By Knut Mørken
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4 = 3 and 5 = 5. 7. For fractional numbers we do need to know how many digits to compute as there may often be infinitely many. 16. It is convenient to have a standard way of writing down the computations involved in converting a fractional number to base-β, and it turns out that we can use the same format as for converting natural numbers. 15 where the fraction 1/5 was converted to base-8. We start by writing the number to be converted to the left of the vertical line. We then multiply the number by β (which is 8 in this case) and write the integer part of the result, which is the first digit, to the right of the line.
Conversely, a binary fractional number can be converted to hexadecimal by converting each group of four successive binary digits to hexadecimal, starting with the most significant digits. A couple of examples will illustrate how this works in general. 18. 3a816 to binary. 0011 1010 12 . 19. 1 that 11002 = c 16 , 10012 = 916 , 01102 = 616 , 46 10002 = 816 . Note that the last group of binary digits was not complete so we added three zeros. c96816 . 4 Properties of Fractional Numbers in Base-β Real numbers in the interval (0, 1) have some interesting properties related to their representation.
In other words we may hire 50 people, give them two numbers each and tell them to compute the sum. And when everybody is finished, we can of course repeat this and ask 25 people to add the 50 results, 50 s= i =1 1 1 1 1 + a 48 + a 49 + a 50 . a i1 = a 11 + a 21 + a 31 + a 41 + a 51 + a 61 + · · · + a 47 a 12 a 22 a 32 17 2 a 24 2 a 25 At the next step we ask 13 people to the compute the 13 sums 25 s= i =1 2 2 2 2 2 + a 22 + a 25 . + a 23 + a 24 a i2 = a 12 + a 22 + a 32 + a 42 + a 52 + a 62 + · · · + a 21 a 13 a 23 a 33 2 a 11 3 a 12 3 a 13 Note that the last person has an easy job; since the total number of terms in this sum is an odd number she just needs to remember the result.