Binomial theorem

“ The student in mathematics experiences a feeling of growing strength and power when he finds, in algebra, that the formula he used in arithmetic in extracting a square root has grown in importance by leading indirectly to a theorem of which it is only one particular case--a theorem with a more definite proof, and a larger capability for use than he had thought possible. When he finds a still simpler proof for the binomial theorem in his study of the calculus, his feeling of increasing power and the desire for still greater results deepens and intensifies. ”

“ By propounding the binomial theorem, the differential calculus, and the integral calculus, he began in 1665 the wonderful series of discoveries in pure mathematics, optics, and physics, which place him in the first rank of the philosophers of all time. ”

“ It is a very simple matter, however, when the binomial theorem has been mastered, or it is a very simple problem in solid geometry. The surveying is practical, and is of course limited to the most elementary problems. There are few boys who do not enjoy it, however, and who do not get something of real educational value out of it. ”

“ There is no more reason for any human being on God's earth to be acquainted with the binomial theorem or the solution of triangles, unless he is a professional scientist,—when he can begin to specialize in mathematics at the same age as the lawyer begins to specialize in law or the surgeon in anatomy,—than for him to be expert in Choctaw, the Cabala, or the Book of Mormon. ”

“ The second class whose merits we propose to discuss are the illustrators of the "Binomial Theorem,"—the double-named families,—who, too hideous to walk alone, conjoin ugliness of equal intensity to scare and appal wherever they make their way. ”

“ Moreover, as usually stated, James Bernoulli's imports something more than the first axiom of probabilities. [73]
26. The generalization of the Binomial Theorem which is called the “Multinomial Theorem” [74] gives the rule when there are more than two alternatives at each trial. For instance, if there are three alternatives, hearts, diamonds or a card belonging to a black suit, the probability that if n cards are dealt there will occur s hearts, t diamonds, and n − s − t cards which are either clubs or spades is
27. ”

“ In schools where practical aims predominate, logarithms should be explained by a comparison of the arithmetical with the geometrical series, and the practical application will immediately follow. But even where recourse is had to Taylor’s Theorem and the Binomial Theorem, the gain to the beginner will not usually be very much greater. Not as though these theorems, together with the elements of differential calculus, could not be made clear; the real trouble lies in the fact that much of what is comprehended is not likely to be retained in the memory. ”

“ About binomial theorem I'm teeming with a lot o' news,
With interesting facts about the square of the hypotenuse.
I'm very good at integral and differential calculus,
I know the scientific names of beings animalculous.
In short, in matters vegetable, animal, and mineral,
I am the very model of a modern Major-Gineral.
[Pg 43]
I know our mythic history—King Arthur's and Sir Caradoc's,
I answer hard acrostics, I've a pretty taste for paradox;
I quote in elegiacs all the crimes of Heliogabalus,
In conics I can floor peculiarities parabolous. ”

“ When he had obtained this method, it occurred to him that the very same process was applicable to the ordinates, and, by following out this idea, he discovered the general method of reducing radical quantities composed of several terms into infinite series, and was thus led to the discovery of the celebrated Binomial Theorem. He now neglected entirely his methods of interpolation, and employed that theorem alone as the easiest and most direct method for the quadratures of curves, and in the solution of many questions which had not even been attempted by the most skilful mathematicians. ”

“ Def.—The exponential function, the sine and the cosine of a dyadic may be defined by infinite series, exactly as the corresponding functions in scalar analysis, viz.,
These series are always convergent. For every value of there is one and only one value of each of these functions. The exponential function may also be defined as the limit of the expression when which is a whole number, is increased indefinitely. That this definition is equivalent to the preceding, will appear if the expression is expanded by the binomial theorem, which is evidently applicable in a case of this kind. ”

“ The bondage of thought to content must be admitted, but generalization and free construction, just because they give us the power to vary conditions as we please, give us thinking in a relative independence of content, and thus show us how thought operates irrespective of, although not independent of, its content. The binomial theorem operates irrespective of the values substituted for its symbols. ”

“ BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term) , in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a + b, a − b. The terms trinomial, quadrinomial, multinomial, &c., are applied to expressions composed similarly of three, four or many quantities. ”

“ And of this nature are all the theorems of the science of number. They assert the identity of the result of different modes of formation. They affirm that some mode of formation from x, and some mode of formation from a certain function of x, produce the same number.
Such, as above described, is the aim and end of the calculus. As for its processes, every one knows that they are simply deductive. In demonstrating an algebraical theorem, or in resolving an equation, we travel from the datum to the quæsitum by pure ratiocination ”

“ Moreover, it is obvious that if a,b,c form a safe combination any two of the numbers determine the remaining one, that is, the system is closed. A particular safe combination which is used later is that in which two piles are equal and the third is zero. In the proofs which follow, the binary scale of notation is used throughout.
Theorem I. If A leaves a safe combination on the table, B cannot leave a safe combination on the table at his next move. B can change only one pile, and he must change one. ”

“ Why not own up honestly and admit that we cannot name such material? We may state that we think it is so and so and where necessary give reasons for the conclusion, but to pass as fact what is only opinion is not the spirit of modern science. The logical solution of the problem is to name subspecifically only such specimens as are humanly demonstrable and use the binomial for the rest. In other words reverse usual practice and instead of using the trinomial regularly and the binomial on occasion use the binomial generally and the trinomial only where necessity or the facts justify its use. ”

“ Hence a selection from them will be a progression; i.e. if is the member of , is a member of , is a member of , and so on; then , , , ... is a progression, and is a sub-class of . Assuming the multiplicative axiom, such a selection can be made. Thus by twice using this axiom we can prove that, if the axiom is true, every non-inductive cardinal must be reflexive. This could also be deduced from Zermelo's theorem, that, if the axiom is true, every class can be well ordered; for a well-ordered series must have either a finite or a reflexive number of terms in its field. ”

“ These ideas are further developed in various papers in the Bulletin and in his L'Anthropométrie, ou mesure des différentes facultés de l'homme (1871) , in which he lays great stress on the universal applicability of the binomial law,—according to which the number of cases in which, for instance, a certain height occurs among a large number of individuals is represented by an ordinate of a curve the binomial) symmetrically situated with regard to the ordinate representing the mean result (average height) . ”

“ The coincidence with theory is pretty close, but considering the large number of throws there is some reason to suspect a tendency in favour of heads.
The special interest of this trial consists in the exhibition, in a practical form, of the results of Bernoulli’s theorem, and the law of error or divergence from the mean to be afterwards more fully considered. It illustrates the connection between combinations and permutations, which is exhibited in the Arithmetical Triangle, and which underlies many important theorems of science. ”

“ Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.
10. A class of numerical equations which needs to be considered is that of the binomial equations xn − a = 0 (a = α + βi, a complex number) .
The foregoing conclusions apply, viz. there are always n roots, which, it may be shown, are all unequal. ”

“ Repassing from the discriminant functions to their coefficients as has just been done, we shall have the nth difference of the primitive function expressed by the development of the nth power of the binomial T — 1, in which we substitute for the powers of T this same function whose index is augmented by the exponent of the power, and for the independent term of t, which is unity, the primitive function, which gives this difference by means of the consecutive terms of this function. ”

“ For this purpose, the number two immediately presents itself; but binary numeration and notation being too prolix for arithmetical practice, it becomes necessary to select for a base a power of two that will afford a more comprehensive notation: a power of two, because no other number will agree with binary gradation. It is scarcely proper to say the third power has been selected, for there was no alternative,—the [Pg 133] second power being too small, and the fourth too large. ”

“ Indeterminateness of Algebraic Integrals.—The theorem that the integral ∫ xa ƒ (z) dz is independent of the path from a to z, holds only on the hypothesis that any two such paths are equivalent, that is, taken together from the complete boundary of a region of the plane within which ƒ (z) is finite and single valued, besides being differentiable. Suppose that these conditions fail only at a finite number of isolated points in the finite part of the plane. ”

“ In the theory of algebraic functions, the method of boundary values (Randwerthaufgabe) serves, as is well known, for the proof of Riemann's existence theorem. In the theory of number fields also, the proof of the existence of just this number offers the greatest difficulty. This proof succeeds with indispensable assistance from the theorem that in the number field there are always prime ideals corresponding to given residual properties. This latter fact is therefore the analogue in number theory to the problem of boundary values. ”

“ But although we may not know how to compute the numerical value of that limit, it is of prime importance to know that a limit does exist, for otherwise the series may be non-convergent. When examining a series to determine whether or not it is convergent, the following theorems, which we state without proofs, are found to be of fundamental importance. [4]
Theorem I. If is a variable that always increases as increases, but always remains less than some definite fixed number , then as increases without limit, will approach a definite limit which is not greater than . ”

“ The binary base, on the contrary, is natural, real, simple, and accords with the tendency of the mind to simplify, to individualize. In business, who ever thinks of a half as two-fourths, or three-sixths, much less as two-and-a-half-fifths, or three-and-a-half-sevenths? For division by two produces a half at one operation; but with any other divisor, the reduction is too great, and must be followed by multiplication. Think of calling a half five-tenths, a quarter twenty-five-hundredths, an eighth one-hundred-and-twenty-five-thousandths! Arithmetic is seldom used as a plaything. ”

“ The theorem of the total differential is immediately applicable to the differentiation of implicit functions. When y is a function of x which is given by an equation of the form ƒ (x, y) = 0, and it is either impossible or inconvenient to solve this equation so as to express y as an explicit function of x, the differential coefficient dy/dx can be formed without solving the equation. We have at once
This rule was known, in all essentials, to Fermat and de Sluse before the invention of the algorithm, of the differential calculus. ”

“ The condition (a) is fulfilled because, by theorem E, a part of can only have one of the cardinal numbers , and therefore, by theorem A, cannot be equivalent to the aggregate . The condition (b) is fulfilled because itself is a part of .
Proof of C.— Let be a cardinal number which is less than . Because of the condition (b) of §2, there is a part of with the cardinal number . By theorem E, a part of can only have one of the cardinal numbers . Thus is equal to one of the cardinal numbers . By theorem B, none of these is greater than . ”

“ We need not think algebra is ended because it gives us rules to form all possible combinations; it remains to find the interesting combinations, those which satisfy such and such a condition. Thus will be formed a sort of indeterminate analysis where the unknowns will no longer be whole numbers, but polynomials. This time it is algebra which will model itself upon arithmetic, following the analogy of the whole number to the integral polynomial with any coefficients or to the integral polynomial with integral coefficients. ”

“ But there is no induction, because there is no inference: the conclusion is a mere summing up of what was asserted in the various propositions from which it is drawn. A case somewhat, though not altogether, similar, is the proof of a geometrical theorem by means of a diagram. Whether the diagram be on paper or only in the imagination, the demonstration (as formerly observed [1] ) does not prove directly the general theorem; it proves only that the conclusion, which the theorem asserts generally, is true of the particular triangle or circle exhibited in the diagram ”