Question

Starting from \[\left( { - 1} \right){\text{ }} \times {\text{ }}5\] write various products showing some pattern to show \[\left( { - 1} \right){\text{ }} \times {\text{ }}\left( { - 1} \right){\text{ }} = {\text{ }}1\]

Answer 1

Hint: We have to show some patterns to show that \[\left( { - 1} \right){\text{ }} \times \left( { - 1} \right){\text{ }} = {\text{ }}1\] using the initial value of \[\left( { - 1} \right){\text{ }} \times {\text{ }}5{\text{ }}\] . We should have the knowledge of concept of multiplication of terms having different signs . We show the value of the product obtained after multiplying the terms having different signs and hence show the multiplication of the required result .

Complete step-by-step answer:
Given :
We have to start from \[\left( { - 1} \right){\text{ }} \times {\text{ }}5{\text{ }}.\]
To prove :- \[\left( { - 1} \right){\text{ }} \times \left( { - 1} \right){\text{ }} = {\text{ }}1\]
Proof :
Using the given multiplication , where a negative number is multiplied by a positive number , we get
 \[\left( { - 1} \right){\text{ }} \times {\text{ }}5{\text{ }} = {\text{ }} - 5\]
Also ,
 \[\left( { - 1} \right){\text{ }} \times {\text{ }}4{\text{ }} = {\text{ }} - 4\]
Similarly ,
 \[\left( { - 1} \right){\text{ }} \times {\text{ }}n{\text{ }} = {\text{ }} - {\text{ }}n\]
Where $n$ is a natural number
Now , Getting the result of a positive term multiplied by a negative term , we get
 \[5{\text{ }} \times {\text{ }}\left( { - 1} \right){\text{ }} = {\text{ }} - 5\]
Also ,
 \[4{\text{ }} \times {\text{ }}\left( { - 1} \right){\text{ }} = {\text{ }} - 4\]
Similarly ,
 \[n{\text{ }} \times {\text{ }}\left( { - 1} \right){\text{ }} = {\text{ }} - n\]
From the above data we can get that the product of a positive number to a negative number or product of a negative number to a positive number always gives a negative number .
Now ,
Multiplying a positive term with a positive term , we get
 \[1{\text{ }} \times {\text{ }}1{\text{ }} = {\text{ }}1\]
Also ,
 \[2{\text{ }} \times {\text{ }}1{\text{ }} = {\text{ }}2\]
Similarly ,
 \[n{\text{ }} \times {\text{ }}1{\text{ }} = {\text{ }}n\]
From this data we can conclude that the product of a positive number to a positive number gives a positive number .
Using the result of the above two expressions , we conclude that the product of a negative number to a negative number gives a positive number .
Hence , \[\begin{array}{*{20}{l}}
  \; \\
  {\left( { - 1} \right){\text{ }} \times {\text{ }}\left( { - 1} \right){\text{ }} = {\text{ }}1}
\end{array}\]
Hence proved that \[\left( { - 1} \right){\text{ }} \times {\text{ }}\left( { - 1} \right){\text{ }} = {\text{ }}1\]

Note: We conclude that the product of two numbers having different signs always gives us a negative number as a result whereas the product of two numbers having same signs always gives us a positive number as a result