The Binary Number System
A computer is an instruction follower. The types of instructions that a computer can follow are instructions to manipulate symbols (data). The symbols that a computer can manipulate may represent
But before the computer can process the information, that information must be stored in the computer's memory. In order to store it, we must be able to transform it into a form the computer can understand. The only form in which the computer can represent data is the digital form of zeros and ones.
So the question is: How can you represent numbers, characters, pictures, and sound using only zeros and ones?????
Numbers can be represented in many different ways. For example, the number 12 can be represented as:
We could represent the number 12 in any of the above forms. However, when we use a computer, our choices are limited. In the early 21st century, we have one choice. The electronic devices that are used to store data in a computer that are the cheapest and most reliable are two-state devices. An example of a two-state device is a light bulb. A light bulb can be either on or off (two states). Furthermore, a light bulb remains in the state in which you place it until you change it. You could say that it "remembers" what state it is in. You can turn a light on, leave for a while, and when you return, the light bulb is in the same state it was in when you left (it remembers).
Computers contain electronic circuit elements which function the same way. Each circuit element in the memory of a computer can be placed in either a high voltage state or a low voltage state and stay in that state (it is a memory device!).
One of these devices by itself can only be in one of two states. It can therefore only "remember" yes or no, on or off, open for business or closed for business, 0 or 1, etc.
But if we group these two-state devices together, our options increase. Consider three light bulbs. The possible states the three bulbs can occupy are (x = bulb off, o = bulb on):
x x o
x o x
x o o
o x x
o x o
o o x
o o o
There are 8 states, which can be numbered 1 through 8. However, it makes sense that all bulbs off should represent state 0. Therefore numbering from 0 to 7 might make more sense. Then a 7 would be represented by all bulbs on. But what about the other states??? How do we decide which number to assign to each state???
If we let a bulb that is ON represent the number 1 and a bulb that is OFF represent the number 0, then our 8 states are:
x x x becomes 0 0 0
x x o becomes 0 0 1
x o x becomes 0 1 0
x o o becomes 0 1 1
o x x becomes 1 0 0
o x o becomes 1 0 1
o o x becomes 1 1 0
o o o becomes 1 1 1
We're used to seeing numbers with the digits 0..9. In a computer, numbers must be represented using only the digits 0 and 1. Fortunately, binary numbers (numbers using only 0 and 1) work the same way as decimal numbers!
Knowing that a number is in BASE 10 tells somebody two things:
EXAMPLE: 12,345 when interpreted as a base 10 number is 5 x 1, plus 4 x 10, plus 3 x 100, plus 2 x 1000, plus 1 x 10,000. Each digit in the number is interpreted according to its place value, and the rightmost digit is always the ones column, and each place as you move left increases by a factor of 10.
Knowing that a number is in BASE 2 tells somebody the same two things:
Example: 10110101 when interpreted as a base 2 number is (starting with the rightmost digit):
1 x 1 = 1
plus 0 x 2 = 0
plus 1 x 4 = 4
plus 0 x 8 = 0
plus 1 x 16 = 16
plus 1 x 32 = 32
plus 0 x 64 = 0
plus 1 x 128 = 128
Add all of these up and you get 181.
The first few numbers in base 2 (with their decimal equivalents in parentheses) are:
In general, knowing that a number is in BASE n tells somebody the same two things:
Once you understand this, you can do numbers in any base. Two other number bases that are often used with computers are base 8 and base 16.
Base 8 has the digits 0 through 7 and the place values (from right to left) are 1, 8, 64, 512, 4096, etc.
Base 16 is a little tricky because we have to come up with 16 digits and we're only used to working with 10. The digits in base 16 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F, where A through F represent the numbers 10 through 15 (in base 10). The place values in base 16 (from right to left) are 1, 16, 256, 4096, etc.
So, when numbers are stored in a computer, they are stored as binary numbers. If you could look at the memory of a computer, all you would see are millions of tiny storage devices, each holding either a high or a low voltage. Each high voltage represents a 1, and each low voltage represents a 0. Each one of these individual storage cells is called a binary digit, or bit for short.
The memory of a computer is divided into cells, each holding a specific number of bits. A standard cell size that is almost universal is 8 bits. Each memory cell on most computers holds 8 bits. Such a cell is called a byte.
BYTE: a group of 8 bits. The standard unit of storage on most computers.
Computer memory is measured in BYTES:
BINARY AND HEXADECIMAL NUMBERS
Since all numbers are stored in a computer in binary, you may wonder why we're bothering with other bases like octal and hexadecimal. Hexadecimal (and octal) numbers may be used as a shorthand notation for binary numbers. Binary numbers, while simple, have two disadvantages:
Hexadecimal overcomes both of these disadvantages. Note that a 4-bit number can be used to represent the numbers 0 (0000) through 15 (1111). Note also that each hexadecimal (or "hex" for short) digit can also represent the numbers 0 (0) through 15 (F). This means that for every 4-digit binary number, an equivalent hex digit can be used. An 8-bit binary number can be represented by 2 hex digits, a 16-bit number can be represented by 4 hex digits, etc. Hex numbers are a fourth as long as their equivalent binary numbers. They are also easier to copy because 16 digits are used instead of 2 and you won't get lost as easily in long strings of 0's and 1's. Hex also has the advantage of representing an 8-bit byte with exactly 2 digits (and 8-bit bytes are now almost universal).
EXAMPLE (Binary to hex):
Represent the binary number 1100101011010001 in hex.
First divide the number into groups of 4 bits: 1100--1010--1101--0001. Then convert each 4-bit number into its hex equivalent: 1100=C, 1010=A, 1101=D, 0001=1. Therefore the number in hex is CAD1.
EXAMPLE (Hex to binary):
Represent the hexadecimal number FACE in binary.
Convert each hex digit into its equivalent binary number: F=1111, A=1010, C=1100, E=1110. Therefore the number in binary is 1111 1010 1100 1110.
BASE 10. Fractions are easy. If you know how decimal fractions work in base 10, you already know how fractions work in other bases. In base 10, the first place to the right of the decimal point is the 1/10 place. The next place is the 1/100 place, followed by the 1/1000 place, etc. Every time you move one place to the right your denominator increases by a factor of 10.
BASE 2. The first place to the right of the "decimal" (actually, it's now a "binary" point, but it's hard to get used to saying that) is the 1/2 place, followed by the 1/4 place, the 1/8 place, 1/16 place, etc. Every time you move one place to the right your denominator increases by a factor of 2.
BASE 16. The first place to the right of the "hex" point is the 1/16 place, followed by the 1/256 place, followed by the 1/4096 place, etc. Every time you move one place to the right, your denominator increases by a factor of 16.
EXAMPLE (converting from base 2 to base 10):
What is the value in base 10 of the binary number .1011?
Multiply each digit by its place value: (1 x 1/2) + (0 x 1/4) + (1 x 1/8) + (1 x 1/16).
This becomes: 1 x .5 + 0 x .25 + 1 x .125 + 1 x .0625 = .6875
EXAMPLE (converting from base 16 to base 10):
What is the value in base 10 of the hexadecimal number .9A2?
Again, multiply each digit by its place value: 9 x 1/16 + A x 1/256 + 2 x 1/4096.
This becomes: 9 x 0.0625 + 10 x 0.00390625 + 2 x 0.000244140625
Which becomes: 0.5625 + .0390625 + 0.00048828125 = 0.60205078125
CONVERTING FROM BASE 10 TO BASE 2
An example is given on page 52 of the textbook, but it's not obvious why this method works. To see why it works, consider the original binary number: 0.110101. Let's multiply it by 2 repeatedly and drop any whole-number that appears on the left of the "binary" point. Note that multiplying by 2 in base 2 simply involves moving the binary point one place to the right (just like multiplying by 10 in base 10 simply involves moving the decimal point one place to the right):
0.110101 x 2 = 1.10101
0.10101 x 2 = 1.0101
0.0101 x 2 = 0.101
0.101 x 2 = 1.01
0.01 x 2 = 0.1
0.1 x 2 = 1.0
Note that the red numbers, when read from top to bottom, give you the original digits in the binary fraction. It should be obvious that this is true because multiplying binary numbers by 2 does not change the digits, it just changes the position of the binary point.
Now, note that if we start out with the same fraction, but in base 10, and do the same multiplications (and again dropping any whole-number that appears to the left of the decimal point), we will get the same digits to the left of the decimal point!
0.828125 x 2 = 1.65625
0.65625 x 2 = 1.3125
0.3125 x 2 = 0.625
0.625 x 2 = 1.25
0.25 x 2 = 0.5
0.5 x 2 = 1.0
A USEFUL TOOL
While it is important to understand how all of this works, it is not necessary to go through all of the tedious conversion routines by hand. If you are running Windows, you have access to a tool that does integer number conversions for you: the Windows calculator desk accessory. Give it a try. Note that you must be in "Scientific View" to enable number bases other than base 10. Also note that it does not do fractional conversions.