Understanding the Termination of 7/8 and Its Decimal Representation

When dealing with fractions, one of the key aspects to understand is whether their decimal representation terminates or repeats.

Is 7/8 a Terminating Decimal?

The fraction 7/8, when converted to a decimal, results in 0.875. This particular decimal does not repeat and terminates after three decimal places. This example is particularly interesting as it aligns with a broader rule in mathematics.

Why 7/8 Terminates

The reason why 7/8 is a terminating decimal is directly related to the properties of its denominator. Specifically, the denominator 8 can be expressed as a power of 2, i.e., (8 2^3). According to the rule mentioned, any fraction where the denominator is composed solely of the prime factors 2 and/or 5 will have a terminating decimal representation. In the case of 7/8, the denominator is (2^3), hence the decimal terminates.

General Rule

A more general statement about the decimal representation of fractions can be made: if a fraction m/n is in its simplest form, and the denominator n has no prime factors other than 2 and 5, then the decimal representation of m/n terminates. For the fraction 7/8, both 7 and 8 are in their simplest form, and the prime factors of 8 are only 2's. Therefore, 7/8 will have a terminating decimal representation.

Examples and Comparisons

Another fraction to consider is 1/8. When converted to a decimal form, 1/8 equals 0.125, which is also a terminating decimal. Notably, 1/8 can be expressed as 125/1000, providing an example of a fraction over a power of ten. Multiplying this by 7 gives 7/8, reinforcing the principle discussed.

Interestingly, the fraction 7/8 may not always appear in its simplest form. The decimal representation can be written as 0.875, but it could theoretically also be expressed as 0.874999999999999…, illustrating the flexibility in decimal representation while still remaining a terminating decimal.

Terminating Decimals in Base 10

In the context of base 10, any fraction in simplest form where the prime factorization of the denominator consists only of 2s and/or 5s will result in a terminating decimal. For instance, 7/8 is in simplest form, and its denominator 8 is expressed as (2^3). Therefore, completing the division 7 by 8 will always yield a terminating decimal, ending in 0.875.

Understanding these principles is crucial for anyone working with decimal representations and fractions, whether in mathematical theory or practical applications. The consistent rule about denominators with prime factors of 2 and 5 ensuring termination provides a solid foundation for these operations.