For all involved in this discussion the devil is a real entity who can really be confronted in the woods on a dark night, the demon world is populated with real creatures, and witches actually can be seen flying through the air.

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 any real number the absolute value or modulus of is denoted by ( a vertical bar on each side of the quantity ) and is defined as

For nearby astronomical objects ( such as stars in our galaxy ) luminosity distance D < sub > L </ sub > is almost identical to the real distance to the object, because spacetime within our galaxy is almost Euclidean.

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 example, if a bank, operating under the Basel I accord, has to hold 8 % capital against default risk, but the real risk of default is lower, it is profitable to securitise the loan, removing the low risk loan from its portfolio.

For example, a boom in the real estate market started around 2003.

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.

For example, the spectrum of an element of a complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

For any real numbers x, r > 0, one has

For every real number x, the cumulative distribution function of a real-valued random variable X is given by

For an atemporal interpretation that “ makes no attempt to give a ‘ local ’ account on the level of determinate particles ”, the conjugate wavefunction, (" advanced " or time-reversed ) of the relativistic version of the wavefunction, and the so-called " retarded " or time-forward version are both regarded as real and the transactional interpretation results.

For example, a perfect crystal of diamond would only contain carbon atoms, but a real crystal might perhaps contain a few boron atoms as well.

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 instance, any continuous function defined on a compact space into an ordered set ( with the order topology ) such as the real line is bounded.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For much of the 1950s the threat of Communist invasion of Taiwan was very real, and required the U. S. Navy to safe guard the Taiwan Straits.

Jones, whose work had been nominated eight times over his career for an Oscar ( winning thrice: For Scent-imental Reasons, So Much for So Little, and The Dot and the Line ), received an Honorary Academy Award in 1996 by the Board of Governors of the Academy of Motion Picture Arts and Sciences, for " the creation of classic cartoons and cartoon characters whose animated lives have brought joy to our real ones for more than half a century.

For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.

For the vertex at Af, these values will be denoted by Af.

For punched-card or tape storage of information all literature values must be conformed to a common language.

For almost one-sixth of the national population discrimination in the free selection of residence casts a considerable shadow upon these values assumed as self-evident by most Americans.

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 N to be constant for all values of w, a ' tan w '/ a tan w must also be constant.

For higher values, the surface diffusion of deposited atomic species would allow for the formation of crystallites with long range atomic order.

For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n ( at most exponentially ).

For example, the graph contains the points ( 1, 1 ), ( 2, 0. 5 ), ( 5, 0. 2 ), ( 10, 0. 1 ), ... As the values of x become larger and larger, say 100, 1000, 10, 000 ..., putting them far to the right of the illustration, the corresponding values of y,. 01,. 001,. 0001, ..., become infinitesimal relative to the scale shown.

For each statistic, two values are given:

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

For example, the formula a AND b is satisfiable because one can find the values a = TRUE and b = TRUE, which make ( a AND b ) = TRUE.

For example, today most major corporations promote their commitment to non-economic values under headings such as ethics codes and social responsibility charters.

For example, one type of " compare " instruction considers two values and sets a number in the flags register according to which one is greater.

For example, an 8-bit CPU deals with a range of numbers that can be represented by eight binary digits ( each digit having two possible values ), that is, 2 < sup > 8 </ sup > or 256 discrete numbers.

For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by

For each variable, the values will normally all be of the same kind.

For the sake of the example ( and this is a gross simplification ), let's assume that he values this particular risk at 5 % per annum ( we could perform a more precise probabilistic analysis of the risk, but that is beyond the scope of this article ).

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

For example, it might study what constitutes upbringing and education, the values and norms revealed through upbringing and educational practices, the limits and legitimization of education as an academic discipline, and the relation between education theory and practice.

For example, today most major corporations promote their commitment to non-economic values under headings such as ethics codes and social responsibility charters.

For example, a moral universalist ( and certainly an absolutist ) might argue that, just as one can discuss what is ' good and evil ' at an individual's level, so too can one make certain " moral " propositions with truth values relative at the level of the species.