Finding the complex conjugate of a complex number
In this article we shall find the complex conjugate of a complex number. Below is a quick summary of what is covered in this article.
Quick summary
- The complex number a – bi is called the complex conjugate of the complex number a + bi
- The complex numbers a + bi and a – bi are called a complex conjugate pair.
- The complex conjugate of z is called z*, so if z = a + bi, z* = a – bi
- Two complex numbers can be divided by using the complex conjugate of the denominator.
We shall look at some examples below;
Example: Write down the complex conjugate of;
- 2 + 3i
- 6 – 2i
- √5 + i
- 1 – i√5
To find the complex conjugate we simply change the sign of the imaginary part, so;
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Below are further examples of working with complex conjugate numbers;
Example: Find z + z* and zz*, given that;
- z = 6 + 5i
- z = 3 – 7i
- z = 2√2 + i√2
Answers
- Let us start with a) The complex conjugate of z = 6 + 5i must be z = 6 – 5i we simply change the sign of the imaginary part
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First we need to add;
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We shall now find zz*
[IMAGE] - If z = 3 – 7i then the complex conjugate must be z* = 3 + 7i
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First we shall add z + z*
[IMAGE] - If z = 2√2 + i√2 then z* = 2√2 – i√2. We shall find the addition first;
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Now we shall find the multiplication;
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Example: Workout; (10 + 5i) ÷ (1 + 2i).
We’re trying to find;
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The denominator is 1 + 2i we must first find the complex conjugate of the denominator.
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We now multiply the numerator and denominator by the complex conjugate;
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…therefore;
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Example; Simplify (4 + 3i) ÷ (2 – 5i)
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…therefore…
Notes & Tips
- If the roots β and α of a quadratic equation are complex, α and β will always be a complex conjugate for example if the first root is α = a + bi the second root will be β = a – bi.
- If the roots of the equation are α and β, the equation is (x – α)(x – β) = 0; …therefore;
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Example: Find the quadratic equation that has roots 2 + 4i and 2 – 4i.
We will form the quadratic equation using;
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The equation is therefore;
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