To determine the type of number represented by [tex]\(\frac{0.\overline{55}}{0.\overline{55}}\)[/tex], let's start by interpreting the repeating decimal [tex]\(0.\overline{55}\)[/tex].
1. Expressing [tex]\(0.\overline{55}\)[/tex] as a Fraction:
The repeating decimal [tex]\(0.\overline{55}\)[/tex] can be expressed as a fraction. To find the exact fraction representation:
- Let [tex]\(x = 0.\overline{55}\)[/tex].
- Multiply both sides by 100 (since the repeating part has two digits): [tex]\(100x = 55.\overline{55}\)[/tex].
- Subtract the original equation from this new equation to get rid of the repeating part: [tex]\(100x - x = 55.\overline{55} - 0.\overline{55}\)[/tex].
- This simplifies to: [tex]\(99x = 55\)[/tex].
- Solving for [tex]\(x\)[/tex]: [tex]\(x = \frac{55}{99}\)[/tex], which can be simplified by dividing the numerator and denominator by their greatest common divisor (GCD):
[tex]\[ \frac{55}{99} = \frac{5}{9}. \][/tex]
2. Evaluating [tex]\(\frac{0.\overline{55}}{0.\overline{55}}\)[/tex]:
Now, let's evaluate the given expression:
[tex]\[
\frac{0.\overline{55}}{0.\overline{55}} = \frac{\frac{5}{9}}{\frac{5}{9}}.
\][/tex]
Dividing a number by itself (given that the number is not zero) yields:
[tex]\[
\frac{\frac{5}{9}}{\frac{5}{9}} = 1.
\][/tex]
3. Determining the Types of the Result [tex]\(1\)[/tex]:
The number [tex]\(1\)[/tex] can be classified in various ways:
- Whole Number: Whole numbers are non-negative integers including zero (0, 1, 2, 3, ...). Since [tex]\(1\)[/tex] is a whole number, it satisfies this classification.
- Integer: Integers include all whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...). Since [tex]\(1\)[/tex] is part of this set, it is an integer.
- Rational Number: A rational number can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]. The number [tex]\(1\)[/tex] can be expressed as [tex]\(\frac{1}{1}\)[/tex], making it a rational number.
- Irrational Number: An irrational number cannot be expressed as a simple fraction (i.e., it cannot be written as [tex]\(\frac{a}{b}\)[/tex] with integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex]). Since [tex]\(1\)[/tex] can be expressed as [tex]\(\frac{1}{1}\)[/tex], it is not irrational.
Therefore, the number [tex]\(1\)[/tex] is:
- A Whole number (Choice A)
- An Integer (Choice B)
- Rational (Choice C)
- Not Irrational (Choice D)
So, the correct answers are:
- A Whole number
- B Integer
- C Rational
