МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
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Том 2, Выпуск 12,
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Декабрь
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EXPANSION OF NUMBERS
Khudaikulova Saida Zakirovna
Teacher of Termez State Pedagogical Institute
Phone: +99890-246-47-47
Niyazova Dildora Karomidinzoda
2nd-year student of
Temez State Pedagogical Institute
Keywords:
expansion of numbers, mathematical expansions, irrational
numbers, decimal fractions, operations on numbers, expansion.
Annotation:
The topic of
expansion of numbers
(or
mathematical expansions
)
involves representing numbers in various forms and studying their
interrelationships. This field is especially relevant in mathematical analysis and
algebra. The study of number expansions primarily focuses on their fractional,
rational, or decimal representations and exploring their logical and analytical
properties. Key concepts in number expansions include decimal expansions,
fractional expansions, and periodic expansions. Decimal expansions are
particularly important for calculating precise values in the real number system.
Fractional expansions, on the other hand, help explore the properties of numbers
expressed in specific forms.
The expansion of numbers is a mathematical method of representing numbers
in various forms. This concept is primarily used to express numbers in decimal,
fractional, or many other formats. To gain a deeper understanding of the
expansion of numbers, let us explore the following key topics.
Decimal Expansion
The decimal expansion is a method of expressing numbers in decimal form.
For example, 3.14 is a decimal expansion consisting of an integer part (3) and a
fractional part (0.14). Decimal expansions are widely used to represent precise
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
282
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values of numbers and can be rational or irrational. Types of decimal expansions
include:
1.
Terminating Decimals
: These have a finite number of digits in their fractional
part. Examples: 0.5, 0.75, 1.25.
2.
Repeating Decimals
: These decimals have a recurring sequence of digits.
Examples: 0.3333... (recurring 3), 0.142857... (recurring sequence 142857).
Fractional Expansion
Fractional expansion involves expressing a number as the ratio of two integers.
For example, the fraction {5}{8} equals the decimal expansion 0.625. Fractions
are useful for defining numbers and understanding relationships between them.
Types of fractional expansions include:
1.
Proper Fractions
: Represented directly by two integers, like {3}{4}.
2.
Improper Fractions
: Can be converted to decimals but may not terminate (e.g.,
repeating or irrational numbers).
Expansion of Irrational Numbers
Irrational numbers have non-terminating, non-repeating decimal expansions.
These cannot be expressed as a ratio of integers. Examples include π\pi
(3.14159...) and (1.41421...). Such numbers produce an infinite sequence of non-
repeating digits in their decimal form.
Periodic Numbers
Periodic numbers are decimals with a recurring sequence of digits. Examples
include 0.3333... 0.3333... (repeating 3) and 0.142857... 0.142857... (repeating
sequence 142857). These are also known as repeating decimals and are
characterized by a repeating block of digits.
Applications of Number Expansions
The expansions of numbers play a significant role in daily life, science,
technology, economics, and engineering. Key applications include:
1.
Finance and Accounting
: Decimal expansions are crucial for monetary
calculations and interest rate computations.
2.
Computer Science and Digital Systems
: Numbers are often converted into
binary, decimal, or other expansions for processing.
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
283
https://universalpublishings.com
3.
Geometry and Trigonometry
: The expansion of π\pi is essential in many
geometric and trigonometric calculations.
Operations with Number Expansions
1.
Converting Fractions to Decimals
: Perform division to obtain the decimal
form.
2.
Converting Decimals to Fractions
: Simplify the decimal into a fraction with a
finite denominator.
3.
Arithmetic Operations
: Decimal and fractional expansions facilitate addition,
subtraction, multiplication, and division by aligning or simplifying forms as
necessary.
Advanced Applications
1.
Mathematical Analysis
: Expansions are used in limits, integrals, and
differentiation. For instance, limits can reveal properties of irrational number
expansions.
2.
Binary Conversions
: Decimal expansions can be converted into binary for
storage and computation in computer systems.
Conclusion
Number expansions form the foundation of mathematical representation and
operations. Understanding various types (decimal, fractional, irrational) and their
properties is essential in mathematics, scientific research, and technological
applications. These expansions are fundamental in solving equations, performing
calculations, and enabling high-precision computations.
References
1.
Xudaykulova, S. (2024). DARAJALI GEOMETRIYA - KO‘PHADLAR VA
NORMAL KONUSLAR. Interpretation and Researches, 1(1). извлечено от
https://interpretationandresearches.uz/index.php/iar/article/view/2496
Mathematical Analysis
(Yu. M. Geller, L. D. Faddeev).
2.
Xudaykulova , S. (2024). TEXNIK IJODKORLIKNING HOZIRGI HOLATI.
Research
and
Implementation.
извлечено
от
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
284
https://universalpublishings.com
3.
Ne’matova , D. (2023). BOSHLANG‘ICH SINF O‘QUVCHILARIDA
TANQIDIY FIKRLASH KO‘NIKMALARINI SHAKLLANTIRISHNING
PEDAGOGIK-PSIXOLOGIK
XUSUSIYATLARI.
Interpretation
and
Researches,
2(1).
извлечено
от
https://interpretationandresearches.uz/index.php/iar/article/view/973
4.
Холмуминова, А. (2023). ОСОБЕННОСТИ И ПРЕИМУЩЕСТВА
ФОРМИРОВАНИЯ
КОМПЕТЕНТНОСТИ
ПОДГОТОВКИ
ИННОВАЦИОННОЙ ДЕЯТЕЛЬНОСТИ У БУДУЩИХ УЧИТЕЛЕЙ
НАЧАЛЬНЫХ КЛАССОВ. Interpretation and Researches, 2(1). извлечено от
https://interpretationandresearches.uz/index.php/iar/article/view/1145
Algebra
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(S. I. Adishchev, Yu. L. Ershov).
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Mathematics
(Educational Textbooks in Uzbekistan).
