To determine if the fraction [tex]\(\frac{1}{2}\)[/tex] is a terminating decimal or not, we need to understand what makes a fraction's decimal representation terminating.
A decimal representation of a fraction is said to be terminating if it has a finite number of digits after the decimal point. For a fraction [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers, the decimal will terminate if and only if the denominator [tex]\(q\)[/tex], after simplifying the fraction, has no prime factors other than 2 and/or 5. These prime factors relate to the base-10 number system.
Let's analyze [tex]\(\frac{1}{2}\)[/tex]:
1. Simplification: First, we note that [tex]\(\frac{1}{2}\)[/tex] is already in its simplest form – the numerator 1 and the denominator 2 share no common factors besides 1.
2. Prime Factors of the Denominator: For this fraction, the denominator is 2. The only prime factor of 2 is itself, which means the denominator does not have any prime factors other than 2 and/or 5.
Since the denominator 2 only has the prime factor 2, this indicates that [tex]\(\frac{1}{2}\)[/tex] will indeed have a terminating decimal representation.
To confirm, we can perform the division:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
This is a simple terminating decimal, as it has a finite number of digits after the decimal point (just one digit, in this case).
Therefore, [tex]\(\frac{1}{2}\)[/tex] is confirmed to be a terminating decimal.
