Is 5.33333….. (3 continuing) rational or irrational?
The correct answer and explanation is:
Correct Answer: Rational
The number 5.33333… (with the digit 3 repeating infinitely) is a rational number.
Explanation (Approximately 300 words):
To determine whether a number is rational or irrational, we must first understand what these terms mean.
A rational number is any number that can be expressed as a ratio of two integers, i.e., in the form a/b, where a and b are integers and b ≠ 0. In contrast, an irrational number cannot be written as a simple fraction. Irrational numbers have non-repeating, non-terminating decimal expansions (like √2 or π).
Now, consider the number 5.33333…, where the digit 3 repeats forever. This is a repeating decimal, and repeating decimals are always rational because they can be converted into a fraction.
Let’s demonstrate how to write 5.33333… as a fraction:
Let x = 5.33333…
We isolate the repeating part by multiplying x by 10 (since the repeating part is just one digit):
10x = 53.33333…
Now subtract the original equation (x = 5.33333…) from this:
10x – x = 53.33333… – 5.33333…
This simplifies to:
9x = 48
Now solve for x:
x = 48 / 9
Simplify the fraction:
x = 16 / 3
So, 5.33333… = 16/3, which is a ratio of two integers. This proves that the number is rational.
It is important not to confuse non-terminating decimals with irrational numbers. Only non-repeating, non-terminating decimals are irrational. Since 5.33333… has a repeating pattern, it fits the definition of a rational number.
In conclusion, 5.33333… is rational because it can be written exactly as the fraction 16/3, and any decimal that repeats is always rational.