What exactly are numbers? I assure you, I won't delve too deeply into philosophical discussions in this post. However, aren't numbers merely a concept that we have created? Why is it that we specifically use the digits 0 through 9 and not a greater set of digits?
And why do we employ fractions and decimals instead of solely relying on whole numbers? This field of mathematics, which explores numbers and the reasons behind their design, is commonly referred to as number theory.
Number theory has its roots in ancient times, when mathematicians devoted their attention to exploring various number systems. This discipline sheds light on the reasons behind our current acceptance of these systems. Here are some examples of number systems that might be familiar to you:
Natural Numbers
In this sequence, we encounter the numbers 1, 2, 3, 4, 5, and so forth. This particular series exclusively comprises positive numbers and represents one of the earliest known number systems. These numbers, known as natural numbers, trace back to ancient times when cavemen would etch tally marks on bones and cave walls as a means of record-keeping.
Additionally, it's worth mentioning that natural numbers serve as the foundation for other number systems and mathematical concepts. They are essential in fields such as arithmetic, counting, and basic calculations.
Whole Numbers
Expanding upon the realm of natural numbers, the notion of "0" was eventually embraced, leading to the introduction of what we now refer to as "whole numbers." Notably, the Babylonians made a valuable contribution by devising a system of place-holding notation.
This system allowed for the representation of empty "columns" in numbers exceeding 9, such as "10," "1,000," or "1,090." The presence of zeros in these numbers denotes the absence of any value occupying those specific columns.
This place-holding notation, which incorporated the concept of zero, proved to be a pivotal development in mathematics, enabling more efficient and concise numerical representation. It laid the foundation for subsequent number systems and paved the way for the evolution of modern mathematics.
Integers
Integers encompass both positive and negative natural numbers, including zero. While we may consider them commonplace today, it is worth noting that ancient mathematicians harbored significant skepticism toward the notion of negative numbers.
However, the utility of negative numbers became evident when faced with situations like subtracting 5 from 3, resulting in -2. This concept proved particularly valuable in realms such as finance, where we quantify profits and losses.
It was in 628 AD that Brahmagupta, an Indian mathematician, played a pivotal role in establishing the acceptance of negative numbers. He demonstrated their necessity in arithmetic, particularly when dealing with the quadratic formula.
Brahmagupta's work highlighted the importance of negative numbers for the advancement of mathematical operations. As a result, integers, incorporating both positive and negative numbers, gained widespread recognition and application.
Rational Numbers
A rational number is any number that can be expressed as a fraction, such as 2/3. This category encompasses not only finite decimals but also integers since they can be represented as fractions as well, for example, 687/100 = 6.87 and 2/1 = 2, respectively. The term "rational" is derived from the fact that these numbers can be expressed as ratios.
The introduction of rational numbers was driven by the recognition that certain quantities, such as time, resources, and other measurements, could not always be neatly quantified in discrete units.
For instance, when dealing with quantities like milk, which may not always be packaged in fixed units like gallons, we may need to measure it as fractions of a gallon. Similarly, if someone runs for 12 minutes, it would be unreasonable to only measure the distance in whole miles, especially if they actually ran 9/10 of a mile.
Rational numbers provide a flexible means of representing and working with such quantities, accommodating measurements that don't neatly align with whole numbers or fixed units.
Irrational numbers
Irrational numbers defy representation as fractions. They encompass well-known examples such as π, the square root of certain numbers like √2, and Euler's number e, which we will explore in detail later. These numbers possess an infinite sequence of decimal digits, exemplified by π as 3.141592653589793238462...
The history surrounding irrational numbers is intriguing. The ancient Greek mathematician Pythagoras staunchly believed that all numbers are rational. He held this conviction so strongly that he established a religion centered around the number 10.
He and his followers would pray, "Bless us, divine number, thou who generated gods and men!" (The significance of "10" remains unclear.) However, according to legend, one of his disciples named Hippasus demonstrated that not all numbers can be expressed as rational fractions, specifically by revealing the existence of the square root of 2. This revelation profoundly challenged Pythagoras's belief system, and in response, he purportedly had Hippasus drowned at sea.
Nevertheless, we now have a clear understanding that not all numbers can be classified as rational. The discovery and acceptance of irrational numbers have expanded our mathematical knowledge and paved the way for further exploration and understanding of the numerical realm.
Real Numbers
Real numbers encompass both rational and irrational numbers. In practical terms, when working with decimals, it is customary to treat them as real numbers. This approach is rooted in the fact that real numbers provide a comprehensive and versatile framework for various mathematical and scientific applications.
By treating decimals as real numbers, we can leverage the rich structure and properties associated with this number system. Real numbers allow for precise measurements and calculations involving continuous quantities, making them indispensable in fields such as physics, engineering, finance, and many other scientific disciplines.
Treating decimals as real numbers offers several advantages. It allows for seamless integration with existing mathematical models and computational algorithms designed for real numbers. It also enables the application of powerful analytical tools and techniques that rely on the properties of real numbers, such as calculus, optimization, and statistical analysis.
In essence, considering decimals as real numbers provides a reliable and convenient approach in practical settings, ensuring that numerical computations and data analysis are conducted effectively and accurately.
Complex and Imaginary Numbers
In addition to real numbers, there exist complex and imaginary numbers, which come into play when taking the square root of a negative number. Imaginary numbers are expressed in the form of "bi," where "b" represents a real number and "i" denotes the imaginary unit (√-1). On the other hand, complex numbers are expressed as the sum of a real part and an imaginary part, usually written as "a + bi," where both "a" and "b" are real numbers.
While complex and imaginary numbers hold significance in specific problem domains, it is worth noting that they might not be extensively discussed or emphasized in this context. However, it is important to acknowledge their existence and their relevance in fields such as electrical engineering, quantum mechanics, and signal processing, where complex numbers play a crucial role in modeling and solving complex phenomena.
For the purpose of this discussion or the focus of the current topic, we will primarily concentrate on real numbers, which form the foundation for many mathematical concepts and practical applications.