Question

Some rational number, such as 1/2, 3/8, and 2/5 have a finite decimal (terminating decimal); while others, such as 1/3, have infinite decimal representation (repeating decimal). For which positive whole numbers, n, is the decimal representation of 1/n finite? For which is it infinite? Describe the pattern, if there is one.

Answer 1

Step-by-step explanation:

Some rational number, such as 1/2, 3/8, and 2/5 are known to have a finite decimal (terminating decimal) as shown

1/2 = 0.5

3/8 = 0.375

2/5 = 0.4

We can see that the values after the decimal points are countable and finite. They are not repeated compare to some fractions like:

1/3 = 0.333333333333333333333333... (The values after the decimal here are infinite i.e they have no end). From the question, we are to look for other values of the denominator n that will make 1/n finite ans infinite.

The value of n that will make 1/n finite is when n = 4, 8, 16... (most of this numbers are even numbers while some are odd e.g 5)

1/4 = 0.25

1/8 = 0.125

1/16 = 0.0625

For infinite, the value of n that will make 1/n infinite are mostly prime numbers e.g 7, 11, 13, 17...

1/7 = 0.142857142857142... (the values 142857 keeps repeating itself)

1/11 = 0.0909090909090909090...(the value 0909 keeps repeating itself)

1/13 = 0.076923076923076923... (the value 076923 keeps repeating itself)

1/17 = 0.0588235294117647... (the value 0588235294117647 will keep repeating itself)