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For example, some unicellular organisms have genomes much larger than that of humans. Cole's paradox: Even a tiny fecundity advantage of one additional offspring would favor the evolution of semelparity. Gray's paradox: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations.
For example, if both the numerator and the denominator of the fraction are divisible by , then they can be written as =, =, and the fraction becomes cd / ce , which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction d / e .
As with fractions of the form , it has been conjectured that every fraction (for >) can be expressed as a sum of three positive unit fractions. A generalized version of the conjecture states that, for any positive k {\displaystyle k} , all but finitely many fractions k n {\displaystyle {\tfrac {k}{n}}} can be expressed as a sum of three ...
By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.
For example, the p i may be the factors of the square-free factorization of g. When K is the field of rational numbers , as it is typically the case in computer algebra , this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition.
For example, the numerators of fractions with common denominators can simply be added, such that + = and that <, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what 5 12 + 11 18 {\displaystyle {\frac {5}{12}}+{\frac {11}{18}}} equals, or whether 5 12 {\displaystyle {\frac {5 ...
This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are: √ 19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...] (sequence A010124 in the ...
By consequence, we may get, for example, three different values for the fractional part of just one x: let it be −1.3, its fractional part will be 0.7 according to the first definition, 0.3 according to the second definition, and −0.3 according to the third definition, whose result can also be obtained in a straightforward way by