The denominator sin a is the quantity well known (after Abbe) as the " numerical aperture."
A fractional number is called a proper fraction or an improper fraction according as the numerator is or is not 3 less than the denominator; and an expression 4 such as 24 is called a mixed number.
Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M.
Hence the value of a fraction is not altered by substituting for the numerator and denominator the corresponding numbers in any other column of a multiple-table (§ 36).
Can be applied to n, n', the denominator n remaining unaltered.
The modern system of placing the numerator above the denominator is due to the Hindus; but the dividing line is a later invention.
To add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index.
And that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x.
When we know the mass of the earth in gravitational measure, its product by the denominator of the fraction just mentioned gives the mass of the sun in gravitational measure.
The pth root of a number (§43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately.
Denominator factors, that the complete system of the quadratic is composed of the form itself of degree order I, 2 shown by az 2, and of the Hessian of degree order 2, o shown by a2.
1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).
Fraction in its Lowest Terms.-A fraction is said to be in its lowest terms when its numerator and denominator have no common the more correct method is to write it a: b.
Hence we can treat the fractional numbers which have any one denominator as 0 o constituting a number-series, as shown in the 2 adjoining diagram.
A simple fraction with ioo for denominator, can be expressed by writing the two figures of the numerator (or, if there is only one figure, this figure preceded by o) with a dot or " point " before them; thus 76 means 76%, or 17 -6 6 o.
A fraction written in this way is called a decimal fraction; or we might define a decimal fraction as a fraction having a power of To for its denominator, there being a special notation for writing such fractions.
The Romans commonly used fractions with denominator 12; these were described as unciae (ounces), being twelfths of the as (pound).
Hindu treatises on arithmetic show the use of fractions, containing a power of io as denominator, as early as the beginning of the 6th century A.D.
Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x=uy, and expanding the denominator in powers of y.
The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.
Where the denominator stands for the same homogeneou~ quadratic function of the qs that T is for the is.
Must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive.
Denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is called the steadiness.
Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.
If we write 74 in the form 47 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.
Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.
- A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a complex fraction (or fractional number), to distinguish it from a simple fraction, which is a fraction having integers for numerator and denominator.
Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.
By means of the present and the preceding sections the rule given in § 63 can be extended to the statement that a fractional number is equal to the number obtained by multiplying its numerator and its denominator by any fractional number.
Except in the case of - and 2, the fraction was expressed by the denominator, with a special symbol above it.
The Babylonians expressed numbers less than r by the numerator of a fraction with denominator 60; the numerator only being written.
Then the denominator of the fraction, the numerical aperture, must be correspondingly increased, in order to ascertain the real resolving power.
The modern method is to deal with fractions which have ioo as denominator; such fractions are called percentages.
A different method was used by Diophantus, accents being omitted, and the denominator being written above and to the right of the numerator.
B,, Y the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that.
In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 60) were followed by', ", "', ", the denominator not being written.
In the case of fractions of the more general kind, the numerator was written first with ', and then the denominator, followed by ", was written twice.
If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur.
(iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent.
We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.
If this arrangement is expressed by a fraction, the numerator of which indicates the number of turns, and the denominator the number of internodes in the spiral cycle, the fraction will be found to represent the angle of divergence of the consecutive leaves on the axis.
For the application of continued fractions to the problem " To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable," the reader is referred to G.
Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o.
This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M.