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Question: What is the conjugate of \(-6-5i\)?...

What is the conjugate of −6−5i-6-5i?

Explanation

Solution

For solving this question you should know about the conjugate and complex conjugates. The conjugates are calculated by only changing the sign between two numbers and the numbers are real and imaginary. The first number is real for this and the second number is imaginary. And if we change the signs by plus to minus or minus to plus then it is made as a conjugate of that number.

Complete step by step answer:
According to our question it is asked that we have to find the conjugate of −6−5i-6-5i. The conjugates only represent the changing of signs between two real and imaginary numbers. If we define the conjugate in math, then ‘the conjugate is two pairs of binomials with identical terms by sharing opposite operations in the middle’.
The conjugate can be found for both numbers, binomial number and complex number. It is easy to find the conjugate for a binomial number. But if we find the conjugate for a complex number then there are many properties for it, which are as follows:
1. z1±z2‾=z1‾±z2‾\overline{{{z}_{1}}\pm {{z}_{2}}}=\overline{{{z}_{1}}}\pm \overline{{{z}_{2}}}
2. z1.z2‾=z1.z2‾\overline{{{z}_{1}}.{{z}_{2}}}=\overline{{{z}_{1}}.{{z}_{2}}}
3. z1‾z2‾=z1‾z2‾\dfrac{\overline{{{z}_{1}}}}{\overline{{{z}_{2}}}}=\dfrac{\overline{{{z}_{1}}}}{\overline{{{z}_{2}}}}
4. z‾=z\overline{z}=z
5. If z=a+ibz=a+ib; then z.z‾=a2+b2=∣z2∣z.\overline{z}={{a}^{2}}+{{b}^{2}}=\left| {{z}^{2}} \right|
6. z+z‾=x+iy+(x−iy)z+\overline{z}=x+iy+\left( x-iy \right)
7. z+z‾=x+iy−(x−iy)z+\overline{z}=x+iy-\left( x-iy \right)
In our question, we have z=−6−5iz=-6-5i and we have to find the conjugate for it. So, here, a=−6a=-6 and b=−5b=-5 if we consider these in the form of a+iba+ib. So, we can write it as,
(−6)+i(−5)\left( -6 \right)+i\left( -5 \right)
For calculating the conjugate of this we have to change the sign form plus to minus. So, the conjugate of a+iba+ib is a−iba-ib. Since a=−6a=-6 and b=−5b=-5, the conjugate will be,
−6−(−5)i ⇒−6+5i \begin{aligned} & -6-\left( -5 \right)i \\\ & \Rightarrow -6+5i \\\ \end{aligned}

So, the conjugate of −6−5i-6-5i is equal to −6+5i-6+5i.

Note: For calculating the conjugate of any term, it can be binomial or it can be complex we always change the sign between them. But we have to be careful because we change the sign outside that digit. If that digit has its own sign then it will multiply by the sign which is available outside that. And then the final conjugate can be written.