Rational number

A number that caimot be expressed as a ratio n/m is called irrational. (Irrational here does not mean insane but rather not expressible as a ratio). The most famous irrational is V2, which did, however, drive the followers of Pythagoras somewhat insane. To prove the irrationality of V2, assume, to the contrary, that it is rational. This implies that V2 = n/m, where n and m are relatively prime. In particular, n and m cannot both be even numbers. Squaring, we obtain 

This implies that the square of n is an even number and therefore n itself is even and can be expressed as n = pjl, where p is another integer. This implies, in turn, that 

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Subsequently Onsager (1948) reported other exponents, and Yang (1952) completed the derivation. The exponents are rational numbers, but not the classical ones. 

The small uncertainties in the calculated exponents seem to preclude the possibility that the d = 3 exponents are rational numbers (i.e. the ratio of integers). (At an earlier stage this possibility had been suggested, since not only die classical exponents, bnt also tlie [Pg.653]

Two simple examples of fields are the rational numbers, Q and the set of integers Zp = (0,1,..., p — 1, where p is prime and addition and multiplication are defined modulo p. The latter is an example of a finite field - that is, of a field containing only a finite number of elements. Since CA are almost always defined so that individual sites take on one of a finite number of values, if those values happen to be elements of a finite field then the dynamics can be well understood by using some of the general properties of that field. An elementary property of finite fields... 

A real number x is said to be computable if there exists a computer program V such that when a positive rational number is inputted, V outputs a rational number r with ] X — r ]< and halts [gerochSG]. Thus, 2, tt, exp(l), Vtt , etc. are all computable numbers. 

In Europe the notion of the zero evolved slowly in various forms. Eventually, probably to express debts, it was found necessary to invent negative integers. The requirements of trade and commerce lead to the use of fractions, as ratios of whole numbers. However, it is obviously more convenient to express fractions in the form of decimals. The ensemble of whole numbers and fractions (as ratios of whole numbers) is referred to as rational numbers. The mathematical relation between decimal and rational fractions is of importance, particularly in modem computer applications. 

As an example, consider the decimal fraction x — 0.616161 . Multiplication by 100 yields the expression 100 x = 61.6161 = 61 + jc and thus, x = 61/99, is a rational fraction. In general, if a decimal expression contains an infinitely repeating set of digits (61 in this example), it is a rational number. However, most decimal fractions do not contain a repeating set of... 

General powers of z are defined by z = e losz. Since log z is infinitely many valued, so too is z unless a is a rational number. 

Familiar fields are the set of real numbers K, the set of complex numbers C, and the set of all rational numbers . The elements of a field are called scalars. A set L of elements (u,v,w,...) is called a vector space4 over a field F if the following conditions are fulfilled ... 

A complete unitary space is called a Hilbert space. The unitary spaces of finite dimension are necessarily complete. For reasons of completeness the vector space of all n-tuplets of rational numbers is not a Hilbert space, since it is not complete. For instance, it is possible to define a sequence of rational numbers that approaches the irrational number y/2 as a limit. The set of all rational numbers therefore does not define a Hilbert space. Similar arguments apply to the set of all n-tuplets of rational numbers. 

One way to determine the common factor of which all 13 numbers are multiples is to first divide all of them by the smallest. The ratios thus obtained may either be integers or they may be rational numbers whose decimal equivalents are easy to recognize. 

Then there axe rational numbers ni,...,nr such that... 

COMMON FRACTION a rational number expressed in the form where a and b are integers, and b 0... 

Once again the logic maps a set of statements onto a range. In this case the range will be the rational numbers from -m to +m, where -m is equivalent to False, and +m is equivalent to True, with complete ignorance at 0. 

If the weights are rational numbers, the network is equivalent in power to the Turing machine model [152-155]. 

Note that Xl and Xh are, by definition, rational numbers, hence we can assume that r and t are integers. 

For example, in mathematics, the word rational refers to a type of number or function. A person is rational if he acts in a controlled, logical way. A number is rational if it acts in a controlled, structured way. If you use the word rational to describe a number, and if the person you re talking to also knows what a rational number is, then you don t have to go into a long, drawn-out explanation about what you mean. You re both talking in the same language, so to speak. 

This expression contains all characteristic structures (([-function, In 2, 7T and a rational number) which one usually encounters in the results of the loop calculations. Let us emphasize that the relative scale of these subleading terms is rather large, of order tt, which is just what one should expect for the constants accompanying the large logarithm. 

When 1/0 is a rational number m/n (m,n integer, undivisible), there are m substrate unit cells per set of n adsorbates a superlattice with unit cell area mS can form, with the superlattice unit cell containing n arbitrarily-positioned adsorbates. It may happen in this case that the adsorbates between themselves (ignoring the substrate) form a structure that has a smaller unit cell than the superlattice unit cell one must then distinguish between the overlayer unit cell (defined in the absence of a substrate) and... 

In theory, there is a resonance horn emerging from every rational number along the abscissa. We have not attempted to show more than a few, just those which are most important in the sense that the corresponding oscillations are relatively easily obtained and exist of a reasonable range of frequencies. 

When we divide one integer by another, we sometimes obtain another integer For example, 6/—3 = —2 at other times, however, we obtain a fraction, or rational number, of the form, where the integers a and b are known as the numerator and denominator, respectively, for example,. The denominator, b, cannot take the value zero because g is of indeterminate value. 

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