Decimal Multiplication Research Articles

Short-cut multiplication and division in automatic binary digital computers, with special reference to a new multiplication process

The paper considers the application of analogues of the well-known decimal short-cut multiplication and division methods, to the control of such operations in automatic binary digital computers.After demonstrating that the simple binary short-cut process leads, on the average, to a slowing down of multiplication, the paper developes a new process, termed the modified-short-cut (m.s.c.) process. This is defined in terms of symbolic equations and is shown to reduce the average number of additions or subtractions required during the execution of a multiplication by more than 17% to some m/3 (for an m-bit number), the maximum number by nearly 50% to (m+2)/2 and the average number of shifts by 30%.Following a discussion of the properties of the process, and, in particular, its application to signed multiplication, comparisons are made between the relative average multiplication times of a number of alternative systems including those using circuits based on ‘carry’ storage or on carry-propagation-detection circuits, or on both.Brief consideration is then given to some aspects of division in automatic binary machines. The discussion is confined to a consideration of what are believed to be ideas not previously published. It is shown, in particular, that short-cut procedures can be easily and cheaply incorporated in machines using either restoring or non-restoring division techniques.

Why “Indent” in Multiplication?

WE MAY BE DOING MORE HARM than good when we teach children deductively to use many of the traditional techniques of computation. “Carrying” in addition; “borrowing” in subtraction; “indenting” in multiplication; “multiplying-subtracting-bringing down” in division; “canceling” in multiplication of fractions; “inverting and multiplying” in division of fractions; “counting decimal places” in multiplication of decimals; and “moving decimal points” in division of decimals are cases in point. To illustrate why this may be so, let us consider “indenting” in multiplication, although any one of the other techniques mentioned rould serve equally well as an example of deductive teaching.

1478. The multiplication and division of decimals and contracted methods

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

An Investigation into Multiplication I. The Multiplication of Decimals

There is probably no other item of the arithmetic syllabus which is confronted by such a diversity of earnestly supported method as is the case with the multiplication of decimals.The problems which form the basis of this investigation are :(a)What are the special advantages and disadvantages which bring about the adoption or rejection of these methods?(b)What is the extent of these advantages and disadvantages?(c)And finally a consideration of the methods of overcoming the disadvantages.

Multiplication and Division of Integers and Decimals

Multiplication and Division of Integers and Decimals

Multiplication and Division of Decimals.

It is perhaps somewhat remarkable that in such frequently recurring and comparatively simple processes as those of multiplication and division of decimals there should still be no final settling down by teachers of Arithmetic into one standardized method and arrangement of work, shown by experience to be the simplest, safest, least laborious, and most easily understood and assimilated by the pupil. Indeed there seems a tendency in some quarters, after trying in succession all the various modifications of method and set-out which have been suggested, each as an improvement on its predecessor, to come round full circle to the older methods in use, say, fifty years ago.

Associate Fellowship

Arithmetic. —The principles and processes of arithmetic applied to whole numbers and vulgar and decimal fractions. The metric system. Approximations to a specified degree of accuracy. Contracted methods of multiplication and division of decimals. Ratio and proportion. Percentage. Averages. Practical applications of arithmetic. Algebra.—Symbolical expressions. Equations of the first or second degree (and problems leading thereto). Square root. Graphs of simple rational integral algebraic functions. Arithmetic and harmonic progression. Geometric progression. Theory of indices. Logarithms. Theory and use of the slide rule. Binomial theorem for a positive integral index. Geometry.— The subjects covered by Euclid I.-IV. with simple deductions, including easy loci and the areas of triangles and parallelograms of which the bases and altitudes are given commensurable lengths. (Euclid proofs will not be insisted upon.) Trigonometry.—Up to and including the solution of triangles, together with the practical solutions of triangles and applications, and numerical examples involving the use of logarithmic and other tables.

Multiplication and Division of Decimals

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Report on the teaching of the multiplication and division of decimals

Report on the teaching of the multiplication and division of decimals

Report of the Public Schools Special Committee

Four meetings of the Committee have been held during the year. A circular has been sent to the Senior Mathematical Master of each of the Public Schools, asking whether he desires that all boys entering his school should be acquainted with the same method for the multiplication and division of decimals, and if so, whether he is in favour of the methods given in the circular. The investigation has been completed, and the result is being sent to the General Committee.

On the Multiplication and Division of Decimals

On the Multiplication and Division of Decimals

Graphic Arithmetic

THIS contribution to the teaching of arithmetic consists of two charts, each 40 in. by 10 in., which can be mounted, or otherwise adapted to class teaching. Chart i., whole numbers, consists of a series of horizontal lines, divided by dots respectively into ones, twos, threes, &c., twelves. By means of vertical lines the multiples of 2, 3, &c., are shown on the line of ones, which is numbered from 0 to 144. In Chart ii., fractions (vulgar and decimal), the series of horizontal lines are 1 metre in length and 1 cm. apart. These lines are divided by dots into halves, thirds, &c., tenths. There are subdivisions into hundredths (cm.) and thousandths (millim). On Chart i. explanations are given of magnitude, unit, number, multiple, &c.; and on ii., of the multiplication and division of decimals, the expression of a vulgar fraction as a decimal, and several other matters. The charts are well adapted for the purpose of giving a sound grounding in the subject, so far as they go.

The Science of Arithmetic The School Arithmetic

THESE books are too well known to mathematical teachers to need detailed notice from us. Both are very good, and stand in the first rank among the scores of arithmetics published in England. The explanations, arrangement and examples, especially in the former book, are generally very good. We will venture, however, to suggest two or three changes to the authors, which we think would render the book better still, and which our experience would make us wish to see universally adopted. The rule for multiplication of decimals given in these books is the old one of counting the decimal places. We think this becomes a rule of thumb. The method ought to be the same as that in multiplication of integers; and it is at once seen by the pupil that as in multiplying by tens and hundreds, the figures are shifted to the left; so in multiplying by tenths and hundredths, they are shifted to the right. The decimal point is brought down straight, and each line in the working has its meaning; as in the example, multiply 712.35 by 15.807:— The Science of Arithmetic. By James Cornwell, and Joshua G. Fitch, M.A. Thirteenth Edition. (Simpkin, Marshall, and Co., 1870.) The School Arithmetic. By the same authors. Eleventh Edition. (Simpkin, Marshall, and Co., 1871.)

XX. Tables for reducing the quantities by weight, in any mix­ture of pure spirit and water, to those by measure; and for determining the proportion, by measure, of each of the two substances in such mixtures

These tables are founded on the experiments of which the results were given in the Report and Supplementary Report on the best method of proportioning the excise on spirituous liquors. They are computed for every degree of heat from 30° to 80°, and for the addition or subtraction of every one part in a hundred of water or spirit; but as the experiments themselves were made only to every fifth degree of heat, and every five in the hundred of water or spirit, the intermediate places are filled up by interpolation in the usual manner, with allowance for second differences. Every table consists of eight columns, and there are two tables for every degree of heat. In the first column of the first of the two tables, are given the proportions of spirit and water by weight, 100 parts of spirit being taken as the constant number, to which additions are made successively of one part of water from 1 to 99 inclusively. The first column in the second table has 100 parts of water for the constant number, with the parts of spirit decreasing successively by unity, from 100 to 1 inclusively. It must be observed, that each of these tables occupying one page, is divided in the middle for adapt­ing it more conveniently to the size of the paper; but the whole of each page is to be considered as one continued table. The second column of all the tables gives the specific gravities of the corresponding mixtures of spirit and water in the first column, taken from the table of specific gravities in the Supplementary Report, the intermediate spaces being filled up by interpolation. In the third column 100 parts by measure of pure spirit, at the temperature marked on the top of every separate table, is assumed as the constant standard number, to which the respective quantities of Water by measure, at the same temperature, are to be proportioned in the next column. The fourth column, therefore, contains the proportion of water by measure, to 100 measures of spirit, answering to the pro­portions by weight in the same horizontal line of the first column. The fifth column shews the number of parts which the quantities of spirit and water contained in the third and fourth columns would measure when the mixture has been completed; that is, the bulk of the whole mixture after the concentration, or mutual penetration, has fully taken place. The sixth column, deduced from the three preceding ones, gives the effect of that concentration, or how much smaller the volume of the whole mixture is, than it would be if there was no such principle as the mutual penetration. The seventh column shews the quantity of pure spirit by measure, at the temperature in the table, contained in 100 measures of the mix­ture laid down in the fifth column. Lastly, the eighth column gives the decimal multiplier, by means of which the quantity by measure of standard pure spirit, of , 825 specific gravity at 60° of heat, may at once be ascertained, the temperature and specific gravity of the liquor being given; pursuant to the idea suggested in the Report, that “the simplest and most equitable “method of levying the duty on spirituous liquors would be, “to consider rectified spirit as the true and only excisable “matter.“