Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Let's analyze each expression step-by-step and determine which sets they belong to. We'll place an [tex]\( X \)[/tex] in the correct columns accordingly.
### 1. Expression: [tex]\(\frac{2}{9}\)[/tex]
- Natural Numbers: These are positive integers starting from 1 onwards (1, 2, 3, ...). [tex]\(\frac{2}{9}\)[/tex] is not a natural number.
- Whole Numbers: These include all natural numbers plus zero (0, 1, 2, 3, ...). [tex]\(\frac{2}{9}\)[/tex] is not a whole number.
- Integers: These include all positive and negative whole numbers and zero (..., -3, -2, -1, 0, 1, 2, 3, ...). [tex]\(\frac{2}{9}\)[/tex] is not an integer.
- Rational Numbers: These numbers can be expressed as the quotient of two integers, where the denominator is not zero. [tex]\(\frac{2}{9}\)[/tex] can be expressed as [tex]\(\frac{2}{9}\)[/tex], so it is a rational number.
- Irrational Numbers: These cannot be expressed as the quotient of two integers. [tex]\(\frac{2}{9}\)[/tex] is a rational number, not irrational.
- Real Numbers: These include both rational and irrational numbers. Since [tex]\(\frac{2}{9}\)[/tex] is rational, it is also a real number.
Thus, for [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text{Expression} & \text{Natural} & \text{Whole} & \text{Integer} & \text{Rational} & \text{Irrational} & \text{Real} \\ \hline 1. & \frac{2}{9} & & & & X & & X \\ \hline \end{array} \][/tex]
### 2. Expression: [tex]\(\sqrt{2}\)[/tex]
- Natural Numbers: [tex]\(\sqrt{2}\)[/tex] is not a whole number, nor is it a positive integer.
- Whole Numbers: [tex]\(\sqrt{2}\)[/tex] is not a whole number.
- Integers: [tex]\(\sqrt{2}\)[/tex] is not an integer.
- Rational Numbers: [tex]\(\sqrt{2}\)[/tex] cannot be expressed as a ratio of two integers, so it is not rational.
- Irrational Numbers: [tex]\(\sqrt{2}\)[/tex] is an irrational number, as it cannot be expressed as the quotient of two integers.
- Real Numbers: [tex]\(\sqrt{2}\)[/tex] falls within the set of real numbers since it is either rational or irrational.
Thus, for [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text{Expression} & \text{Natural} & \text{Whole} & \text{Integer} & \text{Rational} & \text{Irrational} & \text{Real} \\ \hline 2. & \sqrt{2} & & & & & X & X \\ \hline \end{array} \][/tex]
### 3. Expression: [tex]\(\sqrt{(3)^2 - 4(1)(2)}\)[/tex]
First, simplify the expression inside the square root:
[tex]\[ (3)^2 - 4(1)(2) = 9 - 8 = 1 \][/tex]
Then, take the square root of the result:
[tex]\[ \sqrt{1} = 1 \][/tex]
Since [tex]\(1\)[/tex] is obtained, let's see which sets it belongs to:
- Natural Numbers: [tex]\(1\)[/tex] is a natural number.
- Whole Numbers: [tex]\(1\)[/tex] is a whole number.
- Integers: [tex]\(1\)[/tex] is an integer.
- Rational Numbers: [tex]\(1\)[/tex] can be expressed as [tex]\(\frac{1}{1}\)[/tex], so it is a rational number.
- Irrational Numbers: [tex]\(1\)[/tex] is rational, not irrational.
- Real Numbers: Since [tex]\(1\)[/tex] is rational, it is also a real number.
Thus, for [tex]\(\sqrt{(3)^2 - 4(1)(2)}\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text{Expression} & \text{Natural} & \text{Whole} & \text{Integer} & \text{Rational} & \text{Irrational} & \text{Real} \\ \hline 3. & \sqrt{(3)^2-4(1)(2)} & X & X & X & X & & X \\ \hline \end{array} \][/tex]
To summarize everything in one table:
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
& Expression & Natural & Whole & Integer & Rational & Irrational & Real \\
\hline
1. & [tex]\(\frac{2}{9}\)[/tex] & & & & X & & X \\
\hline
2. & [tex]\(\sqrt{2}\)[/tex] & & & & & X & X \\
\hline
3. & [tex]\(\sqrt{(3)^2 - 4(1)(2)}\)[/tex] & X & X & X & X & & X \\
\hline
\end{tabular}
\end{center}
### 1. Expression: [tex]\(\frac{2}{9}\)[/tex]
- Natural Numbers: These are positive integers starting from 1 onwards (1, 2, 3, ...). [tex]\(\frac{2}{9}\)[/tex] is not a natural number.
- Whole Numbers: These include all natural numbers plus zero (0, 1, 2, 3, ...). [tex]\(\frac{2}{9}\)[/tex] is not a whole number.
- Integers: These include all positive and negative whole numbers and zero (..., -3, -2, -1, 0, 1, 2, 3, ...). [tex]\(\frac{2}{9}\)[/tex] is not an integer.
- Rational Numbers: These numbers can be expressed as the quotient of two integers, where the denominator is not zero. [tex]\(\frac{2}{9}\)[/tex] can be expressed as [tex]\(\frac{2}{9}\)[/tex], so it is a rational number.
- Irrational Numbers: These cannot be expressed as the quotient of two integers. [tex]\(\frac{2}{9}\)[/tex] is a rational number, not irrational.
- Real Numbers: These include both rational and irrational numbers. Since [tex]\(\frac{2}{9}\)[/tex] is rational, it is also a real number.
Thus, for [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text{Expression} & \text{Natural} & \text{Whole} & \text{Integer} & \text{Rational} & \text{Irrational} & \text{Real} \\ \hline 1. & \frac{2}{9} & & & & X & & X \\ \hline \end{array} \][/tex]
### 2. Expression: [tex]\(\sqrt{2}\)[/tex]
- Natural Numbers: [tex]\(\sqrt{2}\)[/tex] is not a whole number, nor is it a positive integer.
- Whole Numbers: [tex]\(\sqrt{2}\)[/tex] is not a whole number.
- Integers: [tex]\(\sqrt{2}\)[/tex] is not an integer.
- Rational Numbers: [tex]\(\sqrt{2}\)[/tex] cannot be expressed as a ratio of two integers, so it is not rational.
- Irrational Numbers: [tex]\(\sqrt{2}\)[/tex] is an irrational number, as it cannot be expressed as the quotient of two integers.
- Real Numbers: [tex]\(\sqrt{2}\)[/tex] falls within the set of real numbers since it is either rational or irrational.
Thus, for [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text{Expression} & \text{Natural} & \text{Whole} & \text{Integer} & \text{Rational} & \text{Irrational} & \text{Real} \\ \hline 2. & \sqrt{2} & & & & & X & X \\ \hline \end{array} \][/tex]
### 3. Expression: [tex]\(\sqrt{(3)^2 - 4(1)(2)}\)[/tex]
First, simplify the expression inside the square root:
[tex]\[ (3)^2 - 4(1)(2) = 9 - 8 = 1 \][/tex]
Then, take the square root of the result:
[tex]\[ \sqrt{1} = 1 \][/tex]
Since [tex]\(1\)[/tex] is obtained, let's see which sets it belongs to:
- Natural Numbers: [tex]\(1\)[/tex] is a natural number.
- Whole Numbers: [tex]\(1\)[/tex] is a whole number.
- Integers: [tex]\(1\)[/tex] is an integer.
- Rational Numbers: [tex]\(1\)[/tex] can be expressed as [tex]\(\frac{1}{1}\)[/tex], so it is a rational number.
- Irrational Numbers: [tex]\(1\)[/tex] is rational, not irrational.
- Real Numbers: Since [tex]\(1\)[/tex] is rational, it is also a real number.
Thus, for [tex]\(\sqrt{(3)^2 - 4(1)(2)}\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text{Expression} & \text{Natural} & \text{Whole} & \text{Integer} & \text{Rational} & \text{Irrational} & \text{Real} \\ \hline 3. & \sqrt{(3)^2-4(1)(2)} & X & X & X & X & & X \\ \hline \end{array} \][/tex]
To summarize everything in one table:
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
& Expression & Natural & Whole & Integer & Rational & Irrational & Real \\
\hline
1. & [tex]\(\frac{2}{9}\)[/tex] & & & & X & & X \\
\hline
2. & [tex]\(\sqrt{2}\)[/tex] & & & & & X & X \\
\hline
3. & [tex]\(\sqrt{(3)^2 - 4(1)(2)}\)[/tex] & X & X & X & X & & X \\
\hline
\end{tabular}
\end{center}
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
