Numbers

Arithmetic and Computation

Asok Kumar Mallik, Amit Kumar Das

• Publisher: CRC Press
• Year: 2022
• ISBN:
  9781000595017
• Edition: 1
• Topic: Mathematik
• Subtopic: Arithmetik

This book contains a number of elementary ideas on numbers, their representations, interesting arithmetical problems and their analytical solutions, fundamentals of computers and programming plus programming solutions as an alternative to the analytical solutions and much more.
Spanning seven chapters, this book, while keeping its lucid storytelling verve, describes integers, real numbers and numerous interesting properties and historical references; followed by a good collection of arithmetic problems and their analytical solutions.

Please note: Taylor & Francis does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.

Chapter 1 Introduction

We all start learning mathematics with counting objects as 1, 2,3, .... This set of numbers, called the natural numbers, is denoted by the symbol N. Famous German mathematician Leopold Kronecker (1823–91) once commented “God made the integers, man made the rest”. What he meant is that some means of knowing how many existed long before mathematics was established by human beings. Primitive men knew how many cattle the family possessed. Curious experiments have established that even some animals and birds have the sense of small numbers to determine how many. In fact, it has been found that some of these creatures can be trained to count up to quite a few.

Mathematicians started with these natural numbers and extended the domain of numbers by creating abstract quantities, which exist only in the realm of human imagination. A great leap in the concept of numbers is credited to ancient Indian mathematicians, when they created a number shunya, universally called zero with the symbol 0. This is treated as a whole number representing the absence of an object. The set of natural numbers is also known as the set of positive integers written as ℤ⁺. The set of whole numbers is the union of 0 and ℕ and is denoted by W. As kids we are taught the symbol 0 along with the other counting numbers. Then we are taught addition of two elements of W and we get an element within this set. When teaching subtraction of a bigger number from a smaller one, we introduce negative integers like –1, –2,–3, .... This set of negative integers is denoted by ℤ^–. The set of integers ℤ includes 0, all the positive and negative integers. We will discuss these integers in some detail in Chapter 2.

After learning addition and subtraction, we are taught multiplication (or repeated additions). Multiplying two elements of ℤ, we get an element of the same set. We know that the natural numbers continue forever, and in mathematics this unlimited, endless thing is called infinity and universally expressed by the symbol ∞. It must be pointed out that 0 and ∞ are symbols and abstract ideas and one has to be careful while using these in the midst of operations with counting numbers. This is especially important when applying mathematics to solve real life problems. Mathematically the thickness of a line is zero, but any line drawn in real life will have some non-zero thickness.

The inverse of multiplication or division of an integer by another does not necessarily create another element within ℤ. It is also important to note that division by 0 is not defined. The division of one integer by another (not zero) results in two integers, one we call the quotient and the other remainder. When the remainder is zero, we say the first one is divisible by the second. From this division process, a new type of number is created. This extension of numbers, called rational numbers, we will learn a little later when we learn fractions. Here we describe the process of division of one integer by another again by two integers, but not by the quotient and the remainder. Rather by a rational number (p/q), with q =/ 0, i.e. just write as the dividend by the divisor, both of which are integers.

It may be mentioned at this stage, that in school initially we learn Arithmetic, the rules of handling numbers with different operations, like addition, subtraction, multiplication, division, etc. Then we are taught other branches like Algebra and Geometry, which were developed in different ages and in different civilizations. An interesting feature of mathematics is that different branches originated in different ages and places may be connected at a later date. For example, you may have been exposed to the branch coordinate geometry, developed by the French philosopher and mathematician René Descartes (1596-1650) in the seventeenth century. This branch established connection between geometry, developed much earlier by the Greeks, with Algebra developed by the Arabs around the eighth century.

The Greek geometers were interested in mathematical descriptions and measurements of shapes and sizes. Questions like How big is afield?, How long is a distance?, etc. were more important than how many? Basically they were not concerned with discrete identical elements but something which is continuous. Geometry literally means measurement of the earth. The mathematics of the Greeks did not have any symbol for numbers. They had the idea of a point but not of zero.

It may not be out of place to mention that like the numbers, the ideal geometrical shapes are also imaginary or abstract concepts. No straight line or triangle can be drawn in real life; just as no one has seen a (2/3) donkey. As mentioned earlier, a geometrical line is supposed to have no thickness and we cannot draw a line of zero thickness. Whatever we draw must be at least a molecule thick. No wonder we always started our geometric propositions as Let ABC be a triangle, since a triangle can exist only in imagination, the figure drawn on a piece of paper or slate by us is not an object of geometry.

The whole of plane geometry is based on two figures, the straight line and the circle. Both these figures are defined by two points, say A and B. For draw...