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History of the Computer – How Computers Add – A Logical Approach
We looked at number systems and counting (see It’s a Binary World – How Computers Count) last time. As a quick refresher, we saw that computers are made up of many units of 0s and 1s, the binary system. 1 is the highest possible number, so numbers in the computer are stored as for example 1010 or 10 in decimal. We have also seen that these binary numbers can be seen as octal (8) or hexadecimal (16) numbers – in this case 1010 becomes 15 octal, or A hex.
You probably realize that “standard” PC code is in 8-bit bytes, taking the hex system one step further. You may also know that processors and the Windows software that runs them have gone from 8-bit to 16-bit to 32-bit to 64-bit. Basically, this means the computer can work on 1, 2, 4, or 8 bytes at a time. Don’t worry if this is all just gibberish, you don’t need it to understand how computers add up!
OK now for the math – time to cringe! It’s a little more complicated than last time, but if you think logically, like a computer, realizing that they’re really stupid, you’ll get through it!
We pause here to look at a bit of math you may not have heard of – Boolean algebra. Again it’s very simple, but it shows you how a computer works, and why it’s so pedantic!
Boolean algebra is named after George Boole, a 19th century English mathematician. He designed the logic system used in digital computers over a century before there was a computer to use it!
In Boolean algebra, instead of + and – etc., we use AND and OR to form our logical steps.
x OR y = z means that if x or y is present, we get z.
x AND y = z means that x and y must be present to get z.
We can also consider an XOR (eXclusive OR).
x XOR y=z means either x or y BUT NOT BOTH must be present to get z.
That’s it! That’s all you need to understand how a computer adds. I told you it was easy!
How do we use this logic in the computer? We create a small electronic circuit called Gate with transistors and such, so that we can work on our binary numbers stored in a register – just some memory. (And it’s the last electronics you’ll hear about!). We create an AND gate, an OR gate and an XOR gate.
When we add in decimal, for example 9 + 3, we get 2 ‘units’ and take one to the ten, which gives 10 + 2 = 12
Do you remember binary bit values in decimal – 1,2,4,8 etc. ? We start at 0, then 1 in the first bit position, bit 1. If we add binary 1 + 1, we should end up with 10, which has a 1 bit in the second bit position, and a 0 in the first, giving Decimal 2+0=2. This second bit position is formed by a CARRY from the first bit.
To make an adder, we have to duplicate with a logic circuit the way we add in binary. To add 1 + 1, we need 3 inputs, one for each bit, and a carry – and 2 outputs, one for the result (1 or 0), and a carry, (1 or 0). In this case, the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to constitute the adder for 1 bit.
Now we take another step and forget the gates, because now we have a logic block, an ADDER. Our computer is designed using various combinations of logic blocks. In addition to the adder, we might have a multiplier (a series of adders) and other components.
Our ADDER block takes one bit (0 or 1) of each number to add, plus the carry bit (0 or 1) and produces an output of 0 or 1, and a carry of 0 or 1. A table of the input A , B and carry, and O and Carry output, looks like this:-
AB v OC
0 0 0 0 0
1 0 0 1 0
0 1 0 1 0
1 1 0 0 1
AB v OC
0 0 1 1 0
1 0 1 0 1
0 1 1 0 1
1 1 1 1 1
This is called a truth table, it shows the output state for any given input state.
Let’s add 2+3 decimal. It’s 010 plus 011 binary. We will need 3 ADDER blocks for decimal binary values of 1, 2 and 4)
The first ADDER takes the least significant bit (decimal bit value 1) of each number. Input A will be 0
Input B will be 1
Without carry – 0.
According to the truth table, this gives an output of 1 and a carry of 0 (3rd line).
BIT 1 RESULT = 1
At the same time, the next ADDER (decimal bit value 2) has inputs of A – 1, B – 1 and a carry of 0, giving an output of 0 with a carry bit of 1 (4th row).
RESULT BIT 2 = 0
At the same time, the next ADDER (decimal binary value 4) has inputs of A – 0, B – 0 and a carry of 1, giving an output of uncarried 1 – 0 (5th row).
BIT 4 RESULT = 1.
So we have bits 4,2,1 as 101 binary or 4+0+1=5 decimal.
It seems like a laborious way to do it, but our computer can have 64 or more adders, simultaneously adding two large numbers billions of times per second. This is where the computer scores.
Next time we’ll see how a computer performs more complicated operations, and it’s simple!
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