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Issue 14(49), Volume 1 | ISSN 3030-377X | 15.06.2025
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INTERNATIONAL SCIENTIFIC JOURNAL
THE IMPORTANCE AND APPLICATIONS OF COMPARISON THEORY IN
MODERN SCIENCE AND TECHNOLOGY
Feruza Eshimova Kenjaboyevna
Samarkand campus of the University of Economics and pedagogy
Assistant of the Department "economics and engineering sciences "
Abstract:
This article analyzes the significance and applications of comparison
theory, one of the important branches of algebra and number theory, in modern
science and technology. Although the section is theoretically deep and complex, its
practical application is very wide. Behind today's technological developments lie the
basic principles of this theory. Therefore, in-depth study of the topic is beneficial for
students interested in any mathematical and technological field.
Key words:
Integer, comparison, modulus, remainder, equation, inequality,
degree, root, divisor, row, sequence.
INTRODUCTION
Comparison theory is one of the fundamental and interesting branches of
mathematics, which studies the relationships arising from the remainders of numbers
and is applied in many fields. This theory was first discovered and developed by the
great mathematician Karl Friedrich Gauss in 1801. In recent years, innovations and
research have been carried out based on this theory. Methods for solving systems of
inequalities of two variables using comparisons have been developed, and these
methods are of great importance in the development of students' mathematical
thinking. This theory is the basis of modern cryptographic algorithms and plays an
important role in ensuring information security, and recent research is aimed at
developing new methods and algorithms in this area. In addition, comparisons are
constantly used in solving problems and problems in mathematical olympiads (for
example, Fermat's small theorem, Wilson's theorem, Euler's theorem). This theory
plays an important role in the development of students' mathematical thinking in
solving mathematical proofs, correspondences, and logical problems.
MATERIALS AND METHODS
Looking at the history of comparison theory and its influence on modern
mathematics, we can see that its historical development occurred step by step in the
following form: In the pre-Christian era, ancient Babylonian and Egyptian
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Issue 14(49), Volume 1 | ISSN 3030-377X | 15.06.2025
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mathematicians had some methods for working with residuals, which were mainly
used for practical calculations. For example, the process of finding the GCD using the
"Euclidean algorithm" (300 BC) paved the way for modular arithmetic; Chinese
mathematicians developed the "Chinese Remainder Theorem" (III-V centuries),
which helps to solve a system of comparisons for several modules.
For example:
{
𝑥 ≡ 2 (𝑚𝑜𝑑 3)
𝑥 ≡ 3 (𝑚𝑜𝑑 5)
𝑥 ≡ 2 (𝑚𝑜𝑑 7)
It was during this period that methods for solving such systems appeared.
Diophantus (III century) - studied complex algebraic equations for integers, and later
these works became known as Diophantine equations. Pierre Fermat (17th century)
proved Fermat's small theorem and laid the foundation for comparison theory:
Theorem, "If p is a prime number, then
𝑎
𝑝−1
≡ 1 (𝑚𝑜𝑑 𝑝)
" Leonard Euler (1707-
1783) - Euler developed the basics of modular arithmetic and introduced the Euler
function (
φ
function) to science. In Gauss's work "Disquisitiones Arithmeticae," the
theory of comparisons was reinforced, introduced to science by Karl Friedrich Gauss
(1777-1855), and the concepts of "power series" and "square residues" appeared in
modular arithmetic
.
When solving first-degree comparisons, there are methods of selection, the
method of mastering coefficients, the use of Euler's theorem, and the use of
continuous fractions. Based on these methods, comparisons are carried out and results
are obtained.
REZALTS AND DISCUSSIONS
In the history of mathematics, new research is being conducted on the
development of comparison theory and its influence on modern mathematics. Based
on these innovations, we can see the importance of this theory in modern science and
technology. Comparisons are modified in each area according to their
interrelationships and methods of comparison. To carry out the comparison correctly,
it is necessary to obtain data, analyze it, and derive the results. The proof of
equations, inequalities, and statements often presents difficulties. In this case,
comparisons are useful. When solving examples and problems that lead to
comparisons, we often use properties derived from the definition of a comparison.
Here are some of the basic properties of comparisons:
1
0
. 𝑎 ≡ 𝑎(𝑚𝑜𝑑 𝑚)
2
0
.
If
𝑎 ≡ 𝑏(𝑚𝑜𝑑 𝑚)
, then
𝑏 ≡ 𝑎(𝑚𝑜𝑑 𝑚)
3
0
.
If
𝑎 ≡ 𝑏(𝑚𝑜𝑑 𝑚)
and
𝑏 ≡ 𝑐(𝑚𝑜𝑑 𝑚),
then
𝑎 ≡ 𝑐(𝑚𝑜𝑑 𝑚)
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Issue 14(49), Volume 1 | ISSN 3030-377X | 15.06.2025
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INTERNATIONAL SCIENTIFIC JOURNAL
4
0
.
Comparisons can be added term by term and multiplied term by term, if
𝑎 ≡ 𝑎(𝑚𝑜𝑑 𝑚)
and
𝑐 ≡ 𝑑(𝑚𝑜𝑑 𝑚),
then
𝑎 + 𝑐 ≡ 𝑏 + 𝑑(𝑚𝑜𝑑 𝑚)
and
𝑎𝑐 ≡ 𝑏𝑑(𝑚𝑜𝑑 𝑚)
5
0
.
Both sides of the comparison can be multiplied by an arbitrary number, i.e.,
if
𝑎 ≡ 𝑏(𝑚𝑜𝑑 𝑚)
then
𝑎𝑐 ≡ 𝑏𝑐(𝑚𝑜𝑑 𝑚)
6
0
.
If
𝑎 ≡ 𝑏(𝑚𝑜𝑑 𝑚)
then
𝑎
𝑛
≡ 𝑏
𝑛
(𝑚𝑜𝑑 𝑚).
Here n is an arbitrary natural
number.
7
0
.
An arbitrary part of the comparison can be added to a multiple of the
modulus: a
∈
b (mod m) and if k, l
∈
Z
→
then
𝑎 + 𝑘𝑚 ≡ 𝑠(𝑚𝑜𝑑 𝑚)
and
𝑎 ≡ 𝑠 +
𝑙𝑚(𝑚𝑜𝑑 𝑚)
Example: If n is an odd number, then prove that
𝑛
2
− 1
is divisible by 8.
Solution: If n is an odd number, then we write it in the form n=2k+1, k=1,2,3,...
Now, using the definition of comparison, we can write that, substituting n=2k+1
instead of n, we arrive at the following comparison, i.e.,
(2𝑘 + 1)
2
− 1 ≡ 0(𝑚𝑜𝑑 8)
4𝑘
2
+ 4𝑘 + 1 − 1 ≡ 0(𝑚𝑜𝑑 8)
4𝑘
2
+ 4𝑘 ≡ 0(𝑚𝑜𝑑 8)
4𝑘(𝑘 + 1) ≡ 0(𝑚𝑜𝑑 8)
Since k+1 is even, it follows that the number 4k (k+1) is divisible by 8.
CONCLUSION
In conclusion, through the theory of comparisons, it is possible to develop the
intellectual potential of students and increase the effectiveness of the educational
process. Although this section is theoretically deep and complex, its practical
application is very wide. Behind today's technological developments lie the basic
principles of this theory. Therefore, a deep study of this topic is very useful and
necessary for anyone interested in any mathematical and technological field.
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Issue 14(49), Volume 1 | ISSN 3030-377X | 15.06.2025
SCIENCE SHINE
INTERNATIONAL SCIENTIFIC JOURNAL
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