PROVISIONES PARA PENSIONES Y OBLIGACIONES SIMILARES

One of the challenges of the information age is the transformation of real-world information into data. How can a painting such as the Mona Lisa be stored in a computer? How can an orchestral performance by the London Philharmonic Orchestra be stored on a computer and transmitted around the globe? How can your voice be recorded as data on a voicemail

FIGURE 2.1 A two-dimensional encoding of the phrase “Computational think-

system or how can your smartphone’s password be stored as electronic data within the system?

We first note that there are two different types of data: continuous and discrete. Data is continuous if there are an infinite number of possible val- ues for an individual datum, whereas data is discrete if there are a finite number of possible values. Continuous data is usually associated with measurements involving the physical or real world, whereas discrete data is usually associated with things that can be counted.

As an example, consider measuring the weight of an orange. Although the orange could weigh exactly 200 grams, it might also weigh 229.3 grams or 229.31533 grams or even 229.31533480185993 grams. In other words, the weight of an orange is an example of continuous data since there are an infinite number of possible values that might describe the weight of an orange. On the other hand, consider asking your friends how many biological parents they have who are still living. They may respond with the numbers 0 or 1 or 2. Since no person has more than 2 biological parents, numbers larger than 2 are not possible and it should be obvious that it is not possible to have a fractional number of living parents. The number of living parents is an example of discrete data since there are only a finite number of values that describe this situation.

In electronics, a signal may be either analog or digital. An analog signal is an encoding of continuous data, whereas a digital signal is an encoding of discrete data. In earlier times, electronics were systems that processed analog signals, but modern computing systems almost exclusively utilize digital electronics. The reason that digital systems are generally preferred to analog systems is that there are a finite set of possible values to process in a digital system.

In digital systems, the smallest unit of data is known as a binary digit, or bit. At any point in time, a bit can only take on one of two possible values: ON or OFF. You can think of a bit as an extremely small battery that can

be very quickly charged or discharged. When charged, the bit is ON and when discharged, the bit is OFF. Mathematically speaking, a bit is usually denoted as the value 0 when OFF and the value 1 when ON. Determining whether a bit is ON or OFF is straightforward. Consider, for example, how easy it is to determine wither a lightbulb is on or whether an electric fence is off! Throughout the remainder of this text, we will denote the OFF state as a 0 and the ON state as a 1.

You might be surprised to discover that computing systems encode all information as a sequence of bits. Pictures, sound, textbooks, and video are encoded as long sequences of bits. A sequence of bits is commonly referred to as a bit string. Since the bits in the string are able to vary in the values that they hold, some bits being 1 while other bits are 0, a bit string is able to display a great number of different patterns. For a single bit (i.e., a bit string of length one), there are only two patterns that the string could exhibit at any one point in time. The bit could either be 0 or 1. Consider, however, a string of two lightbulbs. How many different patterns could the string exhibit? Two of the patterns are obvious: both lightbulbs could be 0 or both lightbulbs could be 1. Two other patterns are also possible. The first lightbulb could be 0 and the second lightbulb could be 1. It is also possible that the first lightbulb could be 1 and the second lightbulb could be 0. These four patterns are illustrated in Figure 2.2 where lightbulbs are used to depict a single bit.

Now consider longer bit strings. How many different patterns can a bit string of length three exhibit at any one point in time? If the additional bit is placed as the first bit in the string, it is easy to see that there are twice as many patterns as before. If the first bit is 0 then there are four patterns that the remaining bits can exhibit. If the first bit is 1 (the only other possibility for that bit) there are four patterns that the remaining bits can take on. We conclude that there are eight patterns that a bit string of length three can exhibit at any one point in time. This is illustrated in Figure 2.3.

0 0 0 1 1 0 1 1

FIGURE 2.2 The four patterns that a string of two bits can exhibit are 00, 01, 10, and 11.

Every bit that is added to a bit string doubles the number of patterns that the string can exhibit. This leads us to a very useful generalization to the question of how many patterns a bit string can exhibit. More specifi- cally, we note that the number of patterns that a bit string of length N can exhibit is 2N_{. Figure 2.4 gives insight into this pattern.}

Real-world information can then be encoded as data by arbitrarily associating pieces of information with a particular bit pattern. We might, for example, associate the color red with the pattern 100, the color green with the pattern 010, and the color blue with the pattern 001 in a bit string of length three. As another example, consider encoding all of the sym- bols on a keyboard; including letters, digits, and punctuation symbols. We would make a list of every possible keyboard symbol and then begin to associate each symbol with a unique bit string pattern. We might, for

Length of Bit String Number of Patterns

1 21 _{= 2} 2 22 _{= 4} 3 23 _{= 8} 4 24 _{= 16} 5 25 _{= 32} 8 28 _{= 256} N 2N

FIGURE 2.4 The number of patterns generated by bit strings.

0 0 0 0 0 1 0 1 0 0 1 1

1 0 0 1 0 1 1 1 0 1 1 1

example, associate the letter A with the 8 bit string 01000001 and a period with the 8 bit string 00101110.