If you have used a computer for more than five minutes, then you have heard the words bits and bytes. Both Ram and Hard Disk capacities are measured in bytes, as are file sizes when you examine them in a file viewer.

You might hear an advertisement that says, "This computer has a 32-bit Pentium processor with 64 megabytes of RAM and 2.1 gigabytes of hard disk space.

Another way to express it would be to use powers of 10. Assuming that we are going to represent the concept of "raised to the power of" with the "^" symbol (so "10 squared" is written as "10^2"), another way to express it is like this:

What you can see from this expression is that each digit is a placeholder for the next higher power of 10, starting in the first digit with 10 raised to the power of zero.

That should all feel pretty comfortable -- we work with decimal digits every day. The neat thing about number systems is that there is nothing that forces you to have 10 different values in a digit. Our base-10 number system likely grew up because we have 10 fingers, but if we happened to evolve to have eight fingers instead, we would probably have a base-8 number system. You can have base-anything number systems. In fact, there are lots of good reasons to use different bases in different situations.

So computers use binary numbers, and therefore use binary digits in place of decimal digits. The word bit is a shortening of the words "Binary digIT." Whereas decimal digits have 10 possible values ranging from 0 to 9, bits have only two possible values: 0 and 1. Therefore, a binary number is composed of only 0s and 1s, like this: 1011. How do you figure out what the value of the binary number 1011 is? You do it in the same way we did it above for 6357, but you use a base of 2 instead of a base of 10. So:

You can see that in binary numbers, each bit holds the value of increasing powers of 2. That makes counting in binary pretty easy. Starting at zero and going through 20, counting in decimal and binary looks like this:

 0 =     0
 1 =     1
 2 =    10
 3 =    11
 4 =   100
 5 =   101
 6 =   110
 7 =   111
 8 =  1000
 9 =  1001
10 =  1010
11 =  1011
12 =  1100
13 =  1101
14 =  1110
15 =  1111
16 = 10000
17 = 10001
18 = 10010
19 = 10011
20 = 10100

When you look at this sequence, 0 and 1 are the same for decimal and binary number systems. At the number 2, you see carrying first take place in the binary system. If a bit is 1, and you add 1 to it, the bit becomes 0 and the next bit becomes 1. In the transition from 15 to 16 this effect roles over through 4 bits, turning 1111 into 10000.

With 8 bits in a byte, you can represent 256 values ranging from 0 to 255, as shown here:

  0 = 00000000
  1 = 00000001
  2 = 00000010
   ...
254 = 11111110
255 = 11111111

A CD uses 2 bytes, or 16 bits, per sample. That gives each sample a range from 0 to 65,535, like this: 

    0 = 0000000000000000
    1 = 0000000000000001
    2 = 0000000000000010
     ...
65534 = 1111111111111110
65535 = 1111111111111111

Bytes are frequently used to hold individual characters in a text document. In the ASCII character set, each binary value between 0 and 127 is given a specific character. Most computers extend the ASCII character set to use the full range of 256 characters available in a byte. The upper 128 characters handle special things like accented characters from common foreign languages.

You can see the 127 standard ASCII codes below. Computers store text documents, both on disk and in memory, using these codes. For example, if you use Notepad in Windows 95/98 to create a text file containing the words, "Four score and seven years ago," Notepad would use 1 byte of memory per character (including 1 byte for each space character between the words -- ASCII character 32). When Notepad stores the sentence in a file on disk, the file will also contain 1 byte per character and per space.

Try this experiment: Open up a new file in Notepad and insert the sentence, "Four score and seven years ago" in it. Save the file to disk under the name getty.txt. Then use the explorer and look at the size of the file. You will find that the file has a size of 30 bytes on disk: 1 byte for each character. If you add another word to the end of the sentence and re-save it, the file size will jump to the appropriate number of bytes. Each character consumes a byte.

If you were to look at the file as a computer looks at it, you would find that each byte contains not a letter but a number -- the number is the ASCII code corresponding to the character (see below). So on disk, the numbers for the file look like this:

 F   o   u   r     a   n   d      s   e   v   e   n 
70 111 117 114 32 97 110 100 32 115 101 118 101 110 

Standard ASCII Character Set

The first 32 values (0 through 31) are codes for things like carriage return and line feed. The space character is the 33rd value, followed by punctuation, digits, uppercase characters and lowercase characters.

  0   NUL
  1   SOH
  2   STX
  3   ETX
  4   EOT
  5   ENQ
  6   ACK
  7   BEL
  8   BS
  9   TAB
 10   LF
 11   VT
 12   FF
 13   CR
 14   SO
 15   SI
 16   DLE
 17   DC1
 18   DC2
 19   DC3
 20   DC4
 21   NAK
 22   SYN
 23   ETB
 24   CAN
 25   EM
 26   SUB
 27   ESC
 28   FS
 29   GS
 30   RS
 31   US
 32
 33   !
 34   "
 35   #
 36   $
 37   %
 38   &
 39   '
 40   (
 41   )
 42   *
 43   +
 44   ,
 45   -
 46   .
 47   /
 48   0
 49   1
 50   2
 51   3
 52   4
 53   5
 54   6
 55   7
 56   8
 57   9
 58   :
 59   ;
 60   <
 61   =
 62   >
 63   ?
 64   @
 65   A
 66   B
 67   C
 68   D
 69   E
 70   F
 71   G
 72   H
 73   I
 74   J
 75   K
 76   L
 77   M
 78   N
 79   O
 80   P
 81   Q
 82   R
 83   S
 84   T
 85   U
 86   V
 87   W
 88   X
 89   Y
 90   Z
 91   [
 92   \
 93   ]
 94   ^
 95   _
 96   `
 97   a
 98   b
 99   c
100   d
101   e
102   f
103   g
104   h
105   i
106   j
107   k
108   l
109   m
110   n
111   o
112   p
113   q
114   r
115   s
116   t
117   u
118   v
119   w
120   x
121   y
122   z
123   {
124   |
125   }
126   ~
127   DEL

Name	Abbr.	Size
Kilo	K	2^10 = 1,024
Mega	M	2^20 = 1,048,576
Giga	G	2^30 = 1,073,741,824
Tera	T	2^40 = 1,099,511,627,776
Peta	P	2^50 = 1,125,899,906,842,624
Exa	E	2^60 = 1,152,921,504,606,846,976
Zetta	Z	2^70 = 1,180,591,620,717,411,303,424
Yotta	Y	2^80 = 1,208,925,819,614,629,174,706,176

You can see in this chart that kilo is about a thousand, mega is about a million, giga is about a billion, and so on. So when someone says, "This computer has a 2 gig hard drive," what he or she means is that the hard drive stores 2 gigabytes, or approximately 2 billion bytes, or exactly 2,147,483,648 bytes. How could you possibly need 2 gigabytes of space? When you consider that one CD holds 650 megabytes, you can see that just three CDs worth of data will fill the whole thing! Terabyte databases are fairly common these days, and there are probably a few petabyte databases floating around the Pentagon by now.

  452
+ 751
  ---
 1203

Binary addition works exactly the same way:

  010
+ 111
  ---
 1001

Quick Recap

• Bits are binary digits. A bit can hold the value 0 or 1.
• Bytes are made up of 8 bits each.
• Binary math works just like decimal math, but each bit can have a value of only 0 or 1.

 Have you ever wondered how a computer can do something like balance a check book, or play chess, or spell-check a document? These are things that, just a few decades ago, only humans could do. Now computers do them with apparent ease. How can a "chip" made up of silicon and wires do something that seems like it requires human thought?

If you want to understand the answer to this question down at the very core, the first thing you need to understand is something called Boolean logic. Boolean logic, originally developed by George Boole in the mid 1800s, allows quite a few unexpected things to be mapped into bits and bytes. The great thing about Boolean logic is that, once you get the hang of things, Boolean logic (or at least the parts you need in order to understand the operations of computers) is outrageously simple.

The simplest possible gate is called an "inverter," or a NOT gate. It takes one bit as input and produces as output its opposite. The table below shows a logic table for the NOT gate and the normal symbol for it in circuit diagrams:

NOT Gate
A Q 0 1 1 0

You can see in this figure that the NOT gate has one input called A and one output called Q ("Q" is used for the output because if you used "O," you would easily confuse it with zero). The table shows how the gate behaves. When you apply a 0 to A, Q produces a 1. When you apply a 1 to A, Q produces a 0. Simple.

The AND gate performs a logical "and" operation on two inputs, A and B:

AND Gate
A B Q 0 0 0 0 1 0 1 0 0 1 1 1

The idea behind an AND gate is, "If A AND B are both 1, then Q should be 1." You can see that behavior in the logic table for the gate. You read this table row by row, like this:

AND Gate

A B Q 0 0 0 If A is 0 AND B is 0, Q is 0. 0 1 0 If A is 0 AND B is 1, Q is 0. 1 0 0 If A is 1 AND B is 0, Q is 0. 1 1 1 If A is 1 AND B is 1, Q is 1.

The next gate is an OR gate. Its basic idea is, "If A is 1 OR B is 1 (or both are 1), then Q is 1."

OR Gate
A B Q 0 0 0 0 1 1 1 0 1 1 1 1

Those are the three basic gates (that's one way to count them). It is quite common to recognize two others as well: the NAND and the NOR gate. These two gates are simply combinations of an AND or an OR gate with a NOT gate. If you include these two gates, then the count rises to five. Here's the basic operation of NAND and NOR gates -- you can see they are simply inversions of AND and OR gates:

NOR Gate
A B Q 0 0 1 0 1 0 1 0 0 1 1 0

NAND Gate
A B Q 0 0 1 0 1 1 1 0 1 1 1 0

The final two gates that are sometimes added to the list are the XOR and XNOR gates, also known as "exclusive or" and "exclusive nor" gates, respectively. Here are their tables:

XOR Gate
A B Q 0 0 0 0 1 1 1 0 1 1 1 0

XNOR Gate
A B Q 0 0 1 0 1 0 1 0 0 1 1 1

The idea behind an XOR gate is, "If either A OR B is 1, but NOT both, Q is 1." The reason why XOR might not be included in a list of gates is because you can implement it easily using the original three gates listed. Here is one implementation:

If you try all four different patterns for A and B and trace them through the circuit, you will find that Q behaves like an XOR gate. Since there is a well-understood symbol for XOR gates, it is generally easier to think of XOR as a "standard gate" and use it in the same way as AND and OR in circuit diagrams.

Let's start with a single-bit adder. Let's say that you have a project where you need to add single bits together and get the answer. The way you would start designing a circuit for that is to first look at all of the logical combinations. You might do that by looking at the following four sums:

0	0	1	1
+ 0	+ 1	+ 0	+ 1
0	1	1	10

That looks fine until you get to 1 + 1. In that case, you have that pesky carry bit to worry about. If you don't care about carrying (because this is, after all, a 1-bit addition problem), then you can see that you can solve this problem with an XOR gate. But if you do care, then you might rewrite your equations to always include 2 bits of output, like this:

0	0	1	1
+ 0	+ 1	+ 0	+ 1
00	01	01	10

From these equations you can form the logic table:

By looking at this table you can see that you can implement Q with an XOR gate and CO (carry-out) with an AND gate. Simple.

What if you want to add two 8-bit bytes together? This becomes slightly harder. The easiest solution is to modularize the problem into reusable components and then replicate components. In this case, we need to create only one component: a full binary adder.

The difference between a full adder and the previous adder we looked at is that a full adder accepts an A and a B input plus a carry-in (CI) input. Once we have a full adder, then we can string eight of them together to create a byte-wide adder and cascade the carry bit from one adder to the next.

The logic table for a full adder is slightly more complicated than the tables we have used before, because now we have 3 input bits. It looks like this:

There are many different ways that you might implement this table. I am going to present one method here that has the benefit of being easy to understand. If you look at the Q bit, you can see that the top 4 bits are behaving like an XOR gate with respect to A and B, while the bottom 4 bits are behaving like an XNOR gate with respect to A and B. Similarly, the top 4 bits of CO are behaving like an AND gate with respect to A and B, and the bottom 4 bits behave like an OR gate. Taking those facts, the following circuit implements a full adder:

This definitely is not the most efficient way to implement a full adder, but it is extremely easy to understand and trace through the logic using this method. If you are so inclined, see what you can do to implement this logic with fewer gates.

Now we have a piece of functionality called a "full adder." What a computer engineer then does is "black-box" it so that he or she can stop worrying about the details of the component. A black box for a full adder would look like this:

With that black box, it is now easy to draw a 4-bit full adder:

In this diagram the carry-out from each bit feeds directly into the carry-in of the next bit over. A 0 is hard-wired into the initial carry-in bit. If you input two 4-bit numbers on the A and B lines, you will get the 4-bit sum out on the Q lines, plus 1 additional bit for the final carry-out. You can see that this chain can extend as far as you like, through 8, 16 or 32 bits if desired.

The 4-bit adder we just created is called a ripple-carry adder. It gets that name because the carry bits "ripple" from one adder to the next. This implementation has the advantage of simplicity but the disadvantage of speed problems. In a real circuit, gates take time to switch states (the time is on the order of nanoseconds, but in high-speed computers nanoseconds matter). So 32-bit or 64-bit ripple-carry adders might take 100 to 200 nanoseconds to settle into their final sum because of carry ripple. For this reason, engineers have created more advanced adders called carry-lookahead adders. The number of gates required to implement carry-lookahead is large, but the settling time for the adder is much better.

Memory relies on a concept called feedback. That is, the output of a gate is fed back into the input. The simplest possible feedback circuit using two inverters is shown below:

If you follow the feedback path, you can see that if Q happens to be 1, it will always be 1. If it happens to be 0, it will always be 0. Since it's nice to be able to control the circuits we create, this one doesn't have much use -- but it does let you see how feedback works.

It turns out that in "real" circuits, you can actually use this sort of simple inverter feedback approach. A more useful feedback circuit using two NAND gates is shown below:

This circuit has two inputs (R and S) and two outputs (Q and Q'). Because of the feedback, its logic table is a little unusual compared to the ones we have seen previously:

What the logic table shows is that:

• If R and S are opposites of one another, then Q follows S and Q' is the inverse of Q.
• If both R and S are switched to 1 simultaneously, then the circuit remembers what was previously presented on R and S.

In this circuit, there are two inputs (D and E). You can think of D as "Data" and E as "Enable." If E is 1, then Q will follow D. If E changes to 0, however, Q will remember whatever was last seen on D. A circuit that behaves in this way is generally referred to as a flip-flop.

A very common form of flip-flop is the J-K flip-flop. It is unclear, historically, where the name "J-K" came from, but it is generally represented in a black box like this:

In this diagram, P stands for "Preset," C stands for "Clear" and Clk stands for "Clock." The logic table looks like this:

Here is what the table is saying: First, Preset and Clear override J, K and Clk completely. So if Preset goes to 0, then Q goes to 1; and if Clear goes to 0, then Q goes to 0 no matter what J, K and Clk are doing. However, if both Preset and Clear are 1, then J, K and Clk can operate. The 1-to-0 notation means that when the clock changes from a 1 to a 0, the value of J and K are remembered if they are opposites. At the low-going edge of the clock (the transition from 1 to 0), J and K are stored. However, if both J and K happen to be 1 at the low-going edge, then Q simply toggles. That is, Q changes from its current state to the opposite state.

You might be asking yourself right now, "What in the world is that good for?" It turns out that the concept of "edge triggering" is very useful. The fact that J-K flip-flop only "latches" the J-K inputs on a transition from 1 to 0 makes it much more useful as a memory device. J-K flip-flops are also extremely useful in counters (which are used extensively when creating a digital clock). Here is an example of a 4-bit counter using J-K flip-flops:

The outputs for this circuit are A, B, C and D, and they represent a 4-bit binary number. Into the clock input of the left-most flip-flop comes a signal changing from 1 to 0 and back to 1 repeatedly (an oscillating signal). The counter will count the low-going edges it sees in this signal. That is, every time the incoming signal changes from 1 to 0, the 4-bit number represented by A, B, C and D will increment by 1. So the count will go from 0 to 15 and then cycle back to 0. You can add as many bits as you like to this counter and count anything you like. For example, if you put a magnetic switch on a door, the counter will count the number of times the door is opened and closed. If you put an optical sensor on a road, the counter could count the number of cars that drive by.

Another use of a J-K flip-flop is to create an edge-triggered latch, as shown here:

In this arrangement, the value on D is "latched" when the clock edge goes from low to high. Latches are extremely important in the design of things like Central Processing Units (CPUs) and peripherals in computers.

So how might we implement these gates in real life? Mr. Boole came up with them on paper, and on paper they look great. To use them, however, we need to implement them in physical reality so that the gates can perform their logic actively. Once we make that leap, then we have started down the road toward creating real computation devices.

The easiest way to understand the physical implementation of Boolean logic is to use relays. This is, in fact, how the very first computers were implemented. No one implements computers with relays anymore -- today, people use sub-microscopic transistors etched onto cilicon chips. These transistors are incredibly small and fast, and they consume very little power compared to a relay. However, relays are incredibly easy to understand, and they can implement Boolean logic very simply. Because of that simplicity, you will be able to see that mapping from "gates on paper" to "active gates implemented in physical reality" is possible and straightforward. Performing the same mapping with transistors is just as easy.

Let's start with an inverter. Implementing a NOT gate with a relay is easy: What we are going to do is use voltages to represent bit states. We will define a binary 1 to be 6 volts and a binary 0 to be zero volts (ground). Then we will use a 6-volt battery to power our circuits. Our NOT gate will therefore look like this:

You can see in this circuit that if you apply zero volts to A, then you get 6 volts out on Q; and if you apply 6 volts to A, you get zero volts out on Q. It is very easy to implement an inverter with a relay!

It is similarly easy to implement an AND gate with two relays:

Here you can see that if you apply 6 volts to A and B, Q will have 6 volts. Otherwise, Q will have zero volts. That is exactly the behavior we want from an AND gate. An OR gate is even simpler -- just hook two wires for A and B together to create an OR. You can get fancier than that if you like and use two relays in parallel.

You can see from this discussion that you can create the three basic gates -- NOT, AND and OR -- from relays. You can then hook those physical gates together using the logic diagrams shown above to create a physical 8-bit ripple-carry adder. If you use simple switches to apply A and B inputs to the adder and hook all eight Q lines to light bulbs, you will be able to add any two numbers together and read the results on the lights ("light on" = 1, "light off" = 0).

Boolean logic in the form of simple gates is very straightforward. From simple gates you can create more complicated functions, like addition. Physically implementing the gates is possible and easy. From those three facts you have the heart of the digital revolution, and you understand, at the core, how computers work.