Is 3/10 a non terminating and repeating?
The Correct Answer and Explanation is:
To determine whether the fraction ( \frac{3}{10} ) is non-terminating and repeating, we first need to understand the definitions of terminating and non-terminating decimal representations.
Terminating Decimal: A decimal is considered terminating if it has a finite number of digits after the decimal point. For example, ( 0.25 ) and ( 0.5 ) are terminating decimals.
Non-Terminating Decimal: A decimal is non-terminating if it continues indefinitely without repeating. An example of this is ( \pi ) or ( e ).
Repeating Decimal: A decimal is termed repeating if it has a specific sequence of digits that recurs indefinitely. An example of a repeating decimal is ( 0.333… ), which can be expressed as ( \frac{1}{3} ).
Now, let’s analyze the fraction ( \frac{3}{10} ). When we convert this fraction into a decimal:
[
\frac{3}{10} = 0.3
]
The decimal ( 0.3 ) is a terminating decimal because it has a finite number of digits (only one digit after the decimal point). Therefore, it does not continue indefinitely nor does it repeat.
To further clarify, a fraction can be expressed in decimal form depending on the prime factorization of its denominator. If the denominator can be expressed as ( 2^m \times 5^n ), where ( m ) and ( n ) are non-negative integers, the fraction will result in a terminating decimal. In this case, ( 10 = 2^1 \times 5^1 ), confirming that ( \frac{3}{10} ) meets this criterion.
In conclusion, ( \frac{3}{10} ) is a terminating decimal and, therefore, is not non-terminating or repeating. It has a specific, finite decimal representation: ( 0.3 ).