Title: TTK-007-IMG-EN

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The Complex Conjugate

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The complex numbers \(\color{red}{a+bi}\) and \(\color{blue}{a−bi}\) are called complex conjugates.

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The symbol used to denote the complex conjugate of the number \(z\) is \(\overline{z}\).

Sometimes complex conjugates are written as \(\;5+3i=\overline{5-3i}\;\),

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\(5-3i=\overline{5+3i}\).

Every complex number has associated with it another complex number known as its complex conjugate.

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Example. Find the complex conjugate of \(4 - 5i\), \(3i - 5\), \(9i\) and \(8\).

       Solution:

the complex conjugate of \(4 - 5i\) is \(4 + 5i\),

the complex conjugate of \(3i - 5 =-5 + 3i\) is \(-5 - 3i\),

the complex conjugate of \(9i\) is \(-9i\),

the complex conjugate of \(8\) is \(8\).

To find the complex conjugate simply by changing the sign of the imaginary part of the complex number.

Conjugate Fill in the Blanks

Find the complex conjugate of \(z\).

Q: if \(z = 2+i\) then \(\overline{z}\) = [...]

if \(z = 1-5i\) then \(\overline{z}\) = [...]

if \(z = -2+18i\) then \(\overline{z}\) = [...]

if \(z = -3-i\) then \(\overline{z}\) = [...]
A: 2-i; 1+5i; -2-18i; -3+i

Kaaskompleksarv Fill in the Blanks

Find complex conjugate of \(z = -37i\).

Q: Answer: \(\overline{z}\) = [...]\(i\)
A: 37

Conjugate Fill in the Blanks

Find complex conjugate for \(z = -5\).

Q: Answer: \(\overline{z}=\)[...].
A: -5

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Practice