First, a math lesson...
Computer data is in "digital" format. That is to say, whether it is colors, words, pictures or animation, it is all "coded" in terms of binary digits, that is "0s" and "1s." In the first computers, "0" equaled "off" and "1" equaled "on." This made it easy to program them using a series of "on" and "off" switches. Current computers are still based on this simple system but are exponentially more powerful.
The BINARY SYSTEM is one of numbers at the power of 2:
- 2 to the power of 0 = 1,
- 2 to the power of 1 = 2,
- 2 to the power of 2 = 4,
- 2 to the power of 3 = 8,
- 2 to the power of 4 = 16,
- 2 to the power of 5 = 32,
- 2 to the power of 6 = 64, etc.
These are written as: 2¹, 2², 2³, etc. Specifications—such as how fast it runs and how much memory it contains—are only stated using these sets of numbers.
Every positive integer can be written as sum of powers of 2:
| Number |
Of powers of 2 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
| 45 |
32+8+4+1 |
off |
on |
off |
on |
on |
off |
on |
| 74 |
64+8+2 |
on |
off |
off |
on |
off |
on |
off |
| 77 |
64+8+4+1 |
on |
off |
off |
on |
on |
off |
on |
These on and off switches can be translated into information as follows:
| Decimal System |
Binary System |
| 45 |
0101101
(or simply 101101) |
| 74 |
1001010 |
| 77 |
1001101 |
|
Now For Some Definitions...
- Radix: or "base," or "number base." In a positional representation of numbers, a "radix" is that integer by which the significance of one digit place must be multiplied to give the significance of the next higher digit place. Conventional decimal numbers are radix ten, binary numbers are radix two.
- Bit: stands for a "Binary digIT" (e.g. the number "2" equals the binary digit "01").
- Octal: is shorthand notation for binary code. Octal, which stands for Base 8, uses a number representation using the digits 0 to 7 only, with the right-most digit counting ones, the next counting multiples of 8, then 8^2 = 64, etc. (e.g. octal "177" is digital "127" as seen in the chart below).
| digit |
weight |
value |
| 1 |
8^2 = 64 |
1* 64 = 64 |
| 7 |
8^1 = 8 |
7* 8 = 56 |
| 7 |
8^0 = 1 |
7* 1 = 7 |
| 177 = 127 |
Each octal digit equals 3 Bits (e.g. ["72" = 111010 (111 (7) + 010(2) )]. If the number has a decimal point, start at point and divide into 3s going in both directions (e.g. ["241.46" = 10100001.10011 (10(2) + 100(4) + 001(1) .100(4) + 11(6) )]). NOTE: Octal system used to be widespread back when many computers used 6-bit bytes, as a 6-bit byte can be conveniently written as a 2-digit octal number. These days, a byte is almost always 8-bits long, and the octal system has lost most of its appeal to the hexadecimal system.
Hexadecimal: Base 16. A number representation using the digits 0 to 9, with their usual meaning, plus the letters A to F (or a to f) to represent hexadecimal digits with values of (decimal) 10 to 15. The right-most digit counts ones, the next counts multiples of 16, then 16^2 = 256, etc. (e.g. the hexadecimal for the word "BEAD" is decimal "48813" as seen in the chart below).
| Digit |
Weight |
Value |
| B = 11 |
16^3 = 4096 |
11*4096 = 45056 |
| E = 14 |
16^2 = 256 |
14* 256 = 3584 |
| A = 10 |
16^1 = 16 |
10* 16 = 160 |
| D = 13 |
16^0 = 1 |
13* 1 = 13 |
| BEAD = 48813 |
ASCII (American Standard Code for Information Interchange): The basis of character sets used in almost all present-day computers. US-ASCII uses only the lower seven bits (character points 0 to 127) to convey some control codes, space, numbers, most basic punctuation, and unaccented letters a-z and A-Z. More modern coded character sets (e.g., Latin-1, Unicode) define extensions to ASCII for values above 127 for conveying special Latin characters (like accented characters, or German ess-tsett), characters from non-Latin writing systems (e.g., Cyrillic, or Han characters), and such desirable glyphs as distinct open- and close-quotation marks. ASCII replaced earlier systems, which used fewer bytes.
Bytes: Usually means 8 Bits (e.g. 10001011). Thus, for PCs that used 7-bit ASCII (ASCII is the basis for most operation code), a Byte was 7 bits; for PCs using 6-bit ASCII, a Byte was 6 bits, for PCs using 16-bit technology, a Byte was 16 bits and for current PCs using 32-bit technology, a Byte is 32 bits.]
Nibble: 4 Bits (1/2 of an 8-bit Byte).
Hexadecimal Digit: 4 Bits, a hexadecimal digit stands for a Nibble Byte, which is 8 Bits.
Kilobyte: 1,024 Bytes.
Megabyte: 1,023 Kilobytes. [1,024 x 1,024 = 1,048,576 Bytes]
Gigabyte: 1,024 Megabytes. [1,024 x 1,048,576 = 1,073,741,824 Bytes]
Terabyte: 1,024 Gigabytes. [1,024 x 1,073,741,824 = 1,099,511,627,776 Bytes!]
Microchip: Sometimes just called a "chip," a microchip is a unit of packaged computer circuitry (usually called an "integrated circuit") that is manufactured from silicon at a very small scale. Microchips are made for program logic (logic or microprocessor chips) and for computer memory (memory or RAM—random access memory—chips). Microchips are also made that include both logic and memory and for special purposes such as analog-to-digital conversion, bit slicing, and gateways.
So you can see, when you have a computer with a 20 Gig Hard Drive (20 Gigabytes of memory), you can have room for a tremendous amount of information (21,474,836,480 Bytes to be exact!) |
So How Do Computers Work?
Computers contain thousands of tiny electrical switches located in microchips. As expected, each switch can be either "off" or "on." The various combinations of which switches are "off" and which switches are "on" dictate what the computer is doing at any given moment. You can imagine with thousands of switches perating, there is virtually an infinite number of combinations of "off"s and "on"s. This explains why computers can function in so many different ways—from a adding machine to a word processor!
If you think of a light switch, there are only two distinct conditions which are possible, off or on. What happens if there are two switches controlling two separate lights? We'll call the lights "A" and "B". Now there are four distinct conditions which are possible as listed below:
| Condition |
Light A |
Light B |
| 0 |
off |
off |
| 1 |
on |
off |
| 2 |
off |
on |
| 3 |
on |
on |
Notice that our numbering system starts from "0." This is because with both lights off there is no current flowing to either light, so we call it condition "0." Look around your place and see if you can find a triplet panel—3 switches grouped together to control three separate lights ("A", "B", and "C"). Can you guess how many distinct conditions are possible with three switches at your disposal? The answer is 8 conditions. As discussed before, we start the numbering from "0":
| Condition |
Light A |
Light B |
Light C |
| 0 |
off |
off |
off |
| 1 |
on |
off |
off |
| 2 |
off |
on |
off |
| 3 |
on |
on |
off |
| 4 |
off |
off |
on |
| 5 |
on |
off |
on |
| 6 |
off |
on |
on |
| 7 |
on |
on |
on |
Why did your home builder group these light switches together as a triplet panel? The answer is efficiency and convenience. It is much easier to control all three lights from one position in the room than it would be to move around from wall to wall to control each light separately. Computer designers also found it much more efficient to group computer switches together into panels. Computers started by grouping eight switches on each panel. Would you care to guess how many distinct conditions are possible with eight switches at your disposal? Here's a clue by reviewing what was discussed above:
- with 1 switch, there were 2 distinct conditions possible (Condition 0 to 1)
- with 2 switches, there were 4 distinct conditions possible (Condition 0 to 3)
- with 3 switches, there were 8 distinct conditions possible (Condition 0 to 7)
Do you see a pattern to this yet?
- with 4 switches, there are 16 distinct conditions possible (Condition 0 to 15)
- with 5 switches, there are 32 distinct conditions possible (Condition 0 to 31)
- with 6 switches, there are 64 distinct conditions possible (Condition 0 to 63)
- with 7 switches, there are 128 distinct conditions possible (Condition 0 to 127)
- with 8 switches, there are 256 distinct conditions possible (Condition 0 to 255)
It seems incredible but it is true. With 8 switches grouped together on one panel, there are 256 unique and distinct conditions that can be controlled from this panel! In computer terminology, each switch on the panel is called a bit. The full group of 8 switches is called a byte. A byte can represent any number from 0 to 255. We all know that computers deal with numbers bigger than 255. If a number is bigger than 255, the computer requires yet another byte to make the number.
For example, the number 256 would take up 2 bytes—the first one would hold the maximum amount of 255 and the second one would hold the amount of 1. The sum of 255 + 1 is therefore 256. Let's try another example. How many bytes would the computer need to hold the number 65,025? Did you guess 255 bytes? Incorrect! The computer only requires 2 bytes to hold the number 65,025! This time the computer multiplies the value of 255 in the first byte by 255 in the second byte to give the value of 65,025. To hold a bigger number, like 67,000, the computer would require yet another byte for a total of 3 bytes.
How does the computer know that it should multiply the 2 bytes to create the number 65,025 and add 2 bytes to create a number like 256? This is the responsibility of the computer's Central Processing Unit (CPU) or the brain of the computer. The CPU is contained on a tiny microprocessor (about 1/4-inch square) and is programmed at the factory to be able to look at any of the 8-bit panels (or bytes) and figure out what it is to do with them.
Click here to learn more about HOW COMPUTERS WORK. |
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