CONTINUED FRACTION Meaning and
Definition
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A continued fraction is a mathematical representation of a number or a function, where the number or function is expressed as a sum of a whole number and a fractional part. This representation consists of a sequence of nested fractions, in which each term is formed by dividing one whole number by the sum of another whole number and a smaller fractional part. The nested fractions continue indefinitely, forming a chain-like structure.
In a continued fraction, the terms are often organized in brackets, denoted by square brackets. The first term is written outside the brackets, followed by a plus sign. Inside the brackets, the subsequent terms are written as fractions, separated by a horizontal line. The denominators of these fractions become the numerators of the next fraction, continuing the pattern. The process continues infinitely, unless otherwise specified.
Continued fractions have numerous applications in mathematics, including solving algebraic equations, evaluating transcendental functions, and approximating irrational numbers. They possess unique properties, such as converging rapidly for certain types of numbers, leading to efficient and accurate approximations. Continued fractions also provide an elegant and concise way to represent real numbers, revealing deep relationships between various mathematical concepts.
Overall, a continued fraction can be viewed as a sophisticated mathematical tool for expressing and analyzing numbers and functions, offering insights into their nature and properties.
Common Misspellings for CONTINUED FRACTION
- xontinued fraction
- vontinued fraction
- fontinued fraction
- dontinued fraction
- cintinued fraction
- ckntinued fraction
- clntinued fraction
- cpntinued fraction
- c0ntinued fraction
- c9ntinued fraction
- cobtinued fraction
- comtinued fraction
- cojtinued fraction
- cohtinued fraction
- conrinued fraction
- confinued fraction
- conginued fraction
- conyinued fraction
- con6inued fraction
Etymology of CONTINUED FRACTION
The term "continued fraction" is derived from the Latin word "fractus", meaning "broken" or "fractured", and the word "continuus", meaning "continuous" or "uninterrupted".
In mathematics, a continued fraction is a way of representing a real number using a sequence of nested fractions. It is called a "continued fraction" because the representation continues indefinitely, without any interruption or end. The term was first introduced by the Swiss mathematician Leonhard Euler in his work "Introductio in analysin infinitorum" published in 1748.
The concept of continued fractions has ancient origins. It can be traced back to ancient Egyptian and Greek mathematics, where approximations of irrational numbers were expressed as a sum of fractions. However, the modern terminology and formalization of continued fractions can be attributed to Euler.
