International projectionist (Oct 1931-Sept 1933)

Not a few projectionists find the way to a better understanding of developments affecting their daily work barred by an unfamiliarity with mathematics, a subject of increasing importance to the projectionist. Many articles of decided help to the projectionist are hurriedly glossed over because of the difficulty experienced in correctly interpreting mathematical terms and symbols and, in particular, in applying mathematics to one's work. The accompanying article is the first of a series which is intended to meet this acute need within projectionist ranks. The series begins with consideration of fundamentals (by which is meant simple arithmetic), and progressively works along toward and deals with algebra, geometry, trigonometry and the calculus. The articles are infinitely more interesting than one might suspect from this description of their content, the transition from one stage of the work to another being accomplished in easy step-by-step fashion. The value of this series to the projectionist can be measured beforehand in terms of the degree of determination which he exhibits in applying himself thereto. Unquestionably the series should prove exceedingly helpful; but this publication can do no more than present it in these pages. Following each installment will be a group of questions bearing on the information given immediately preceding. Projectionists are invited to send in to International Projectionist their answers to these problems. The names of those who submit correct papers will be printed along with the answers in the following issue. — Editor. MATHEMATICS FOR THE PROJECTIONIST Gordon S. Mitchell THERE is probably nothing which seems more complicated or is more discouraging than to look ahead in a mathematics course, with its complex formulae and intricate text explanations. By the same token there is nothing that gives one more real satisfaction than to glance back over the problems and the text of a course well done, realizing that those things which seemed to be most involved have proven to be rather simple after all. To shorten and simplify many of the principles and operations of the more advanced mathematical subjects, a knowledge of the four arithmetical fundamentals is necessary. Individuals react in different ways to numbers and numerical calculations, and while there are countless shortcuts and tricks which may be used in yrithmetical calculations, many of these require as much or more mental effort as would a straightforward solution of the problem at hand. In adding a column of figures, such as might be necessary in order to determine the total resistance of a set of resistors connected in series, it is well to carry only the totals mentally as the columns are added. For instance, in adding: 749 987 765 579 682 224 374 from the bottom to top of the righthand column of figures, the mental operation would be somewhat as follows: 4, 8, 10, 19, 24, 31, and 40. The 0 should be placed as the right-hand figure of the sum, the 4 carried to the next column, which will add somewhat as follows: 4, 11, 13, 21, 28, 34, 42, and 46. Placing the 6 in the sum and carrying the 4 to the next column, and so on, we obtain a total for the column of 4,360. As a method for shortening the mental labor involved in adding a column of figures, and after some dexterity has been obtained in handling numbers, groups of numbers in any column which total 10 may be mentally grouped and considered during the adding process as one number. Referring to the example above, we would by this method group the lower three figures in the right-hand column (the 4, 4, and 2), mentally and add them as 10, 19, 24, 31, and 40. In like manner, the second column would be added, 4, 11 (grouping the 8 and 2), 21, 28, 34, 42, and 46. As additional practice in adding increases mental skill, it will become possible to group numbers which add to ten, although they may not appear together in the column, and thus increase the speed of addition. The second column of the foregoing example would then be added 4, 11, 21, 28 (grouping the 6 and the 4), 38, and 46. [15] Subtraction of numbers probably presents as little difficulty as any mathematical operation, but a single example will be solved in order that a complete set of fundamental processes may be had for any reference or practice which may be desired. Given the number 1,234, which is to be subtracted from 4,321 : 4,321 1,234 it can be easily determined that 4 cannot be taken away from 1, consequently it becomes necessary to "borrow" 10 from the adjacent number, making our first subtraction 4 from 11, leaving 7. Inasmuch as we have borrowed 10 from the 2, we find in the next column a subtraction of 3 from 1, which is again impossible, and necessitates borrowing 10 from the adjacent 3. This subtraction then becomes 3 from 11, leaving 8. In the next column, our 3, having by reason of the borrowing of 10 during the previous subtraction, become reduced to 2, leaves a subtraction of 2 from 2, which is 0. A straightforward subtraction of 1 from 4, leaving 3, completes the operation, giving an answer of 3,087. This may be readily checked by adding the answer, 2,087 to the number being subtracted, 1,234, which gives the sum of 4,321. An important point, and one which often proves puzzling, is the problem of adding or subtracting unlike units.