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.

For real values of, we have and for purely imaginary we have hence has a limit at 0 ( i. e., ƒ is complex differentiable at 0 ) if and only if.

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 a brief period each year, the rays of the sun are warm enough to melt some of the snows piled a mile deep at the base of the headwalls, and then the pinnacles glisten in the daytime at high noon, and billions of gallons of water begin their slow seepage under the glaciers and across the rockstrewn hanging valleys on their long, meandering journey to the sea -- running east past the sky-carving massifs of Gurla Mandhata and Kemchenjunga, then turning south and curling down through the jungles of Assam, past the Khasi Hills, and into Bengal, past Sirinjani and Madaripur, until the hard water of the melting snows mingles with the soft drainage of fields and at length fans out to meld with the teeming salt depths of the Bay of Bengal.

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

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

For some, aesthetics is considered a synonym for the philosophy of art since Hegel, while others insist that there is a significant distinction between these closely related fields.

3: 17 For though the fig tree doesn ’ t flourish, nor fruit be in the vines ; the labor of the olive fails, the fields yield no food ; the flocks are cut off from the fold, and there is no herd in the stalls: 3: 18 yet I will rejoice in Yahweh.

For video, there are two frame rate standards: NTSC, which shoot at 30 / 1. 001 ( about 29. 97 ) frames per second or 59. 94 fields per second, and PAL, 25 frames per second or 50 fields per second.

For Husserl this is not the case: mathematics ( with the exception of geometry ) is the ontological correlate of logic, and while both fields are related, neither one is strictly reducible to the other.

For example, oscillating charges produce electric and magnetic fields that may be viewed in a ' smooth ', continuous, wavelike fashion.

For each of the prime fields, one elliptic curve is recommended.

For each of the binary fields, one elliptic curve and one Koblitz curve was selected.

For example, in quantum field theory " locality " means that quantum fields at different points of space do not interact with one another.

For s-polarization, a positive r or t means that the electric fields of the incoming and reflected or transmitted wave are parallel, while negative means anti-parallel.

For p-polarization, a positive r or t means that the magnetic fields of the waves are parallel, while negative means anti-parallel.

For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.

For moderate magnetic fields the Hall coefficient is

For example, a Hall sensor integrated into a ferrite ring ( as shown ) can reduce the detection of stray fields by a factor of 100 or better ( as the external magnetic fields cancel across the ring, giving no residual magnetic flux ).

For many villeins, the wheat must have run low in the days before Lammas, and the new harvest began a season of plenty, of hard work and company in the fields, reaping in teams.

For these reasons, minimally invasive surgery has emerged as a highly competitive new sub-specialty within various fields of surgery.

( For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.

For example the Boricua Popular Army are unofficially called Macheteros because of the machete-wielding laborers of sugar cane fields of past Puerto Rico.

For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations

In fact, for many fields K one does not know in general precisely which finite groups occur as Galois groups over K. This is the inverse Galois problem for a field K. ( For some fields K the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.