Unit 2.1: Binary Numbers

Last week, we learned how to manipulate bits in a computer.

We have yes, we have no, 1, 0, and we know how to manipulate them.

But what can we do with only 1 and 0?

Can we represent more sophisticated things with them?

Of course we, we must, if we want to use a computer to do useful things.

But what can you do with only two values, 0 and 1?

Well, you definitely can put two of them together, and

then you get four possibilities.

Three of them together, you get eight possibilities.

In general, if you have n of them together,

you have 2 to the n possibilities and now you could pre,

represent any 2 to the n different things that you may want to.

For example, you can represent numbers.

And this is what we're going to be interested in.

How are you going to represent integer numbers?

Well, 0 is easy.

0, 1 is easy.

1.

2 is already the first problem, basically because we don't have yet

another possibility, so we have to use two bits.

0, 10.

Three is 11.

For the next one, we already need another bit, 110 for 4.

And in general,

it seems we can represent any number that we want with some sequence of bits.

Now how did we co, connect the sequence of bits,

the binary representation of an integer number, to the integer number?

What's going on?

What's the general system?

In order to understand that, we have to go back to second grade.

When we learned about decimal numbers.

When you see 789, what does that mean?

Where, what does the digit 7, 8 and 9 have to do with the value of the number 789?

Well, we learned that we have the positional system, where the right-most

digit is the ones, the next one is the tens, the next one is the 100s.

So we know that 789 is really 9 plus eight 10 plus seven 100s, and so on.

In general, the case position from the right is 10 to the k.

So now exactly the same thing that was going to happen also with binary numbers,

but it's going to be much simpler because we only have 0 and

1 rather than all the numbers, digits between 0 and 9.

Thus in binary notation, the different positions will be powers of 2.

1, 2, 4, 8 and so on.

Suppose you want to know, for example, what is a value of 1 0 1,

in binary notation?

Well, we know the rightmost bit is basically the 1s.

The next one, the 0, correspond to 2s.

And the third, the left-most bit here, is the 4s.

Altogether, we have 4 plus 2 plus 2 times 0, plus 1 times 1.

Altogether, that's 5, so our number is 5 and in general, what do we do in general?

Exactly the same thing.

You have any sequence of bits and we are going to number them from the right.

Most bit, which is going to be b 0, the next one b 1, and so

on, all the way to bn, if we have n plus 1 bits.

And the value of this thing is going to be b0 times 2 to

the 0 plus d1 times 2 to the 1, plus b2 time 2 to the 2,

all the way until the end of it, which we have, where we have bn time 2 to the n.

And, this is going to be a number.

And, that's how we can covert any sequence of bits,

any binary representation of a number to the value of a number.

One thing that we should note at this point that if we look at the maximum

number we can represent with k bits.

Well we sum up from 2 to the 0 all the way to the k minus 1.

Remember that the case bit if we start counting from 0,

the last bit is not counted as index k, k minus 1.

So we have the sum of 1 plus 2 plus 4, all the way to the 2k minus 1.

All together, we have 2 to the k of that minus 1.

And that's a range of bits, we can actually represent with k.

The range of numbers we re, can represent with k bits.

Now, so far, we assumed,

we have an arbitrary length number of bits that we can use to represent.

And of course, if we want to represent an arbitrary long number,.

We will need an arbitrary number of bits.

In computers, you usually have a fixed number of bits that is allocated.

And then of course you can only be able to represent a fixed number,

a fixed range of integer numbers.

So for example, eh, if we only have eight bits, what are the,

what are the possible numbers that you can represent?

Well you can represent that something starting with 000 eight times.

000 ending with a 1 all the way up to eight 1s.

The smallest number is, we have all together 256, 256 such possibilities.

The first one is index 0, the last one is index 255.

Now, that's not exactly true.

Really, when we have eight bits, usually we want to, to, want to reserve

part of these, all the possibilities to actually represent negative numbers.

We're not going to actually talk about this now, but rather in a unit from now.

But in general, half of the indi, half of the 256 possibilities are going to be re,

be reserved for negative numbers, and we're only going to be able to use

the numbers between 1, between 0 and 127, and these are the positive numbers

that we're going to be able to use out of these 256 possibilities of 8 bits.

So far we saw, basically, if you better or

a string of bits, how can you convert it to decimal numbers?

Now we are going to do the opposite thing,

suppose you were given a number in decimal, 87 for example.

How can you represent it as a sequence of bits?

This is also something which we should be able to do, if we're going back and

forth between decimal notation and binary numbers.

That they're actually going to be eh, represented as in the computer.

Well remember that we know that really the way that we get the decimal from

the binary is by summing up powers of 2.

So we start by figuring out,

figuring out what is the largest power of 2 that fits into our 87 number.

And that is going to be 64.

And then, what is the next one?

After we have 64, what is the next power of 2 that we can add to 64 and

we still remain under 87?

That turns out to be 16.

And so on, we can keep on going, and we write the number 87 as a sum

of binary pow, of powers of 2.

And it turns out that 87 is exactly 64 plus 16 plus 4 plus 2 plus 1 and once we

have that, from this representation of the decimal number, as a sum of powers of 2,

we can quite directly actually get the binary representation.

How do you do it?

Well every time,

we have a power that the peer is in the sum we need to put a 1 in the bit there.

And whenever is a power is not part of the sum, we put a 0 there.

So for example, look at the right-most bit, that corresponds to the 1s.

We do have a 1, so that's going to be a 1.

On the other hand,

if you look at that third bit from the right, which is correspond to the 8s.

Since we don't have 8 in the sum there, we're going to have 0s there.

And that's basically a gener, general way how you can take any number and

convert it to binary.

So this concludes this unit where basically we discussed

how you can represent integer numbers within a binary system.

In the next unit,

we're going to actually discuss how can we actually perform arithmetic operations.

In particular, addition and these represented binary numbers.

Once we get that under our belt, we will go back and

discuss the issue of negative numbers in unit 3.

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