Use "Continued Fraction" in a sentence | "Continued Fraction" sentence examples
Continued fraction is a mathematical concept that represents a real number as an infinite sequence of fractions. It is a powerful tool used in various branches of mathematics, including number theory, analysis, and approximation theory. In this article, we will explore the meaning of continued fraction and provide tips on how to use this term effectively in sentences.
1. Definition and Explanation: A continued fraction is an expression of the form [a0; a1, a2, a3, ...], where a0 is an integer and ai is a positive integer for i ?
1. The semicolon (;) separates the whole number part (a0) from the fractional part (a1, a2, a3, ...). The ellipsis (...) indicates that the sequence continues indefinitely. Each term in the sequence is called a partial quotient.
2. Using "continued fraction" in a sentence: - "The continued fraction [3; 1, 4, 1, 5, ...] represents the irrational number ? (pi)." - "To approximate the square root of 2, we can use the continued fraction [1; 2, 2, 2, ...]." - "The continued fraction expansion of the golden ratio ? is [1; 1, 1, 1, ...]."
3. Explaining the concept: - "Continued fractions provide an alternative way to represent real numbers, especially irrational ones, with remarkable properties." - "The beauty of continued fractions lies in their ability to capture the essence of a number by expressing it as an infinite sequence of rational approximations." - "By truncating a continued fraction at a certain point, we obtain increasingly accurate rational approximations of the original number."
4. Applications and significance: - "Continued fractions have applications in number theory, where they are used to solve Diophantine equations and study the properties of quadratic irrationals." - "In analysis, continued fractions are employed to study the convergence of series and to approximate functions." - "Continued fractions are also used in cryptography, as they provide a secure method for encoding and decoding messages."
5. Historical context: - "The concept of continued fractions dates back to ancient Greece, with early contributions from mathematicians like Euclid and Archimedes." - "The study of continued fractions gained significant attention during the 18th and 19th centuries, with mathematicians such as Leonhard Euler and Joseph Lagrange making important discoveries." - "Continued fractions continue to be an active area of research, with modern mathematicians exploring their connections to other fields and developing new algorithms."
6. Related terms and concepts: - "Convergents: In continued fractions, the convergents are the rational numbers obtained by truncating the sequence at a certain point." - "Periodic continued fractions: Some numbers have a repeating pattern in their continued fraction expansion, resulting in a periodic continued fraction." - "Best rational approximations: Continued fractions provide a systematic way to find the best rational approximations of a given number."
In conclusion, continued fractions are a fascinating mathematical concept that allows us to represent real numbers as infinite sequences of fractions. By understanding the definition, applications, and historical context of continued fractions, you can confidently use this term in various sentences related to mathematics, number theory, and approximation theory.
In the remaining portion of this article, additional example sentences are presented to demonstrate the usage of the term "Continued Fraction" within sentences.
"Continued Fraction"
(1) The continued fraction of e
(2) Continued fraction is often used in number theory.
(3) Determine the convergents of the continued fraction.
(4) The continued fraction of pi is [3; 7, 15, 1, 292, ...].
(5) The concept of continued fraction dates back to ancient times.
(6) The continued fraction expansion of π is an infinite sequence.
(7) The continued fraction of the golden ratio is [1; 1, 1, 1, ...].
(8) The continued fraction of e is [2; 1, 2, 1, 1, 4, 1, 1, 6, ...].
(9) The concept of continued fraction is widely used in mathematics.
(10) The continued fraction representation of a number can be infinite.
Sentence For "Continued Fraction"
(11) Continued fraction can be used to solve certain types of equations.
(12) The continued fraction of the square root of 5 is [2; 4, 4, 4, ...].
(13) The continued fraction of the square root of 10 is [3; 6, 6, 6, ...].
(14) Convergents can be used to estimate the value of a continued fraction.
(15) The continued fraction of the golden ratio squared is [2; 2, 2, 2, ...].
(16) The continued fraction expansion of a number can be computed iteratively.
(17) If the convergents approach a limit, the continued fraction is convergent.
(18) Continued fraction can be used to solve problems in finance and economics.
(19) Continued fraction is a powerful tool for approximating irrational numbers.
(20) Continued fraction can be used to solve problems in physics and engineering.
"Continued Fraction" In A Sentence
(21) Continued fraction can be used to analyze the convergence of certain series.
(22) Continued fraction is a topic of study in both pure and applied mathematics.
(23) The continued fraction of the square root of 3 is [1; 1, 2, 1, 2, 1, 2, ...].
(24) Convergents can be used to determine the periodicity of a continued fraction.
(25) The continued fraction expansion of the square root of 2 is [1; 2, 2, 2, ...].
(26) Continued fraction is a useful tool for studying the behavior of real numbers.
(27) The continued fraction representation of a number can be unique or non-unique.
(28) The continued fraction algorithm allows for efficient computation of fractions.
(29) The continued fraction expansion of the number provided a series of convergents.
(30) The continued fraction expansion of the square root of 2 is known to be infinite.
"Continued Fraction" Sentence
(31) The continued fraction of the Euler's number is [2; 1, 2, 1, 1, 4, 1, 1, 6, ...].
(32) If the convergents are converging to an integer, the continued fraction is exact.
(33) The continued fraction expansion of the golden ratio is particularly fascinating.
(34) The continued fraction representation of e is an important mathematical constant.
(35) If the convergents are equal to the actual value, the continued fraction is exact.
(36) If the convergents are getting farther apart, the continued fraction is diverging.
(37) The continued fraction representation of a number can reveal interesting patterns.
(38) The continued fraction of the square root of 7 is [2; 1, 1, 1, 4, 1, 1, 1, 4, ...].
(39) Convergents can be found by applying a recursive formula to the continued fraction.
(40) Continued fraction can be used to find the best rational approximation of a number.
"Continued Fraction" Sentence Examples
(41) The convergents of the continued fraction were used to approximate the square root.
(42) The continued fraction representation of the golden ratio is particularly interesting.
(43) If the convergents are converging slowly, the continued fraction is converging slowly.
(44) Continued fraction can be used to solve problems in computer science and cryptography.
(45) If the convergents are converging to a rational value, the continued fraction is exact.
(46) If the convergents are converging rapidly, the continued fraction is converging rapidly.
(47) The continued fraction expansion of √2 is an example of an infinite periodic sequence.
(48) Continued fraction can be used to find the best rational approximation of a square root.
(49) The continued fraction expansion of pi is an irrational number with no repeating pattern.
(50) The continued fraction of the Euler-Mascheroni constant is [0; 1, 1, 2, 1, 2, 1, 4, ...].
Sentence With "Continued Fraction"
(51) The continued fraction of the natural logarithm of 2 is [0; 1, 2, 1, 1, 4, 1, 1, 6, ...].
(52) The continued fraction of the square root of 13 is [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...].
(53) College students should be able to convert an irrational number into a continued fraction.
(54) If the convergents are oscillating between two values, the continued fraction is periodic.
(55) If the convergents are converging to a repeating value, the continued fraction is periodic.
(56) The continued fraction representation of a number can be used to calculate its convergents.
(57) The convergents of the continued fraction were used to approximate the value of a square root.
(58) The mathematician studied the convergents of the continued fraction to understand its behavior.
(59) The continued fraction representation of the imaginary unit j is used in electrical engineering.
(60) The convergents of the continued fraction were used to solve a challenging mathematical problem.
Use "Continued Fraction" In A Sentence
(61) The continued fraction expansion of a number can be used to determine its irrationality measure.
(62) The concept of a continued fraction is often used in mathematics to represent irrational numbers.
(63) If the convergents are converging to an irrational value, the continued fraction is non-periodic.
(64) Continued fraction is a fascinating subject that offers insights into the nature of real numbers.
(65) The continued fraction expansion of a number can be used to determine its periodicity properties.
(66) The continued fraction expansion of the golden ratio is closely related to the Fibonacci sequence.
(67) If the convergents are getting closer to each other, the continued fraction is converging rapidly.
(68) Continued fraction is a versatile tool that finds applications in various branches of mathematics.
(69) The convergents of the infinite continued fraction were used to approximate the irrational number.
(70) If the convergents are converging to a non-repeating value, the continued fraction is non-periodic.
Sentence Using "Continued Fraction"
(71) Continued fraction can be used to find the best rational approximation of a trigonometric function.
(72) The convergents of the infinite continued fraction were computed to estimate the irrational number.
(73) The convergents of the continued fraction expansion were calculated to approximate the square root.
(74) The continued fraction representation of the Euler's number is an interesting mathematical curiosity.
(75) The continued fraction representation of the imaginary unit i is an interesting mathematical concept.
(76) The continued fraction algorithm allows us to compute the best rational approximations of a given number.
(77) If the convergents are converging to a non-integer rational value, the continued fraction is approximate.
(78) The continued fraction expansion of a number can be used to calculate its continued fraction convergents.
(79) The continued fraction representation of a number can be used to study its continued fraction coefficients.
(80) The continued fraction representation of the Euler-Mascheroni constant is an important mathematical constant.
Sentences With "Continued Fraction"
(81) The continued fraction expansion of the square root of 3 is another example of an infinite continued fraction.
(82) The continued fraction expansion of the square root of 5 is another example of an infinite continued fraction.
(83) The continued fraction representation of a number can be used to study its diophantine approximation properties.
(84) The continued fraction expansion of the square root of 7 is yet another example of an infinite continued fraction.
Learning English Faster Through Complete Sentences With "Continued Fraction"
Sentences are everywhere.
Without sentences, language doesn’t really work.
When you first started learning English, you may have memorized words such as English meaning of the word "Continued Fraction"; But now that you have a better understanding of the language, there’s a better way for you to learn meaning of "Continued Fraction" through sentence examples.
True, there are still words that you don’t know. But if you learn whole sentences with "Continued Fraction", instead of the word "Continued Fraction" by itself, you can learn a lot faster!
Focus Your English Learning On Sentences With "Continued Fraction".
Why Is Focusing on Sentences Important?
Sentences are more than just strings of words. They’re thoughts, ideas and stories. Just like letters build words, words build sentences. Sentences build language, and give it personality.
Again, without sentences, there’s no real communication. If you were only reading words right now, you wouldn’t be able to understand what I’m saying to you at all.
- The Word "Continued Fraction" in Example Sentences.
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- 10 examples of sentences "Continued Fraction".
- 20 examples of simple sentences "Continued Fraction".
All the parts of speech in English are used to make sentences. All sentences include two parts: the subject and the verb (this is also known as the predicate). The subject is the person or thing that does something or that is described in the sentence. The verb is the action the person or thing takes or the description of the person or thing. If a sentence doesn’t have a subject and a verb, it is not a complete sentence (e.g., In the sentence “Went to bed,” we don’t know who went to bed).
Four Types Of Sentence Structure.
Simple Sentences With "Continued Fraction"
A simple sentence with "Continued Fraction"contains a subject and a verb, and it may also have an object and modifiers. However, it contains only one independent clause.
Compound Sentences With "Continued Fraction"
A compound sentence with "Continued Fraction" contains at least two independent clauses. These two independent clauses can be combined with a comma and a coordinating conjunction or with a semicolon.
Complex Sentences With "Continued Fraction"
A complex sentence with "Continued Fraction" contains at least one independent clause and at least one dependent clause. Dependent clauses can refer to the subject (who, which) the sequence/time (since, while), or the causal elements (because, if) of the independent clause.
Compound-Complex Sentences With "Continued Fraction"
Sentence types can also be combined. A compound-complex sentence with "Continued Fraction" contains at least two independent clauses and at least one dependent clause.