0
If the bit is 0 then it becomes 1

If you like, the NOT converts a bit into its opposite.

So NOT 1 = 0
And NOT 0 = 1

Why do we use the word NOT? Well, mathematicians often use the number 1 to mean TRUE and 0 to mean FALSE.

So NOT 1 means NOT TRUE, which means FALSE, which is 0. That is, NOT 1 is 0. And, as NOT FALSE is most certainly TRUE, NOT 0 is 1.

If we are to NOT a binary number consisting of several bits, we simply apply the rule for NOT to each bit individually.

So NOT 10110010
becomes 01001101

Some people think of this process as "turning the number on its head" — so it's sometimes called inverting. Others call it taking the complement of the number.

NOT just works on a single binary

number. However, there are other sums or operations that have a set of rules for combining two binary numbers.

For instance we can AND two binary numbers. Let's look at the rules for ANDing a single bit with another bit. When you think about it, there are four possible combinations of bits that we could AND: 0 with 0,0 with 1,1 with 0 and 1 with 1.

We write that we are ANDing, say, 0 with 1 as 0 AND 1.

The rules for ANDing are:

0 AND 0 = 0 (case a)
0 AND 1 = 0 (case b)
1 AND 0 = 0 (case c)
1 AND 1 = 1 (case d)

Notice that the only time the result is 1 (TRUE) is when the two bits ANDed are both 1 (TRUE). This helps us to see why we use the word AND to describe the operation.

If you think of the first bit as "this" and the second bit as "that", what we're doing when we're ANDing is asking whether "this and that" is true

"This and that" can only be true when both "this" is true AND "that", is true - hence the use of AND to describe the process.

For example, consider the statement that it is dry and sunny

This is true only if dry is true and sunny is true (case d).

If either of the two (or both) are false (cases a, b, c) the whole statement is false, since it isn't both dry and sunny.

We can AND pairs of binary numbers of more than one bit - just apply the rules of ANDing to each bit individually. For example

AND 10010110
AND 10110011
gives 10010010

We can also OR two binary numbers. The rules for ORing a single bit with another bit are as follows (again there are four possible combinations):

0 OR 0 = 0 (case e)
0 OR 1 = 1 (case f)
1 OR 0 = 1 (case g)
1 OR 1 = 1 (case h)

In this case, you only get a FALSE result (0) when both bits are FALSE. If either or both bits are TRUE (1) the result is TRUE. It's easy to see why we use OR to describe this. If one, OR the other, OR both is true the whole thing is true!

Let's use the meteorological analogy again. Let's consider the statement that it is dry or sunny

This is only FALSE when it is NOT dry and NOT sunny (case e), otherwise it is TRUE (cases f, g, h).

To sum up, with OR, the whole thing is true if either or both the things being ORed is true.

As we did with AND, we can OR pairs of numbers with more than one bit - we just apply the rules of ORing to each bit individually.

For example:

10010110
OR 10110011
gives 10110111

• In the next article in this occasional series, we'll look at EOR, and the use of masks.