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Conjugate of a complex number is equal to the complex number where a and b values are equal but with a negative sign in between. A complex number is represented as a + ib where a is the real part and b is the imaginary part of the number. a and b are real numbers. i is an imaginary number known as iota whose value is equal to the square root of -1. For example: If 5 + 3i is our complex number then our complex conjugate is 5 -3i. In the argand plane, a - ib is the reflection of a + ib about the real axis (X-axis). A complex number's complex conjugate is used to rationalize that number.
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Definition of Conjugate of a Complex Number
When two complex numbers are considered, if only the sign of the imaginary part is different then in that case, they are known as a complex conjugate of each other.
To find the conjugate of a complex number, just substitute the letter i with the letter '-i' in the original complex number. The complex conjugate of p + iq is p - iq and the complex conjugate of p - iq is p + iq.
Properties of the Conjugate of a Complex Number
Let's look at a few aspects of complex conjugate that can help us simplify our computations. Consider the two complex integers z and w, as well as their complex conjugates and .
- The product of two complex numbers' complex conjugates equals the product of the two complex numbers' complex conjugates → \(\bar{ab} = \bar{a}.\bar{b}\)
- The quotient of two complex numbers' complex conjugates is equal to the quotient of the two complex numbers' complex conjugates → \((\bar{a/b}) = \bar{a}/\bar{b}\)
- The total of two complex numbers' complex conjugates equals the sum of the two complex numbers' complex conjugates → \(\bar{a+b} = \bar{a} + \bar{b}\)
- The difference of two complex numbers' complex conjugates is the difference of the two complex numbers' complex conjugates → \(\bar{a-b} = \bar{a} - \bar{b}\)
- The difference between a complex number and its complex conjugate is twice the complex number's imaginary portion → \(a-\bar{a} = 2Im(a)\)
- The real component of a complex number is equal to double the sum of the complex number and its complex conjugate → \(a-+\bar{a} = 2Re(a)\)
- The square of the magnitude of a complex number is equal to the product of the complex number and its complex conjugate → \(a.\bar{a} = |a^2|\)
- The imaginary part of a complex number is equal to the negative of the imaginary part of its complex conjugate, and the real part of a complex number is equal to the real part of its complex conjugate → \(Re(a) = Re (\bar{a}) and Im(a) = Im (\bar{a})\)
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Points to Remember
- All the properties of complex numbers and their conjugates must be on your tips as they will come in handy while solving all your questions. You have to set up your complex number in a way that it fits one of the properties and the answer can be found out easily.
- The conjugate of a complex number is just the same number with a negative imaginary part.
- The conjugate of the complex number must be used to multiply and divide it and rationalize the complex number, this leads to easier calculations.
- In Argand's plane, the conjugate of a complex number is a means to describe the reflection of a 2D vector that has been split down into its vector components using complex numbers. The conjugate of a complex number aids in the determination of a 2D vector around two planes, as well as the angles between them.
- The complex conjugate root theorem says that if f(x) is a polynomial with real coefficients and a + ib is one of its roots, then the complex conjugate a - ib is likewise a root of f(x) (x). This implies that non-real roots, i.e. a polynomial's complex roots, occur in pairs.
- The geometric explanation of the conjugate of a complex number is important as there can be questions where you may have to plot points on a graph or find the angle.
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Sample Questions
Ques 1.Express (5 – 3i)3 in the form a + ib (2 Marks)
Ans. (5 – 3i)3 = 53 - 3 × 52 × (3i) + 3 × 5 (3i)2 – (3i)3
= 125 – 225i – 135 + 27i
= -10 - 198i
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Ques 2. Find the modulus of the complex number 6+8i. (2 Marks)
Ans. √62 + 82 = √36 + 64
= √100
= 10
Therefore, |6 + 8i| = 10
Ques 3. | p-\(\frac{4}{p}\)| = 2. What is the maximum value of |p|? (2 Marks)
Ans. .| p-\(\frac{4}{p}\)| \(\geq\)| p | - | \(\frac{4}{p}\)| 2\(\geq\) | p | - \(\frac{4}{|p|}\) 2 | p | \(\geq\)| p |2 - 4 | p |2 -2 | p | - 4 \(\leq\)
0 | p | \(\leq\) \(\sqrt{5} +1\)
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Ques 4. Find the multiplicative inverse of 2 – 3i. (3 Marks)
Ans. Let p = 2 -3i
Then \(\bar{p}\)= 2 + 3i and |p| = 22 + (-3)2 = 13
Therefore, the multiplicative inverse of 2 - 3i is given by
\(p^{-1} = \frac{\bar{p}}{|p^2|}\)
= \(\frac{2 + 3i}{13} = \frac{2}{13}+ \frac{3}{13}i\)
Ques 5. Express the complex number p/q in the form of z + iw if p = 4 - 5i and q = -2 + 3i, where z and w are real numbers. (3 marks)
Ans. To simplify p/q = (4 - 5i)/(-2 + 3i), rationalizing the denominator by multiplying p/q by the complex conjugate -2 - 3i of -2 + 3i.
\(\frac{p}{q}\)= \(\frac{4 - 5i}{-2 + 3i} X \frac{-2 - 3i}{-2 - 3i}\)
\(= \frac{-8-12i+10i-15}{2^2 +3^2}\)=\( \frac{-23 - 2i}{13}= \frac{-23}{13} + \frac{-2}{13}i\)
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Ques 6. Show that ( 2 + i √3)10 - ( 2 - i √3)10 is purely imaginary. (3 Marks)
Ans. Let \(x\) = \({( 2 + i \sqrt{3})^{10} - ( 2 - i \sqrt{3})^{10}}\)
\(\bar{x}\)= \(\bar{{( 2 + i \sqrt{3})^{10} - ( 2 - i \sqrt{3})^{10}}}\)
= \(\bar{{( 2 + i \sqrt{3})^{10}} - ( 2 - i \sqrt{3})^{10}}\)
\(\bar{x}\)= -x ⇒ x = – \(\bar{x}\)
Therefore, x is purely imaginary.
Ques 7. Express i-35 in terms of a + ib. (2 Marks)
Ans. i -35 →
Now, rationalising it,
=\(\frac{1}{-i}\)x \(\frac{i}{i}\)
\(=\frac{i}{-i^2}\)
= i
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Ques 8. p is a complex number, then what is the minimum value of |p| + |p − 1|? (2 Marks)
Ans. We know that |-p| = |p|.
We also know that |p1 + p1| ≤ |p1| + |p1|
Therefore, |p| + |p − 1| = |p| + |1 − p| ≥ |p + (1 − p)|
= |1|
= 1
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