Hierarchy of Numbers

In mathematics, the hierarchy of numbers refers to the ordering and classification of different types of numbers. There are several different levels in the hierarchy of numbers, including:

Natural Numbers:

The natural numbers are a set of positive integers used for counting and enumeration. They are the numbers 1, 2, 3, 4, 5, and so on. The natural numbers are also known as the counting numbers or positive integers. They play a crucial role in mathematics and are used in a variety of mathematical applications, including number theory, combinatorics, and set theory.

In mathematics, the set of natural numbers is often denoted by the symbol "N". The natural numbers have many properties that make them unique among other number sets. For example, they have a property known as the "well-ordering principle", which states that any non-empty set of natural numbers has a smallest element. Additionally, the natural numbers are the building blocks for the other number sets, such as the integers, rational numbers, and real numbers.

Whole Numbers:

Whole numbers are a subset of the set of integers and include the positive integers (1, 2, 3, 4, etc.) as well as zero. In other words, whole numbers are the numbers that you would use to count a set of objects and include 0.

The set of whole numbers is often denoted by the symbol "W". Whole numbers play a significant role in many mathematical applications, including arithmetic, algebra, and number theory. Additionally, whole numbers are used in computer programming and other technological applications, as they are often used to represent discrete values in a digital system.

Integers:

The integers are the set of whole numbers and their negatives. The set of integers includes positive whole numbers (1, 2, 3, 4, etc.), negative whole numbers (-1, -2, -3, -4, etc.), and zero. The set of integers is often denoted by the symbol "Z".

Integers are used in many mathematical applications, including arithmetic, algebra, and number theory. In addition to the properties shared by the sets of whole numbers and natural numbers, integers also have additional properties related to their positive and negative values. For example, the sum of two positive integers is always positive, while the sum of a positive and negative integer can be positive, negative, or zero.

The set of integers is well-ordered, meaning that it has a first and a last element, and any non-empty subset of the integers has a smallest element. The set of integers also has the property of closure under addition and subtraction, meaning that the sum or difference of any two integers is always an integer.

Rational Numbers:

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not equal to zero. A fraction is a rational number if and only if both the numerator and the denominator are integers. The set of rational numbers is often denoted by the symbol "Q".

Examples of rational numbers include 1/2, -3/4, and 5. The number zero is also a rational number, as it can be expressed as 0/1.

Rational numbers play a significant role in many areas of mathematics, including algebra, geometry, and number theory. They are used to represent quantities that can be expressed as a ratio of two integers, such as lengths, distances, and angles.

Rational numbers have several properties that make them unique among other number sets. For example, they are closed under addition, subtraction, multiplication, and division (provided that the denominator is not equal to zero). Additionally, the set of rational numbers is dense, meaning that between any two distinct rational numbers, there exists an infinite number of other rational numbers.

In mathematics, rational numbers are considered as "basic" numbers, as they can be used to construct other number sets, such as the real numbers and the complex numbers. The set of rational numbers is a subset of the set of real numbers, but not all real numbers are rational.

Irrational Numbers:

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers, and therefore cannot be expressed as a simple fraction. The decimal representation of an irrational number is non-repeating and non-terminating. Some famous examples of irrational numbers include the `\sqrt2`, `\sqrt3`, and π (pi).

Irrational numbers play an important role in many areas of mathematics, including geometry, calculus, and number theory. They are used to represent quantities that cannot be expressed as a simple fraction, such as the length of a diagonal line segment in a square, the circumference of a circle, and the value of certain trigonometric functions.

The set of irrational numbers is a subset of the set of real numbers and has several important properties. For example, irrational numbers are dense in the set of real numbers, meaning that between any two distinct real numbers, there exists an infinite number of irrational numbers. Additionally, the set of irrational numbers is uncountable, meaning that there is no one-to-one correspondence between the irrational numbers and the positive integers.

Real Numbers:

Real numbers are numbers that represent quantities on the number line, including both rational and irrational numbers. The set of real numbers is often denoted by the symbol "R".

Rational numbers are real numbers that can be expressed as a ratio of two integers, such as 1/2, -3/4, and 5. Irrational numbers are real numbers that cannot be expressed as a ratio of two integers, such as the `\sqrt2`, `\sqrt3`, and π (pi).

Real numbers have several important properties that make them useful in many areas of mathematics, including geometry, calculus, and analysis. For example, the set of real numbers is complete, meaning that any non-empty set of real numbers that has an upper bound also has a least upper bound. Additionally, the set of real numbers is ordered, meaning that for any two real numbers a and b, either a < b, a = b, or a > b.

The set of real numbers is also uncountably infinite, meaning that there is no one-to-one correspondence between the real numbers and the positive integers. This property makes the set of real numbers much larger than the set of rational numbers, and allows for the representation of quantities that cannot be expressed as a simple fraction.

Complex Numbers:

Complex numbers are numbers that consist of a real part and an imaginary part, and are represented in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers are used to represent quantities in the complex plane, which is a two-dimensional graph with the real part represented on the x-axis and the imaginary part represented on the y-axis.

Complex numbers have several important properties and applications in many areas of mathematics, including algebra, calculus, and engineering. For example, they can be used to solve equations that have no real solutions, and they play a significant role in the study of alternating current (AC) electrical circuits.

The set of complex numbers is closed under addition, subtraction, multiplication, and division, meaning that the sum, difference, product, and quotient of two complex numbers is also a complex number. Additionally, complex numbers have a notion of magnitude and phase, which can be represented as a vector in the complex plane.

History of Numbers

 Why we need numbers ?

Numbers are essential to our daily lives and play a crucial role in many aspects of human existence. Some of the reasons why we need numbers include:

• Measurement: Numbers are used to measure quantities, such as length, weight, volume, and time. This allows us to quantify and compare objects, events, and phenomena in a consistent and meaningful way.

• Calculation: Numbers are used to perform mathematical operations, such as addition, subtraction, multiplication, and division. This allows us to make quantitative predictions, solve problems, and make decisions.

• Communication: Numbers are used to communicate information, such as telephone numbers, addresses, and dates. They are also used to represent data and statistics, which help us to make informed decisions and understand the world around us.

• Science and Technology: Numbers play a crucial role in many fields of science, including physics, chemistry, and engineering. They are used to describe natural phenomena and to model complex systems, such as the weather, the economy, and the behavior of living organisms.

• Commerce and Finance: Numbers are used to represent monetary values, to keep track of financial transactions, and to make investment decisions. They are also used to calculate taxes, loans, and other financial obligations.

In short, numbers provide us with a way to quantify, describe, and understand the world, making them an essential tool in many aspects of human life.

History of Numbers

The history of numbers, or numerals, dates back to the earliest civilizations. People have been counting and recording numbers for thousands of years in various forms.

Tally Marking

Bone artifacts bearing incised markings seem to indicate that the people of the Old Stone Age had devised a system of tallying by groups as early as 30,000 B.C. The most impressive example is a shinbone from a young wolf, found in Czechoslovakia in 1937; about 7 inches long, the bone is engraved with 55 deeply cut notches, more or less equal in length, arranged in groups of ve. (Similar recording notations are still used, with the strokes bundled. Voting results in small towns are still counted in the manner devised by our remote ancestors.) For many years such notched bones were interpreted as hunting tallies and the incisions were thought to represent kills. The markings on bones discovered in French cave sites in the late 1880s are grouped in sequences of recurring numbers that agree with the numbers of days included in successive phases of the moon. One might argue that these incised bones represent lunar calendars.

Ancient Egyptians Number System (Hieroglyphic)

The ancient Egyptians used a hieroglyphic numeral system to represent numbers. This system was based on the use of symbols to represent quantities, similar to the way we use numerals today. In the Egyptian numeral system, there were symbols for the numbers 1, 10, 100, and 1,000. To write larger numbers, these symbols were combined in a similar way to the way we use place value in our modern decimal numeral system. For example, the symbol for 100 was used to represent 100, while the symbol for 10 was used to represent 10, 20, 30, etc.

For example:- the number 142,136 = 1*100,000 + 4*10,000 + 2*1,000 + 1*100 + 3*10 + 6*1

is represented in hieroglyphic as

The Egyptian numeral system was a decimal system, meaning that it was based on the number 10. This made it possible for the ancient Egyptians to perform mathematical operations, such as addition, subtraction, multiplication, and division, using the symbols.

Lets see the example of addition

It is worth noting that while the hieroglyphic numeral system was used for mathematical purposes, it was also used to represent quantities in other contexts, such as measuring quantities of grain, recording the number of people in a population, or counting the number of objects in a collection.

Overall, the hieroglyphic numeral system was an important tool for the ancient Egyptians, allowing them to perform calculations, record information, and communicate quantitative information in a consistent and meaningful way.

Greek Alphabetic Numeral System

Around the fifth century B.C., the Greeks of Ionia also developed a ciphered numeral system, but with a more extensive set of symbols to be memorized. They ciphered their numbers by means of the 24 letters of the ordinary Greek alphabet, augmented by three obsolete Phoenician letters. The resulting 27 letters were used as follows. The initial nine letters were associated with the numbers from 1 to 9; the next nine letters represented the next nine integral multiples of 10; the next nine letters were used for the next nine integral multiples of 100. The following table shows how the letters of the alphabet (including the special forms) were arranged for use as numerals.

Because the Ionic system was still a system of additive type, all numbers between 1 and 999 could be represented by at most three symbols. The principle is shown by ψπδ = 700 + 80 + 4 =784

Multiplication example

Hindu-Arabic Numeral

The Hindu-Arabic numeral system is a decimal numeral system that is used widely in the world today. It is also known as the base-10 numeral system and the decimal system. The Hindu-Arabic numeral system uses ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, to represent numbers. The symbols are combined to represent larger numbers and to perform mathematical operations. The use of zero is indeed one of the key features of the Hindu-Arabic numeral system that sets it apart from other numeral systems. The concept of zero as a number in its own right was first introduced by Indian mathematicians, and it was incorporated into the Hindu-Arabic numeral system as a symbol that represented the absence of value.

The Hindu-Arabic numeral system originated in India and was developed from the earlier Indian numeral system. It was later transmitted to the Islamic world and eventually to Europe, where it was adopted and became widely used. The Hindu-Arabic numeral system has several advantages over other numeral systems, such as the ability to perform arithmetic operations quickly and easily, the flexibility to represent very large and very small numbers, and the ease of use and compactness of the symbols.

The Hindu-Arabic numeral system has been adopted as the standard numeral system in most countries and has become an essential tool in commerce, science, and technology. Its widespread use has also helped to promote communication and collaboration between people and cultures, and has made it possible to perform complex calculations and to analyze data in a way that was not possible with earlier numeral systems.

Above we show some of the civilization and their numeral system but in the human history we see many civilization that use their own numeral system like

• One of the earliest known numeral systems was developed by the Sumerians in Mesopotamia around 4000 BCE. They used a system of cuneiform symbols to represent numbers.

• The Babylonians, who lived in Mesopotamia around 2000 BCE, used a sexagesimal (base-60) numeral system, which was later adopted by the ancient Mayans in Central America.

• The ancient Romans used a system of letters and symbols, known as Roman numerals, which is still used today for certain purposes, such as in the numbering of chapters in books and in the numbering of movie sequels.

• The Chinese numeral system is a traditional numeral system used in China and other East Asian countries, such as Japan and Korea. It is a non-positional numeral system, which means that the value of a symbol depends on its position relative to other symbols. One unique aspect of the Chinese numeral system is that it has two sets of symbols: one for counting objects, and one for representing numbers in financial transactions. This is because the characters for the numbers 1 to 9 have different forms in the two sets of symbols, with the financial forms being more complex and ornate.

Throughout history, people have continued to develop new methods of representing numbers and calculating with them. The invention of computers and the development of digital technology has allowed us to perform complex mathematical operations on a massive scale, revolutionizing the way we use and understand numbers.