Question

without actually performing the long division state whether the following rational number why have a terminating decimal expansion or non-terminating repeating decimal expansion: 1 upon 7 , 1/11​

Answer 1

We know that

Let x be a rational number whose decimal expansion terminates then x can be expressed in the form of p/q where p and q are coprime in the prime factorization of q is of the form 2^n× 5^m for the sum non-negative integer m and n

let x = p/q be a rational number such that the prime factorization of q is not of the form 2^n× 5^m where n and m non-negative Integers. Then x has a decimal expansion which is non- terminating repeating

1.) 1/7

Notice that the denominator here 7 is clearly not of the form 2^n and 5^m. Therefore, it is a non- terminating repeating decimal expansion

2.) 1/11

Notice that the denominator here 11 is clearly not of the form 2^n and 5^m. Therefore, it is a non- terminating repeating decimal expansion