For one thing, there is a natural belt of rock across the river bed ; ;

For the only time in the opera, words are not set according to their natural inflection ; ;

For so long as death was not violent, it was natural and to be welcomed, making a funeral a feast.

For the Aztec the count by 20s was completely natural.

For example, Pyrethrum ( refers to several Old World plants of the genus Chrysanthemum ) is a natural insecticide with minimal environmental impact.

For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

For example, the positrons produced in natural radioactive decay quickly annihilate themselves with electrons, producing pairs of gamma rays, a process exploited in positron emission tomography.

For Hume, philosophy can be split into two general parts: natural philosophy and the philosophy of human nature ( or, as he calls it, " moral philosophy ").

For most of human history, pearls were the ultimate precious beads of natural origin because of their rarity, although the pearl-culturing process has now made them far more common.

For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

Some sets are infinite ; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size.

For example, Georg Cantor ( who introduced this concept ) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers ( non-negative integers ), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

* For every natural number, the-sphere is compact.

For each specific consistent effectively represented axiomatic system for the natural numbers, such as Peano arithmetic, there exists a constant N such that no bit of Ω after the Nth can be proven to be one or zero within that system.

For example, the cofinality of ω² is ω, because the sequence ω · m ( where m ranges over the natural numbers ) tends to ω² ; but, more generally, any countable limit ordinal has cofinality ω.

For this reason, its practice violated the natural laws of medicine, transforming health care practitioners into those who harm rather than heal.

For example, by inoculating in the months of milder climate, one had a better chance of fighting the infection and becoming immune instead of the alternative: natural exposure to the disease during harsher weather, when the body's defenses were already challenged.

For example, philosopher John D. Kenyon writes: Reason might manage to raise a doubt about the truth of a conclusion of natural inductive inference just for a moment in the study, but the forces of nature will soon overcome that artificial skepticism, and the sheer agreeableness of animal faith will protect us from excessive caution and sterile suspension of belief.

For instance, we could define natural number as follows ( after Peano ):

For example, Darwinism may be used to refer to Darwin's proposed mechanism of natural selection, in comparison to more recent mechanisms such as genetic drift and gene flow.

For example, while burning an area of fairly thick forest and thus turning it into a more open, grassy environment might reduce the viability of a large browser ( an animal that eats leaves and shoots rather than grasses ), the reverse could also be true: removing the browsing animals ( by eating them, or by any other means ) within a few years produces a very thick undergrowth which, when a fire eventually starts through natural causes ( as fires tend to do every few hundred years ), burns with greater than usual ferocity.

For example, in accordance with the fashion of the times, most transitional societies have announced economic development plans of varying numbers of years ; ;

For one thing, although considerable numbers of men have been trained, bureaucracies are still deficient in many respects ; ;

For they considered the odd numbers as male and the even ones as female, equating the two groups with the Yang and Yin principles in Nature ; ;

For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

For example, suppose that X is the set of all non-empty subsets of the real numbers.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

For, the letters associated with those numbers are K, L, M, N, O, ..., respectively.

For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.

For socket numbers, a few well-known numbers were reserved for special purposes specific to the AppleTalk protocol itself.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

For example, if the numbers involved in a computation may be arbitrarily large, the time required by a single addition can no longer be assumed to be constant.

For this reason, parallel computing is only useful for either small numbers of processors, or problems with very high values of P: so-called embarrassingly parallel problems.

For example, the smallest pair of amicable numbers is ( 220, 284 ); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284 ; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.

For small numbers of at-bats, it is possible ( though unlikely ) for a player's on-base percentage to be lower than his batting average ( H / AB ).

For instance, division of real numbers is a partial function, because one can't divide by zero: a / 0 is not defined for any real a.