International projectionist (Oct 1931-Sept 1933)

16 INTERNATIONAL PROJECTIONIST September 1933 Any group of numbers, in order that they may be correctly operated upon mathematically, must either originally be all of the same unit or converted to the same unit before any calculations are made. Thus, it is impossible to add bananas to apples to obtain any sura other than, say 5 bananas and 6 apples. In like manner, it is not possible to add volts to amperes, or frequency to decibels. Units which are multiples of some other unit under consideration (as for instance, volts and milli-volts), may be added after the proper conversion factor has been introduced. For instance, to add 117 milli-volts to 23 volts, it would be necessary to convert both numbers to milli-volts, and the sum would become 117 milli-volts added to 23,000 millivolts, or 23,117 milli-volts, total. Multiplication has been described as merely a short-cut method of addition, — as, similarly, is division a short-cut method of subtraction. It can easily be understood that if we were to add seven fives together, we would arrive at the sum of 35; but to considerably shorten this process we multiply the five by seven to arrive at the product 35. Let us consider the number 9,745, which is to be multiplied by 486. This, for purposes of calculation, may be considered as 9,745 multiplied by 6, plus 80, plus 400. If we multiply the 9,745 by 6, we obtain the product 58,470 (6 times 5 equals 30, putting down the 0 and carrying the 3 ; 6 times 4 is 24 plus the 3 carried over, making 27 — put down the 7 and carry the 2 ; 6 times 7 is 42, plus the 2 carried over is 44 — put down the 4 and carry the 4; and 6 times 9 is 54, plus the 4 carried over makes 58). Similarly, multiplying the 9,745 by 80, we obtain the product 779,600, and when we multiply the same number by 400 we obtain the product 3,898,000. Adding these three products together, we see that the total is 4,736,070. 9,745 486 58,470 779,600 3,898,000 4,736,070 Many times, in electrical formulas, a number or a letter will appear followed by an exponent, such as 5^ or F^. These are read as "5 squared" or "F cubed," and mean that the number so modified is to be, in the case of the exponent 2, squared, (multiplied by itself) or in the case of exponent 3, cubed (multiplied by itself twice). Higher exponents, as 5* or F^ are read as ''five to the fourth power," or "F to the seventh power," respectively, and merely mean that the number is to be multiplied by itself a number of times one less than the numerical value of the exponent. Numbers so modified may be substituted for in any formulae in the following manner, — 5^ may be replaced by 5 x 5, or F'^ may be replaced byFxFxFxFxFxFxF. rhe actual method of handling such numbers, which may be considerably simplified beyond an actual multiplication of the number by itself for any given exponent, will be taken up subsequently. The fourth arithmetical operation with which we are concerned is division, which is probably the most difficult of the group. Division should be thought of always as the reverse of multiplication; for instance, 5x6 being 30, — 30 divided by 6 will equal 5, as will similarly 30 divided by 5 equal 6. When the number to be divided (the divisor), into any larger number (the dividend) is smaller than 10, it is customary to use the process known as short division. Short division is entirely a mental process, and should present no difficulty, after some practice, to one who has mastered the multiplication table. If, for example, it is desired to divide 544 by 8, by short division 8 will go into 54, 6 times with a remainder of 6 left over (6 times 8 being 48, and 54 less 48 being 6). Carrying the remainder of 6 to our next figure, 8 will go into 64 exactly 8 times, giving a quotient (the result of the division operation being known as the quotient) of 68. The process of long division is essentially similar to that of short division, except that certain figures are set down on the paper to simplify the mental eftort connected with the operation. Considering the number 2,431 divided by 17: 143 (a) 17)2431 17 73 68 51 51 As the number 17 obviously is too large to be divided into 2, we try the next figure, successfully making our division 24 by 17. As 17 will go into 24 once only, we place our 1 in the space provided for the quotient (line a, in the example above), multiply 17 by 1, placing this result below the 24. Subtracting, we obtain a remainder of 7. Bringing down the 3 from the dividend above, we again determine by trial and error that 17 will go into 73 approximately 4 times. Placing the 4 in the quotient line above, we multiply 17 by 4, obtaining 68, which is subtracted from 7Z, leaving a remainder of 5. Bringing down from the quotient above the figure 1, we determine that 17 will go into 51 approximately 3 times. Placing the 3 in the quotient, and multiplying, the product equals 51 and there is no remainder. However, should the dividend have been, instead of 2431, 2435, it can be determined that after the final operation above there would have been a remainder of 4. This may be handled in a variety of ways, but for present purposes it will be best to carry any remainder as a proper fraction, in this case 4/17, making the quotient 143 and 4/17. For technical calculations, the use of decimals has become universally standard, and all technical formulas, units, and tables are based upon the decimal system, which is merely a method for expressing and handling numbers less than 1, in terms of tenths. Calculations and use of the system should present no difficulty, in that the entire monetary system of the United States is based upon the decimal. In adding or subtracting decimals, it should be borne in mind that the decimal point should always be kept in the same vertical line, as follows : 8.754 9,023.02 .1111 97.5 9,129.3851 In the multiplication or division of decimals, this is not the important factor. In multiplication, the actual multiplication is done exactly as if there were no decimals, and the total number of figures ("places") to the right of the decimal in both multiplier and multiplicand is determined, and the decimal placed the same number of "places" to the left in the product. An example : 45.73 22.2 9146 9146 9146 1015206 which after correctly placing the decimal point, becomes 1,015.206. In dividing decimals, the common practice is to, in effect, multiply both divisor and dividend by a factor of 10 which will make the divisor a whole number, and then proceed in the usual way. For example, if it is desired to