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In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. [14] [15] It is said to be an improper fraction, or sometimes top-heavy fraction, [16] if the absolute value of the fraction is greater than or equal to 1 ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
Michael Stifel published the following method in 1544. [3] [4] Consider the sequence of mixed numbers,,,, … with = + +.To calculate a Pythagorean triple, take any term of this sequence and convert it to an improper fraction (for mixed number , the corresponding improper fraction is ).
In other words, a fraction a / b is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. [2]
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: = = + + + + +. The first n {\displaystyle n} terms of the series sum to approximately ln n + γ {\displaystyle \ln n+\gamma } , where ln {\displaystyle \ln } is the natural logarithm and γ ≈ 0.577 {\displaystyle \gamma \approx 0.577 ...
Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ...
If this infinite continued fraction converges at all, it must converge to one of the roots of the monic polynomial x 2 + bx + c = 0. Unfortunately, this particular continued fraction does not converge to a finite number in every case. We can easily see that this is so by considering the quadratic formula and a monic polynomial with real ...
Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero. In 830, Mahāvīra unsuccessfully tried to correct the mistake Brahmagupta made in his book Ganita Sara Samgraha : "A number remains unchanged when divided by zero."
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