In this chapter, we will learn method to square the negative number along with solved examples.
Try understand the lesson, you should have basic understanding of the concept of square of numbers. Click the red link to read about the same.
Can we square a negative number ?
Answer is YES !!!
You can square any possible number available in Math domain.
When we square a number, we basically multiply a number by itself.
So on squaring a negative number, we multiply the same number twice to get the positive number.
Consider the below example for better clarity.
Let’s square the number -2.
\mathtt{\Longrightarrow \ ( -2)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ -2\ \times \ -2}\\\ \\ \mathtt{\Longrightarrow \ 4}
Conclusion
Squaring of negative number results in positive number.
Representation of square of negative number
If (-a) is the given number, it’s square is represented as follows;
\mathtt{\Longrightarrow \ ( -a)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ -a\ \times \ -a}\\\ \\ \mathtt{\Longrightarrow \ a^{2}}
Hence, the whole calculation is summarized to below formula;
\mathtt{Hence,\ ( -a)^{2} =a^{2}}
Square of negative fraction
On squaring fraction, we multiply numerator and denominator by itself twice.
In this case also, the square of negative fraction results in positive fraction.
Let (-a/b) is the given fraction.
Squaring the number, we get;
\mathtt{\Longrightarrow \ \left(\frac{-a}{b}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{-a\ \times -a}{\ b\ \times \ b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a^{2}}{b^{2}}}
The above calculation is summarized by following formula;
\mathtt{\left(\frac{-a}{b}\right)^{2} =\ \frac{a^{2}}{b^{2}}}
I hope you understood the above concepts. Let us solve some problems related to this chapter.
Squaring negative numbers – Solved examples
(01) Square the below numbers.
(i) -9
(ii) -0.2
(iii) -5
(iv) -2/3
(v) -6/11
(vi) -10
(vii) -2.5
(viii) -15/13
(ix) -21
(x) -30
Solution
(i) -9
\mathtt{\Longrightarrow \ ( -9)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -9\ \times \ -9}\\\ \\ \mathtt{\Longrightarrow \ 81}
Hence, 81 is the solution
(ii) -0.2
\mathtt{\Longrightarrow \ ( -0.2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -0.2\ \times \ -0.2}\\\ \\ \mathtt{\Longrightarrow \ \frac{-2}{10} \times \frac{-2}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{100}}\\\ \\ \mathtt{\Longrightarrow \ 0.04}
Hence, 0.04 is the solution.
(iii) -5
\mathtt{\Longrightarrow \ ( -5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -5\ \times \ -5}\\\ \\ \mathtt{\Longrightarrow \ 25}
Hence, 25 is the solution.
(iv) -2/3
\mathtt{\Longrightarrow \ \left(\frac{-2}{3}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-2}{3} \times \frac{-2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{9}}
Hence, 4/9 is the final solution.
(v) -6/11
\mathtt{\Longrightarrow \ \left(\frac{-6}{11}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-6}{11} \times \frac{-6}{11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{36}{121}}
Hence, 36/121 is the solution.
(vi) -10
\mathtt{\Longrightarrow \ ( -10)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -10\ \times \ -10}\\\ \\ \mathtt{\Longrightarrow \ 100}
Hence, 100 is the solution.
(vii) -2.5
\mathtt{\Longrightarrow \ ( -2.5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -2.5\ \times \ -2.5}\\\ \\ \mathtt{\Longrightarrow \ \frac{-25}{10} \times \frac{-25}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{625}{100}}\\\ \\ \mathtt{\Longrightarrow \ 6.25}
Hence, 6.25 is the final solution.
(viii) -15/13
\mathtt{\Longrightarrow \ \left(\frac{-15}{13}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-15}{13} \times \frac{-15}{13}}\\\ \\ \mathtt{\Longrightarrow \ \frac{225}{169}}
Hence, 225/169 is the solution.
(ix) -21
\mathtt{\Longrightarrow \ ( -21)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -21\ \times \ -21}\\\ \\ \mathtt{\Longrightarrow \ 441}
Hence, 441 is the solution.
(x) -30
\mathtt{\Longrightarrow \ ( -30)^{2}}\\\ \\ \mathtt{\Longrightarrow \ -30\ \times \ -30}\\\ \\ \mathtt{\Longrightarrow \ 900}
Hence, 900 is the solution.