396 Mental Multiplication and Division of Large Numbers. 



Then 9 is unit digit of answer and " carry zero."" 



Uncover 8 and proceed with " "+ (8 x 3) x 2 = 48. 



Then 8 is tens digit, and " carry 4." 



Uncover and proceed with " 4 "+ (0 x 3) x 2 + 8 2 = 68. 



Then 8 is hundreds digit and " carry 6." 



Uncover 2 and proceed with " 6 " + (2 x 3 + x 8) X 2 = 18, 

 and so on. When we have uncovered the whole o£ the 

 number, we next begin covering up the digits, beginning 

 from the right end. 



By reversing the above process in a manner analogous to 

 mental division we can obtain square roots to any degree of 

 accuracy desired. 



Suppose we require s/12'1. The first figure is plainly 3. 

 Work out 12 — 3 2 = 3. To the 3 tack on the next integer in 



31 

 the square, viz. 1, making 31. Then — - =4 and 7 over, the 



6 being 2x(lst digit in the root), which will be our only 

 divisor throughout the calculation. To the 7 tack on the 

 next digit in the square, viz. 0, and proceed as follows (line 

 after line), the corresponding stage of the answer (minus its 

 first digit) being shown on the extreme right : — 



31 



54 



A & 7 over 



70-42=54 ^ =7 & 12 over 

 6 



120-2(4 x7) = 64 



160-2(4 X 8) -7 2 =47 



170-2(4x5)-2(7x8)=18 



180-2(4 x0)-2(7x5)-8 2 =46 



160-2(4xo)-2(7x0)-2(8x5)=40 



= 8& 16 over 

 = 5 & 17 over 



=0 & 18 over 

 = 5 & 16 over 

 =4 & 16 over 



4 



47 



478 



4785 



47850 



478505 



4785054 



64 



Of course we have here left out trials like -=- =9 and 



18 



10 over or — =1 and 12 over, which would have given 



o 



us negative remainders on the left. Hence 



^121 = 3-4785054. 

 This is easily verified to be correct, for 



VT2 r l = l-l Vl0 = l-1(3-1622776602). 



