Real Numbers

*What is the importance of mathematics in our daily lives? – Reference*

Definition

Real Numbers

Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to represent a complex number. Some of the examples of real numbers are 23, -12, 6.99, 5/2, π, and so on. In this article, we are going to discuss the definition of real numbers, the properties of real numbers and the examples of real number with complete explanations. ¹³

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as ${\sqrt {2}}$ (1.41421356…, the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as π (3.14159265…).¹⁰

*Real numbers can be thought of as points on an infinitely long number line – Wikipedia*

Complex Numbers

In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted i, called the imaginary unit, and satisfying the equation i² = −1. Moreover, every complex number can be expressed in the form a + bi, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols $C$ or C. Despite the historical nomenclature “imaginary”, complex numbers are regarded in the mathematical sciences as just as “real” as the real numbers and are fundamental in many aspects of the scientific description of the natural world.⁶

$N$ Denotes the set of natural numbers {1, 2, 3, …}, or sometimes {0, 1, 2, 3, …}. It is often denoted by $N$ .

$Z$ Denotes the set of integers {-2 ,-1, 0, 1, 2, …}. It is often denoted by $Z$ .

$Q$ Denotes the set of rational numbers (fractions of two integers). It is often denoted by $Q$ .

$R$ Denotes the set of real numbers. It is often denoted by $R$ .

$C$ Denotes the set of complex numbers. It is often denoted by $C$

Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the **modulus** or **absolute value** of z. The angle φ is the **argument** of z. – Wikipedia

Algebraic Numbers

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, $(1+{\sqrt {5}})/2$ , is an algebraic number, because it is a root of the polynomial x² – x – 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number $1+i$ is algebraic because it is a root of x⁴ + 4.

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.⁸

Irrational Numbers

In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no “measure” in common, that is, there is no length (“the measure”), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio π of a circle’s circumference to its diameter, Euler’s number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.

Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.

As a consequence of Cantor’s proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.²

Transcendental Numbers

In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known, in part as it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare.¹

Rational Numbers

In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. For example, −3/7 is a rational number, as is every integer (e.g. 5 = 5/1). The set of all rational numbers, also referred to as “the rationals“, the field of rationals or the field of rational numbers is usually denoted by a boldface Q or $Q$ ; it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for “quotient“, and first appeared in Bourbaki’s Algèbre.⁷

Integers

An integer (from the Latin integer meaning “whole”) is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, …). The set of integers is often denoted by the boldface Z or $Z$ letter “Z”—standing originally for the German word Zahlen (“numbers”).

$Z$ is a subset of the set of all rational numbers $Q$ , which in turn is a subset of the real numbers $R$ . Like the natural numbers, $Z$ is countably infinite.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.⁴

Natural Numbers

In mathematics, the natural numbers are those numbers used for counting (as in “there are six coins on the table”) and ordering (as in “this is the third largest city in the country”). In common mathematical terminology, words colloquially used for counting are “cardinal numbers“, and words used for ordering are “ordinal numbers“. The natural numbers can, at times, appear as a convenient set of codes (labels or “names”), that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense.

Some definitions begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, … Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).⁵

Whole Numbers

Whole number is a colloquial term in mathematics. The meaning is ambiguous. It may refer to either:⁹

Natural number, an element of the set {0, 1, 2, …} or of the set {1, 2, 3, …}
Integer, an element of the set {…, −2, −1, 0, 1, 2, …}

Who

Any student taking mathematics needs to understand the hierarchy of numbers.

What

Just like different family members live in different homes, different numbers are of the same family but have different types. Over time, different patterns of ten digits have been classified into a variety of number types. These patterns of numbers are different from each other due to different representations and properties.¹²

*Some Important Number Sets – iitjeegaming*

Why

In science, the habit of either proving a general concept or of quantifying an observed phenomenon requires a sophisticated use of numbers. Making sense of the data available to astronomers, chemists and biologists among others would be impossible without a system of assigning values and weighing variables. Many scientific ideas, such as the speed of light or the gravitational constant, are impossible to comprehend without the concept of numbers.

Numbers play a role in shaping human societies. Very few national governments are entirely without means of tracking the behavior and opinions of their citizens, and numbers such as social security numbers, resident alien ID numbers and the results of opinion polls are closely watched as an aid to social control and central planning.

Economically, numbers are central to most human activities. An hourly wage is expressed as a number. Units of measurement and quantities of goods are also necessarily expressed as numbers. Speeds and dosages are most efficiently expressed with a discreet numerical value, and it is difficult to picture a commodities market, stock market or futures exchange operating without price data.¹¹

See Theoretical Knowledge Vs Practical Application.

How

Many of the References and Additional Reading websites and Videos will assist you with understanding numbers.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

¹ “Transcendental Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Transcendental_number.

² “Irrational Number – Wikipedia”. 2013. en.wikipedia.org. https://en.wikipedia.org/wiki/Irrational_number.

3 “Whole Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Whole_number.

⁴ “Integer – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Integer.

⁵ “Natural Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Natural_number.

⁶ “Complex Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Complex_number.

⁷ “Rational Number – Wikipedia”. 2013. en.wikipedia.org. https://en.wikipedia.org/wiki/Rational_number.

⁸ “Algebraic Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Algebraic_number.

⁹ “Whole Number – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Whole_number.

¹⁰ “Real Number – Wikipedia”. 2016. en.wikipedia.org. https://en.wikipedia.org/wiki/Real_number.

¹¹ “Why Are Numbers Important?”. 2015. reference.com. https://www.reference.com/world-view/numbers-important-d1523f16da8cc2e4.

¹² “Types of Numbers – Difference and Classification”. 2022. The Story of Mathematics. https://www.storyofmathematics.com/types-of-numbers.

¹³ “Real Numbers (Definition, Properties And Examples)”. 2022. BYJUS. https://byjus.com/maths/real-numbers/.

Additional Reading

“List Of Types Of Numbers – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/List_of_types_of_numbers.

“Number Theory – Types Of Math Numbers | Math Goodies”. 2022. mathgoodies.com. https://www.mathgoodies.com/articles/numbers.

“Types Of Numbers – Definition, Properties, And Examples”. 2022. BYJUS. https://byjus.com/maths/types-of-numbers/.

“Where Transcendental Numbers Hide In Everyday Math | Quanta Magazine”. 2021. Quanta Magazine. https://www.quantamagazine.org/where-transcendental-numbers-hide-in-everyday-math-20211027/.

Videos

What is an Imaginary Number?

Types of Numbers 1 : Natural Numbers, Whole Numbers, Integers

Introduction to rational and irrational numbers | Algebra I | Khan Academy

The Best Explanation of Pi

Pi is IRRATIONAL: animation of a gorgeous proof

What is the Number e?

What do numbers like e, pi, the square root of 2 and the golden ratio all have in common? And why are they different from other numbers?

Transcendental numbers powered by Cantor’s infinities

In today’s video the Mathologer sets out to give an introduction to the notoriously hard topic of transcendental numbers that is both in depth and accessible to anybody with a bit of common sense. Find out how Georg Cantor’s infinities can be used in a very simple and off the beaten track way to pinpoint a transcendental number and to show that it is really transcendental. Also find out why there are a lot more transcendental numbers than numbers that we usually think of as numbers, and this despite the fact that it is super tough to show the transcendence of any number of interest such as pi or e. Also featuring an animated introduction to countable and uncountable infinities, Joseph Liouville’s ocean of zeros constant, and much more.