8/10/2019 Notes on Arithmetic for the Practical Man (Fr Understndng in Gnrl Nd Algrthms of Arthmtc Esp Dvsn Nd Mre)

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For dividends and divisors consisting of several figures use is madeof such a division table and the complete division is carried out insteps or stages corresponding to the successive steps in multiplication.According as the divisor is less than 21 or greater than 20 (whenTable I is used) one or the other of two general methods is used.These two methods are known as short division and long division.

NOTE ON TWO METHODS OF DIVISION: Thus there is long division and short division.

In short division the divisor is divided directly into each successivefigure or pair of figures of the dividend, beginning at the left, and thesuccessive separate quotients written down one after the other to forma number which is the complete quotient.

As an illustration, let us divide 129636 by 3. As 3 is not con-tained in 1 a whole number of times the first two figures, 12, aretaken as a partial dividend. Then 12 + 3 = 4, 9 + 3 = 3, 6 + 3 = 2,3 + 3 = 1, and the complete quotient is 43212. That is, 129636 +3 = 43212. This operation is usually written out in the following

3)129636 ..form: 43212 and the separate quotients are wntten below the cor-responding separate dividends.

If any of the separate dividends (figures of the complete dividend)are smaller than the divisor, or do not contain the divisor exactly awhole number of times, a slightly different procedure is followed, asillustrated by the following example: Divide 189576 by 9. We get18 + 9 = 2 and 9 + 9 = 1, but 9 is not contained in 5, so a zero iswritten for the next figure of the quotient. The next individual divi-dend is then taken as 57 instead of 7, and 57 + 9 = 6 with a re-mainder of 3 (since 6 X 9 = 54). This 3 is written (or imagined

written) before the next figure, 6, giving 36, and this 36 is used as thenext dividend. The final figure of the quotient is then 36 + 9 = 4,the successive figures of the complete quotient are 2, 1, 0, 6 and 4.The result is, therefore, 189576 + 9 = 21064. The complete opera-. .. fi II 9 )189576tlOn IS written as 0 ows: 21064'

NOTE VERY IMPORTANT ON HOW SHORT DIVISION IS PERFORMED AND THE LOGIC BEHIND IT AND WHAT TO DO IF NUMBER DOES NOT GO INTO EVENLY 13!: Thus in this examplewe start from the left (that is we start at the most significant digits, if we have 1243, we start at 1 instead of 3).

1.1 Also notice that if a digit of the dividend is too small, then we take the next number with the number that was too small.1.1.1 We might ask while we do this, well if we consider the principle of decima

8/10/2019 Notes on Arithmetic for the Practical Man (Fr Understndng in Gnrl Nd Algrthms of Arthmtc Esp Dvsn Nd Mre)

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l place and value, the 1 in 129636 actually represents 100,000 and the 2 represents 20,000.To divide 100,000 by 3 would be messy and not able to algorithmatize. Thus 120,000 however is very clean and hence we perform that division first.1.2 This leads us to the concept of "regrouping" for short division similar to that which we do in addition and subtraction.If We divide 189576 by 9, we get 18/9 = 2, 9/9 = 1, but 9 is not contained in 5

and so we put a 0 for this figure in the quotient, the next individual dividendtaken will thus be 57 insetad of 7.57/9 = 6 with a remainder of 3. This 3 is written (or imagined wirtten) beforethe next figure, which is 6, givign 36 ,and this 36 is used as the next divident. 36/9 = 4.1.2.1 We may say that the 5 represented 500 and not a "5" so why not just only take 57, instead of writing "0", since writing "0" seems to indicate that it can't go into that.1.2.2 The "0" actually indicates that instead of taking 9/500, that instead we will take 9/570, but still why does it makesense to put a "0" there in the algorithm?????????????????

????????????????1.2.5 When we got remainder "3" , that remainder 3 was because of the "7", sinceif it was 54 there would be no remainder. Thus that "3" is really a 30, whichis why we add it to the "6" in the one's placeto get 36.1.3 Thus in short division what we're actually doing is using the distributive property of division. We are actually taking apart the number from most significant digitto least significant digit and dividing each of those parts by the divisor one at a time. Then we put this partial result below the proper decimal place in ouranswer becauseit correlates to the significance of the digit we were dealing with at each step.