Answer :
Sure, let's fill in the table with detailed steps for each operation.
### Table-1
#### Row 1
1. Rational Numbers: [tex]\( \frac{-1}{2}, \frac{3}{4} \)[/tex]
2. Operation: +
3. Result Calculation:
[tex]\[ \frac{-1}{2} + \frac{3}{4} = \frac{-2 + 3}{4} = \frac{1}{4} \][/tex]
4. Closure Property: Rational number (since the result [tex]\( \frac{1}{4} \)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{-1}{2} + \frac{3}{4} = \frac{3}{4} + \frac{-1}{2} \][/tex]
(Addition is commutative)
#### Row 2
1. Rational Numbers: [tex]\( \frac{-3}{8}, \frac{2}{3} \)[/tex]
2. Operation: -
3. Result Calculation:
[tex]\[ \frac{-3}{8} - \frac{2}{3} = \frac{-3 \times 3 - 2 \times 8}{8 \times 3}= \frac{-9 -16}{24} = \frac{-25}{24} \][/tex]
4. Closure Property: Rational number (since the result [tex]\(\frac{-25}{24}\)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{-3}{8} - \frac{2}{3} \neq \frac{2}{3} - \frac{-3}{8} \][/tex]
(Subtraction is not commutative)
#### Row 3
1. Rational Numbers: [tex]\( \frac{4}{5}, \frac{-1}{5} \)[/tex]
2. Operation: +
3. Result Calculation:
[tex]\[ \frac{4}{5} + \frac{-1}{5} = \frac{4 - 1}{5} = \frac{3}{5} \][/tex]
4. Closure Property: Rational number (since the result [tex]\(\frac{3}{5}\)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{4}{5} + \frac{-1}{5} = \frac{-1}{5} + \frac{4}{5} \][/tex]
(Addition is commutative)
#### Row 4
1. Rational Numbers: [tex]\( \frac{8}{9}, \frac{-5}{7} \)[/tex]
2. Operation: [tex]\(\times\)[/tex]
3. Result Calculation:
[tex]\[ \frac{8}{9} \times \frac{-5}{7} = \frac{8 \times -5}{9 \times 7} = \frac{-40}{63} \][/tex]
4. Closure Property: Rational number (since the result [tex]\(\frac{-40}{63}\)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{8}{9} \times \frac{-5}{7} = \frac{-5}{7} \times \frac{8}{9} \][/tex]
(Multiplication is commutative)
### Final Table
\begin{tabular}{|c|c|c|c|c|c|}
\hline \multicolumn{2}{|c|}{ Rational Numbers } & Operation & Result & Closure Property & Commutative Property \\
\hline[tex]$\frac{-1}{2}$[/tex] & [tex]$\frac{3}{4}$[/tex] & + & [tex]$\frac{1}{4}$[/tex] & Rational number & Yes \\
\hline[tex]$\frac{-3}{8}$[/tex] & [tex]$\frac{2}{3}$[/tex] & - & [tex]$\frac{-25}{24}$[/tex] & Rational number & No \\
\hline[tex]$\frac{4}{5}$[/tex] & [tex]$\frac{-1}{5}$[/tex] & + & [tex]$\frac{3}{5}$[/tex] & Rational number & Yes \\
\hline[tex]$\frac{8}{9}$[/tex] & [tex]$\frac{-5}{7}$[/tex] & [tex]\(\times\)[/tex] & [tex]$\frac{-40}{63}$[/tex] & Rational number & Yes \\
\hline
\end{tabular}
This table provides a clear overview of the operations performed on the given rational numbers, the results, and examines their closure and commutative properties.
### Table-1
#### Row 1
1. Rational Numbers: [tex]\( \frac{-1}{2}, \frac{3}{4} \)[/tex]
2. Operation: +
3. Result Calculation:
[tex]\[ \frac{-1}{2} + \frac{3}{4} = \frac{-2 + 3}{4} = \frac{1}{4} \][/tex]
4. Closure Property: Rational number (since the result [tex]\( \frac{1}{4} \)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{-1}{2} + \frac{3}{4} = \frac{3}{4} + \frac{-1}{2} \][/tex]
(Addition is commutative)
#### Row 2
1. Rational Numbers: [tex]\( \frac{-3}{8}, \frac{2}{3} \)[/tex]
2. Operation: -
3. Result Calculation:
[tex]\[ \frac{-3}{8} - \frac{2}{3} = \frac{-3 \times 3 - 2 \times 8}{8 \times 3}= \frac{-9 -16}{24} = \frac{-25}{24} \][/tex]
4. Closure Property: Rational number (since the result [tex]\(\frac{-25}{24}\)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{-3}{8} - \frac{2}{3} \neq \frac{2}{3} - \frac{-3}{8} \][/tex]
(Subtraction is not commutative)
#### Row 3
1. Rational Numbers: [tex]\( \frac{4}{5}, \frac{-1}{5} \)[/tex]
2. Operation: +
3. Result Calculation:
[tex]\[ \frac{4}{5} + \frac{-1}{5} = \frac{4 - 1}{5} = \frac{3}{5} \][/tex]
4. Closure Property: Rational number (since the result [tex]\(\frac{3}{5}\)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{4}{5} + \frac{-1}{5} = \frac{-1}{5} + \frac{4}{5} \][/tex]
(Addition is commutative)
#### Row 4
1. Rational Numbers: [tex]\( \frac{8}{9}, \frac{-5}{7} \)[/tex]
2. Operation: [tex]\(\times\)[/tex]
3. Result Calculation:
[tex]\[ \frac{8}{9} \times \frac{-5}{7} = \frac{8 \times -5}{9 \times 7} = \frac{-40}{63} \][/tex]
4. Closure Property: Rational number (since the result [tex]\(\frac{-40}{63}\)[/tex] is a rational number)
5. Commutative Property:
[tex]\[ \frac{8}{9} \times \frac{-5}{7} = \frac{-5}{7} \times \frac{8}{9} \][/tex]
(Multiplication is commutative)
### Final Table
\begin{tabular}{|c|c|c|c|c|c|}
\hline \multicolumn{2}{|c|}{ Rational Numbers } & Operation & Result & Closure Property & Commutative Property \\
\hline[tex]$\frac{-1}{2}$[/tex] & [tex]$\frac{3}{4}$[/tex] & + & [tex]$\frac{1}{4}$[/tex] & Rational number & Yes \\
\hline[tex]$\frac{-3}{8}$[/tex] & [tex]$\frac{2}{3}$[/tex] & - & [tex]$\frac{-25}{24}$[/tex] & Rational number & No \\
\hline[tex]$\frac{4}{5}$[/tex] & [tex]$\frac{-1}{5}$[/tex] & + & [tex]$\frac{3}{5}$[/tex] & Rational number & Yes \\
\hline[tex]$\frac{8}{9}$[/tex] & [tex]$\frac{-5}{7}$[/tex] & [tex]\(\times\)[/tex] & [tex]$\frac{-40}{63}$[/tex] & Rational number & Yes \\
\hline
\end{tabular}
This table provides a clear overview of the operations performed on the given rational numbers, the results, and examines their closure and commutative properties.
