Rational Numbers Decimal Expansions

To determine whether a decimal expression is terminating or non-terminating, you can use the following criteria:

1. If the decimal expression is a fraction with a numerator and denominator that are both integers, it will be terminating if the denominator is divisible by no primes other than 2 and 5. For example, the fraction 3/2 is a terminating decimal because 2 is divisible by only 2, and the fraction 3/4 is a terminating decimal because 4 is divisible by only 2.
2. If the decimal representation of number is a repeating decimal, it is non-terminating. A repeating decimal is a number in which one or more digits repeat indefinitely. For example, 0.373737… is a repeating decimal because the digits 37 repeat indefinitely.
3. If the decimal expression is a non-repeating decimal that is not a fraction with a denominator that is divisible by only 2 and 5, it will be non-terminating. For example, 0.1 is a non-terminating decimal because it is not a fraction and the digits do not repeat.

These criteria apply to decimal expressions that are expressed in denominator divisible by 10. In other cases, the rules for determining whether a decimal expression is terminating or non-terminating may be different.

To determine whether a fraction is terminating or non-terminating, you can use the following criteria:

1. If the fraction is expressed in such a way that the denominator is divisible by no primes other than 2 and 5, the fraction will be terminating. For example, the fraction 7/4 is a terminating decimal because 4 is divisible by only 2, and the fraction 3/40 is a terminating decimal because 40 is divisible by only 2 and 5.
2. If the fraction is expressed in such a way that the denominator is divisible by primes other than 2 and 5 also, the fraction will be non-terminating. For example, the fraction 1/6 is a non-terminating decimal because the denominator has a factor of 3 also which is differ from 2 or 5.
3. These criteria apply to fractions that are expressed in denominator as a fraction of 10. In other bases, the rules for determining whether a fraction is terminating or non-terminating may be different.

To check whether a fraction is terminating or non-terminating, you can divide the denominator by 2 and 5 and see if there is a remainder or not. If there is no remainder, the fraction is terminating. If there is a remainder, the fraction is non-terminating.