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THE HISTORY OF IRRATIONAL NUMBERS
Tashkhodjayev Abdugaffor
Rakhmonova Nilufar
Lecturer, Department of Digital Technologies and Mathematics, Kokand University
Number is one of the fundamental concepts in mathematics and has emerged from
practical human needs. The origin and development of numbers can be described in the
following early stages:
Natural numbers arose from the need to measure and distribute quantities.
Positive numbers were created due to the needs of mathematics itself, namely, to solve
and justify algebraic equations. Zero appeared as a result of introducing negative
numbers. This list could be extended further, but we will now turn to the history of
irrational numbers, which appeared after the aforementioned types of numbers.
In the Pythagorean school (5th century BC), it was proven that rational numbers are not
sufficient to precisely measure all line segments; there exist segments that are
incommensurable. For instance, the side of a square with area 2 is not commensurable
with its diagonal. This is proven through contradiction in Euclid’s "Elements".
This discovery contradicted Pythagorean doctrine, which held that any quantity could
be expressed through whole numbers and their ratios. Initially, they attempted to keep
this discovery secret.
Hippasus of Metapontum (5th century BC) continued this work, and by the end of the
same century, Theodorus of Cyrene demonstrated that the sides of squares with areas 3,
5, 6, 7, 8, 10, 11, 12, 13, 14, 15, and 17 are not commensurable with the side of a unit
square—i.e., they are irrational. Theaetetus generalized this idea by proving the
irrationality of
N
for any whole number N that is not a perfect square. Realizing that
infinitely many segments and geometric quantities cannot be measured using whole or
fractional numbers, the Pythagoreans attempted to base geometry and algebra not on
numbers but on geometry itself. Thus, geometric algebra was created and developed.
Based on this, mathematicians began to represent whole numbers and any quantity
using line segments, rectangles, and other geometric shapes.
In the Arab East, mathematics began to develop from the 7th century onwards. In this
period, many Central Asian scholars made significant discoveries related to the concept
of numbers, such as: Al-Khwarizmi (783–850), Abu Rayhan Biruni(973–1048),
Avicenna (Ibn Sina) (980–1037), Abu Nasr al-Farabi (873–950), Omar Khayyam
(1048–1131), Some of their contributions include:
1. Development of methods for extracting square roots from numbers
2. Discovery of decimal fractions
3. Expansion of the concept of positive real numbers
Although Al-Khwarizmi, in his work On the Calculation with Hindu Numerals,
provided a detailed explanation of the decimal system, it only started being widely used
300 years later.
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Negative numbers were first explicitly mentioned in the French mathematician Nicolas
Chuquet’s (1445–1500) work Le Triparty en la science des nombres (1484; published in
Lyon in 1848). However, initial notions of negative numbers already existed in the
works of Indian and Chinese mathematicians. For example, Chinese mathematicians
used negative numbers implicitly when solving systems of five linear equations with
five unknowns.
The Indian mathematician Brahmagupta (598–660) described negative numbers as
“debts.” He stated the following rules: “The sum of two debts is a debt.” “The sum of
zero and a debt is a debt.” He referred to a positive number as a “property,” thus
defining the sum of “property” and “debt” as their difference. If they are equal, the
result is zero.
Arab mathematicians used metaphors: negative signs as “enemies” and positive signs as
“friends,” and they interpreted the signs of the product of numbers with real-life rules.
In the field of irrational numbers, Persian mathematician al-Karaji (died 1016) in his
book Al-Fakhri discussed evaluating polynomials containing square and cube roots. He
also performed transformations on simple cube roots, such as simplifying expressions
like √a + √b.
The term “rational” comes from the Latin ratio, meaning “ratio,” and “irrational” means
not rational. Originally, these terms were applied to measurable and immeasurable
quantities. Roman mathematicians Martianus Capella and Cassiodorus in the 5th and
6th centuries translated these terms into Latin as rational and irrational, respectively.
In Euclid’s Elements, irrational numbers are discussed from a geometric perspective.
By the beginning of the Common Era, unlike Greek geometric algebra, in the Eastern
countries both geometry and arithmetic-based algebra began to develop rapidly. Plane
and spherical trigonometry and the computational methods needed for astronomy were
also improved.
Despite the fact that Eastern mathematicians in India, Central Asia, and the Near East
could not work without irrational numbers while developing algebra, trigonometry, and
astronomy, they still hesitated to fully accept these numbers. The Greeks called
irrational quantities alogos (unspeakable), and the Arabs referred to them as
asami(mute).
In the 16th century, Italian mathematician Rafael Bombelli (1526–1572) and Dutch
mathematician Simon Stevin (1548–1620) considered irrational numbers to be even
more powerful than rational ones.
Even before them, many mathematicians of the Near and Far East had widely used
irrational numbers in algebra. For example, Omar Khayyam, in his work Commentaries
on Difficult Postulates of Euclid, introduced the idea of divisible units and a
generalized number concept, referring to them as “numbers.” This generalized concept
included both rational and irrational numbers.
Thus, Omar Khayyam modernized the ancient concept of numbers, defining ratios of
quantities as numbers themselves. These ratios were the new kind of numbers—rational
in the old sense but general numbers in the new sense.
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Overall, Khayyam showed that there is no essential difference between irrational
quantities and numbers, thereby expanding the concept of numbers to positive real
numbers.
In this field, the Azerbaijani mathematician Nasir al-Din al-Tusi (1201–1274)
also made significant contributions. In his works Treatise on the Complete
Quadrilateral and Commentary on Euclid, he further developed the theory of
proportions and teachings about numbers.
His book Commentary on Euclid (Tahrir Uqlidis), which was renowned in both the East
and later in medieval Europe, exists in two versions: one brief and another extended
version with 10 books, published in Rome in 1594. In it, the scholar elaborates on
square irrationalities and gives the following definition of a rational quantity: “Any
quantity that is in ratio with a given quantity is called rational, wherein a number is in
ratio with another number.” Otherwise, it is called irrational. An irrational quantity, in
relation to another quantity, is like the ratio of a number to another when the first is
irrational. For example:
√2 or √3
In Europe, Simon Stevin wrote about decimal fractions about 150 years after Al-Kashi,
in 1585. In 1594, in another work Algebraic Supplements, he developed the ideas from
his earlier work and showed that decimal fractions could be used to approximate real
numbers infinitely closely. Thus, in the 16th century, the introduction and formal
justification of the concept of irrational numbers led to the creation of the idea of
decimal computation.
The publication of the book Geometry (1637) by the great French philosopher,
mathematician, physicist, and physiologist René Descartes (1596–1650) helped clarify
the link between irrational numbers and the measurement of arbitrary segments. On the
number line, irrational numbers were represented as points, just like rational numbers.
This geometric representation made it easier to understand the nature of irrational
numbers and facilitated their acceptance.
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