 Number 197 - October 1999 |
| Back to Basics - Binary Numbers and Computers | ||
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by By Alex Dumestre 1960 PC Users Group, January 1999 As Seen in NOCCC Orange Bytes | ||
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It seems that every
article that attempts to explain computers to beginners has to include a
statement to the effect that computers naturally work with binary
numbers. They then go on to explain that binary numbers are made up of
ones and zeros. Many of the articles drop it at that and you may wonder
why they even brought up the subject in the first place. If that level
of explanation satisfies you (or if you already know a lot about number
systems other than the decimal system you grew up with, then feel free
to skip to the next article in this news letter. If you want to learn a
little about binary numbers and number systems in general then read on.
The Development of Number Systems Early civilizations experimented with all sorts of ways to say and write numerical quantities. Many started with simple tally marks four baskets of oats were recorded as four marks on a wet clay tablet, e.g., ////. Easy enough! 14 baskets of oats already started getting rather cumbersome: //////////////. 322 baskets went far beyond the pain threshold. Some civilizations decided to use tally marks for small numbers but to invent different marks for larger groups of objects. For example one could use a / for one item but perhaps a ^ for ten items. One very familiar system built on this concept is known to us as Roman numerals. In this system a unit is represented by I, five is represented by V, ten by X, 50 by L, 100 by C, 500 by D and 1000 by M. String together as many of these symbols as it takes to represent the number you have in mind. As an example, the year 1066 is written as MLCVI. This is not too bad as a way to record numbers but just try doing arithmetic with Roman numerals! Try dividing MCDIX by XLIII without converting to Arabic numerals. This is not an article about Roman numerals so we will leave it at that except for a couple of interesting things: Note that larger groups that received their own symbols came in multiples of 5 and 10. There is no way to write the number zero in Roman Numerals. If the fact that 5 and 10 have some special place in Roman numerals doesn't strike you as strange, that is probably because of two things. People have 5 fingers on each hand and 10 fingers total and fingers were the first counting aids. Our own familiar number system--the decimal system--uses 10 in a special way. Some civilizations used other numbers as the base of their number systems. 60 was popular with the Babylonians and remnants of that still show up in our time and angular systems. 12 was another base sometimes used and also remains in our clock and dozen and gross, etc. 20 shows up in some systems. But for human convenience, ten has won out, hands down. Positional Systems Mention was made earlier about the number zero. The whole concept of zero was missing from most of the primitive number systems, including Roman numerals. Zero did show up in the Babylonian and in the Mayan systems, though. The reason I mention it is that zero is fundamental in any positional number system. Our customary number system is decimal (or base ten) and, more to the point, it is a positional base ten system. Rather than try to define that abstractly let's explain by example. Our base ten system is made up of only 10 distinct digits. These are not the numbers 1 through 10, but rather the numbers 0 through 9. In this system one counts 0 1 2 ...7 8 9, thus using up all of the available digits. To go higher one just reuses the same set of digits but positions them one place to the left and they, thereby, take on values ten times their normal value. That carries us to 99 at which point we simply shift one more place to the left and continue up to 999. Once established, this scheme has no upper limit. With it we can write numbers in the millions or write the national debt or the distance to the edge of the universe in centimeters! This is powerful! According to this scheme, the digit 3 can have the value 3 or the value 30 or the value 300 or 30000 depending only on where it is positioned in the number. In the general case, we require the digit 0 to allow us to identify that position. Reverting to Roman numerals, we can write the value one thousand and three as MIII. Look ma, no zero. But in our traditional system, if we attempted the same thing, we would get 13 or 1 3 if we did not have zeros to mark empty positions. We are much more comfortable with 1003. Let's review In a positional number system, we first use up all of the available digits, in order. Then we carry 1 (move over one place left and put a 1 or increment whatever digit is already there) and run through all of the digits again. Continue this as long as necessary to count up to the value we need. |
Base ten is convenient
for us but not for computers. They don't have ten fingers; they have
transistors and transistors have only two fingers (really two states
on/off or, by convention, 0/1). Bummer. Not very exciting. The saving
grace for computers is that although transistors can only count to two
there are an awful lot of transistors. So if we try to count using
transistor states we can count 0, 1 on our first transistor but to count
up one more we must carry to the next transistor. Simplistic as this is
we can continue this indefinitely as long as we can carry to yet other
transistor. Since only two digits are available this is called the
binary system and each binary digit is called a bit.
Let's list the first several counting numbers in both the traditional base ten system and in the binary system. The bold numbers in the formula shown below are the positional multipliers. To illustrate the use of the positional multiplier, take the decimal number 13. Its value is (1 x 10) + (3 x 1) = 13. In like manner the binary number 1101 has a value of (1 x 8) + (1 x 4) + (1 x 1) = 13. [ That is 1000+100+010]. In the decimal system the positional multipliers are 1, 10, 100, 1000, 10000 all powers of ten, i.e. 1 x 10 = 10, x 10 = 100, x 10 = 1000.1 see below Powers of Two Now look at the positional multipliers in the binary table above. They progress 1, 2, 4, 8, 16. Each number is double the previous one. If we continued this process, we would get 32, 64, 128, 256, 512, 1024, 2048, 4096, etc. These numbers are all powers of two. That is, 2 squared (2 x 2) is 4, 2 cubed (2 x 2 x 2) is 8, 2 to the 4th power is 16, etc. Notice that the series of numbers in the preceding paragraph contains many of the numbers that you routinely encounter in dealing with and reading about your computer. Some examples: 8 bits to the byte; 16 bit soundboards; 32-bit software for Windows 95/98; 64 Mb of RAM; etc. Let's wrap things up with a little computer math review. How do we determine that 8 bits will give us 256 values? Each bit provides two choices and each additional bit doubles the total number of values. Therefore 8 bits = 28 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256. Where did the numbers 65,000 (Hi-Color) and 16.7 million (True Color) come from? Both of these are just approximations to simplify talking about them. Let's take the big one first. Remember that 24-bit color depth is actually made up of three 8-bit channels, R, G and B. Since each of 256 shades of red can be combined with each of the 256 shades of green, we can end up with 256 x 256 = 2562 = 65,536 possible red-green combinations. Since each of these combinations can be combined with any of the 256 shades of blue we end up with 256 x 65,536 = 2563 = 16,777,216 which we refer to as 16.7 million. A more direct calculation is 224 = 16,777,216. Finally, the hi-color mode on PCs uses a total of 16 bits, rather than 24 bits, to store all three-color channels. This leads to the calculation of 216 = 65,536, which is commonly referred to as 65 thousand or 64K. (Although we have a tendency to think that K stands for 1,000, it actually stands for 210 = 1,024. 64K = 64 x 1,024 = 65,536.) [Editor's note: K does stand for 1000, except in special Computer circumstances. A Kilo Hertz is still 1000 cycles per second.] Working with computers, we can never escape "powers of two." By the same token, we tend to think that Meg stands for one million, whereas it actually stands for 220 = 1,048,576. [Ed note: See above.] So, 16 Mega bytes of memory is really not 16 million bytes but rather 16 x 1,048,576 = 16,777,216 bytes. Now, aren't you glad you asked? There are other number bases that are used around computers but these are important mainly to programmers; however, users may sometimes encounter them. We'll defer a discussion of hexadecimal and octal numbers until later. Alex Dumestre has been associated with computers for over 30 years, most of that time writing geophysical applications for use on main frame, minicomputers, and workstations. He is, however, a novice on PCs. Mr. Dumestre is a member of the 1960 PC Users Group, and can be contacted via e-mail at dumestrea@pdq.net
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Number 197 - October 1999
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