AA What Is the Decimal Form of 1 12

AA What Is the Decimal Form of 1 12

What Is the Decimal Form of 1 12

This decimal is written as "1. 12". It is represented with an underline.



We said earlier that all fractions when put into decimal form either terminate or recurr. This is true for our fractions whose decimal fractions correspond to a particular number series. They cannot go on for ever in the decimal digits! The reason lies in the fact that the numbers (the powers of 2, say) appear with two digits each. When we get to a power greater than 100, there will be an "overflow" into the 2-digit power before it. In fact what happens is that we do include every number from 0 upwards but the overflows eventually cause the decimal to get into a recurring sequence. There are lots of other series such as powers of 2: 0, 1, 2, 4, 8, 16, ... or the Fibonacci numbers 0,1,1,2,3,5,8,13,21,... which appear as the decimal form of some special fractions, but not all series! There is no fraction that gives the primes numbers for example. So which series can we find in our decimal fractions and which can we not find?

We usually call the base-B digits in the expansion of a base-B fraction the decimal form of the fraction even if the base is not 10. We ought really to talk about the bimals for base 2 and the trimals for base 3, etc, but this sounds strange, so we opt for the easier base 2 decimal or base 3 decimal, etc. (On my Fibonacci pages, in the section The Fibonacci Series as a Decimal Fraction we saw how to prove that 1/89 was the Fibonacci series-in-a-fraction by using a generating function, which is a polynomial which encoded the Fibonacci numbers as its coefficients: To move a series, we merely multiply the fraction by a power of 10 to move it to the left in the decimal, or divide it by a power of 10 to move it to the right, introducing zeroes as the new decimal places after the decimal point. We can also just add an integer before we divide too. (Source: www.maths.surrey.ac.uk)


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