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By Aaron Bakst

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Thus we can clear up the mystery of the regular order of the digits in the products of 142,857 by 1, 2, 3, 4, 5, and 6. When we multiply 142,857, say by 4, we may think of this product as 4/7. Now, if we transform 4/7 into a decimal fraction, we note that 4 is one of the remainders obtained when we transformed 1/7 into a decimal fraction. Naturally the same remainders must be obtained and in the same order, but they must begin with some other digit, and not 1, as in the case of 1/7. This means that when a fraction with a denominator 7, but with a numerator other than 1, is transformed into a decimal fraction we obtain the same period, except that some of the digits of 142,857 will be transposed to the right.

Write Write Write Write 2 and carry 9. 1 and carry 9. 1 and carry 9. ge. Short Turns in Long Division Division, too, follows the same pattern in all systems of numeration. The reader is advised to go back over the sections on subtraction and multiplication before proceeding. Below are examples of division: Two-System 11011011 [111 111 11111 1101 111 1100 111 1011 111 1001 111 10 10 Thus, 11,011,011 -;- 111 = 11111·111 +ITl' 24 Mathematics-Its Magic & Mastery Thus, 2,122 + 12 = Three-System 2122 ~ ~ 112 22 12 102 101 1 112 ·12 + 1.

Multiplication in the two-system is so easy that it bears out the statement that this is the simplest of all systems. As you know, there are only two digits, 0 and 1, and multiplica~ion by 1 naturally results in the same number. Thus: 10011 101 10011 10011 1011111 The product of two numbers in the three-system is obtained as follows: 112 221 112 1001 1001 110222 The multiplication by 1 results in the original number 112. Multiplying 112 by 2 is done this way: 2·2=11, 2·1 = 2, and 2 2· 1 = 2, and 2 +1= +1= 10.

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Mathematics: Its Magic and Mystery by Aaron Bakst


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