Is a terminating decimal a rational number? This question often arises in the realm of mathematics, particularly when discussing the classification of numbers. In this article, we will explore the relationship between terminating decimals and rational numbers, providing a clear understanding of why terminating decimals are indeed rational numbers.
Terminating decimals are those that have a finite number of digits after the decimal point. For example, 0.25, 3.14, and 12.56 are all terminating decimals. On the other hand, rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero. This means that rational numbers can be either terminating decimals, repeating decimals, or integers.
To prove that a terminating decimal is a rational number, we can demonstrate that it can be expressed as a fraction with an integer numerator and a power of 10 as the denominator. Let’s take the example of 0.25. This terminating decimal can be written as 25/100, which simplifies to 1/4. Since 1 and 4 are both integers, 0.25 is a rational number.
The same process can be applied to any terminating decimal. By multiplying the decimal by a power of 10, we can move the decimal point to the right until it becomes an integer. This integer will serve as the numerator, while the power of 10 will serve as the denominator. The resulting fraction will always have integer numerator and denominator, thus proving that the terminating decimal is a rational number.
It is important to note that terminating decimals are a subset of rational numbers. While all terminating decimals are rational, not all rational numbers are terminating decimals. For instance, the fraction 1/3 is a rational number but cannot be expressed as a terminating decimal. Instead, it is a repeating decimal (0.333…).
In conclusion, a terminating decimal is a rational number because it can be expressed as a fraction with an integer numerator and a power of 10 as the denominator. This relationship highlights the connection between the two mathematical concepts and provides a clear understanding of their interdependence.
