The length of the decimal expression of a fraction depends on therelationship be

Question

The length of the decimal expression of a fraction depends on therelationship between its denominator and the base of the placesystem.
a) Which fractions can be expressed as decimals of no more than 3places (after the decimal point)? Which can be expresed asdecimals of no more than 5 places? Explain.
b)Which fractions can be expressed exactly as finite decimals?Explain.
c) Give three examples of fractions that cannot be expressedexactly as finite decimals. Justify your answers.
d) Are there fractions that can be expressed in finitely manyplaces in the Babylonian sexagesimal (base 60) system, but not inout decimal system?
If so, give three examples (with justification). If not, whynot?
e) Are there fractions that can be expressed in finitely manyplaces in our decimal system, but not in the Babylonian sexagesimalsystem? If so, give three examples (withjustification). If not, why not?

Explanation / Answer

a. Any fraction which can be written in the form x/1000 can beexpressed as decimals to 3 places. So, 352/1000 can bewritten as .352. 4697/1000 can be written as 4.697. Any fraction which can be written in the form x/100000 can beexpressed as decimal to 5 places, by the same logic as thefirst part of this question. b. A fraction can be expressed as a finite decimal if it can bewritten as x/10^y, where x and y are finite, rational numbers. c) 1/3: 31 will give you 3, with a remainder of1. Dividing again will again give 3 with a remainder of1. The cycle will never end. 2/3 is similar, but you'll keep getting 6 with a remainder of 2,infinitely /10. Since cannot be written as a finitedecimal, neither can /10 . d) Are there fractions that can be expressed in finitely manyplaces in the Babylonian sexagesimal (base 60) system, but not inout decimal system? Sure! There are all sorts of numbers that don't give finitedecimals when divided by powers of 10, but do give them whendivided by 60. 20/60 = 1/3 is infinitely repeating in oursystem, but in the babylonian system, you would get 20! Same with 2/3, 1/6, 5/6, etc. e) nope! anything that can be written with a power of 10 as adenominator can be written with a power of 60 as a denominator.

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