Mathematics for Digital Science 2 (eBook)

Digital Information

Gérard-Michel Cochard, Mhand Hifi (Autoren)

eBook Download: EPUB

2025
538 Seiten
Wiley-Iste (Verlag)
978-1-394-37282-9 (ISBN)

Infos

Over the past century, advancements in computer science have consistently resulted from extensive mathematical work. Even today, innovations in the digital domain continue to be grounded in a strong mathematical foundation. To succeed in this profession, both today's students and tomorrow's computer engineers need a solid mathematical background.
The goal of this book series is to offer a solid foundation of the knowledge essential to working in the digital sector. Across three volumes, it explores fundamental principles, digital information, data analysis, and optimization. Whether the reader is pursuing initial training or looking to deepen their expertise, the Mathematics for Digital Science series revisits familiar concepts, helping them refresh and expand their knowledge while also introducing equally essential, newer topics.

Gérard-Michel Cochard is Professor Emeritus at Université de Picardie Jules Verne, France, where he has held various senior positions. He has also served at the French Ministry of Education and the CNAM (Conservatoire National des Arts et Métiers). His research is conducted at the Eco-PRocédés, Optimisation et Aide à la Décision (EPROAD) laboratory, France.

Mhand Hifi is Professor of Computer Science at Université de Picardie Jules Verne, France, where he heads the EPROAD UR 4669 laboratory and manages the ROD team. As an expert in operations research and NP-hard problem-solving, he actively contributes to numerous international conferences and journals in the field.

Textauszug

1
Representation of Numbers

CONCEPTS COVERED IN THIS CHAPTER.–

This first part, devoted to the digital representation of information, primarily addresses numbers. Starting with numbers is appropriate because computers were originally designed for calculation purposes.

Since computers operate using only 0s and 1s, it is essential to understand the binary system and the conversions between the decimal and the binary systems, as well as vice versa. In addition, two other systems commonly used in computing, the octal system and the hexadecimal system, are discussed.

The approach to the machine representation of numbers involves working within a fixed range, as computers cannot recognize infinity. This necessitates methods for representing negative numbers and rational numbers. The common solutions used for these cases are described.

References: [STA 07, VEL 19, LIP 83].

Using a computer assumes that the information transmitted to it is in a format or language understandable by its components. Transforming data from a human-readable format into a format understandable by the computing machine is called “information coding”.

Modern computers work exclusively with the binary system, which uses a reduced alphabet of {0, 1}. This raises the question of how to represent of common types of information (such as numbers, texts, images, sounds and videos), in binary. The ellipsis suggests that in the future, it might also be possible to represent other sensory experiences, such as smells and tastes, in binary form.

The study will begin with the coding of numbers, as computers were originally designed for computation. The following chapter will then cover the coding of other types of information, including texts, images, sounds and videos.

1.1. Representation of numbers

The decimal system is commonly used for manipulating numbers today, but historically, some civilizations used sexagesimal numbering systems. This approach is still evident in the measurement of time and angles: 1 hour = 60 minutes; 1 minute = 60 seconds, and angles are measured such that 1 flat angle = 180° = 3×60°.

Roman numerals were not very suitable for calculation, serving primarily for counting. The decimal system, which is quite ancient (originating in Egypt in the 3rd millennium BCE and later refined with the invention of 0), is based on the 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and uses positional numbering. As a result, the number 1234 is distinct from the number 3124, despite using the same digits, because the position of each digit is important for determining the number’s value.

1.1.1. Position numbering

EXAMPLE 1.1.–

Consider the number 1234. This number can be understood as follows: 1×1,000 + 2×100 + 3×10 + 4, or 1234 = 1×103 + 2×102 + 3×101 + 4×100.

In the decimal system, which as a base of 10 (with 10 digits), the position of each digit in a number corresponds to a power of 10.

In the binary system, which uses only the digits 0 and 1 and has a base of 2, a decimal number like 1234 is written as: 10011010010. This binary representation is understood as follows:

10011010010 = 1×210 + 0×29 + 0×28 + 1×27 + 1×26 + 0×25 + 1×24 + 0×23 + 0×22 + 1×21 + 0×20.

Just like in the decimal system, the position numbering is used in the binary system. In the binary system, the base is 2, and the digits are 0 and 1, which are referred to as bits in computing. The binary can be verified to correspond to the number 1234 (from Example 1.1) by using powers of 2 (see Figure 1.1).

It should be noted that 210 = 1,024 is referred to as a kilobyte (KB) in computing, commonly abbreviated as “K”. In this context, 1,024 bytes are equivalent to 1 KB.

Figure 1.1. Powers of 2

Unlike human thinking, computers cannot process infinite numbers and are limited in the range of numbers they can manipulate. To represent an n-bit number, the largest integer that can be represented is 2n − 1. To represent the largest integer with m decimal digits in the binary system, which is 10m − 1, how many n bits are needed to represent this number in a computer?

To represent the largest integer with m decimal digits, the 10m − 1 = 2n − 1 must be satisfied (assuming this is possible). Taking logarithms gives:

m × log10(10) = n × log10(2).

Since log10(10) = 1, this simplifies to

m = n × log10(2)

Solving for n:

It therefore takes approximately (3.32 × m) bits to represent an integer of m decimal digits in the binary system.

1.1.2. The binary system

As noted previously, the binary system is based on two symbols (bits): 0 and 1. Unlike the decimal system, operations in binary are extremely simple. Some may recall learning decimal multiplication tables in elementary school!

This is because, in the binary system, basic operations such as addition and multiplication are much more direct and less complex compared to the decimal system. Binary addition is performed bit by bit (see column 1 of Figure 1.2), where 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 (without carry) and 1 + 1 = 10 (with carry). Binary multiplication follows simple rules (see column 2 of Figure 1.2), where 0 × 0 = 0, 0 × 1 = 0, 1 × 0 = 0 and 1 × 1 = 1.

Figure 1.2. Addition and multiplication in binary

This simplicity in binary operations is one of the reasons why computers use this system for internal calculations. This allows for efficient data manipulation and rapid calculations at the electronic circuit level.

The primary problem to solve is converting between decimal and binary systems. To convert a decimal number into a binary number, a straightforward method based on powers of 2 is used.

Consider the example of covering the decimal number 1234 to binary. Start by finding the largest power of 2 less than or equal to 1234, which is 1,024 = 210. Subtract 1,024 from 1234 to get 210. Next, find the largest power of 2 closest to 210, which is 128 = 27. Subtract 128 from 210 to get 82. Continue this process by finding the closest power of 2 to each resulting value and performing the corresponding subtractions. The sequence of operations is shown in Figure 1.3.

Figure 1.3. Decimal-to-binary conversion

The powers of 2 present in this decomposition are 21, 24, 26, 27 and 210 and the missing powers of 2 are 20, 22, 23, 25, 28 and 29. To write the binary number, use the position numbering rule: place a 1 for each present power of 2 and a 0 for each absent power of 2. Thus, the binary representation is:

When necessary, the subscript 2 indicates a number in the binary system, and the subscript 10 indicates a number in the decimal system.

Now let us move on to the reverse problem: converting a binary number into a decimal number. It is actually simpler because you just write it down with the position numbering rule and do the math. Therefore, to convert this binary number to decimal, each bit of the binary number is multiplied by 2 raised to the power of its position, starting from zero at the far right.

Now, consider how to represent a rational number in base 2. Take the example of the decimal number (14.80078125)10. To express this in base 2 follow this rule for a binary number of the form (0. xyzt … )2 where x, y, z, t, … are bits:

This method can be used to convert a decimal number with fractions into base 2. Applying this method will illustrate the conversion process for the example provided.

EXAMPLE 1.2.–

Consider the decimal number 14.80078125 in base 10. First, convert the integer part (14)10 into binary, which is (1110)2. For the decimal part (0.80078125)10, perform successive multiplications by 2,while recording the integer part of the result at each step:

The decimal part can be expressed as a sum of fractions:

This corresponds to the binary representation (0.11001101)2.

Combining this with the integer part (14)10 = (1110)2, the full binary representation of (14.80078125)10 is:

The calculation performed in the previous example will be useful later. By converting the decimal number (14.80078125)10 to binary, the complete binary representation is (1110.11001101)2. This conversion illustrates the general method for converting decimal numbers to binary numbers using the position numbering rule. This method will be explored in more detail later.

1.1.3. Octal system and hexadecimal system

In the octal system , the base is 8, so there are eight digits available: 0, 1, 2, 3, 4, 5, 6 and 7. Converting a decimal number to octal follows the same principle as converting to binary. Using powers of 8, the number (1234)10 can be represented in octal base. Given that

Break down (1234)10 as follows:

Converting to octal:

The conversion process involves starting with the binary representation of the number, grouping the into packets of 3 from the right, and then using the table in Figure 1.4 to convert each group to octal. For example:

Group the binary...

Produkt-Details

• Erscheint lt. Verlag: 16.4.2025
• Reihe/Serie: ISTE Invoiced
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Schlagworte: Computer Science • data analysis • Data Analysis • digital information • Digital Science • Mathematics • Optimization
• ISBN-10: 1-394-37282-5 / 1394372825
• ISBN-13: 978-1-394-37282-9 / 9781394372829