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Q13:
Find the multiplicative inverse of the complex number $-i$.
Solution :
Step 1: Definition of Multiplicative Inverse
For any non-zero complex number $z = a + bi$, the multiplicative inverse is denoted by $z^{-1}$ or $\frac{1}{z}$ and is defined such that $z \cdot z^{-1} = 1$. [Per the algebraic property of fields in the complex number system $\mathbb{C}$].
Step 2: Representation of the Given Complex Number
Given the complex number $z = -i$. In standard rectangular form $a + bi$, this is represented as: $z = 0 - 1i$ Where $a = 0$ and $b = -1$.
Step 3: Deriving the Inverse Formula
The inverse is calculated as follows: $z^{-1} = \frac{1}{-i}$
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator, or more simply, multiply by $\frac{i}{i}$ to simplify the imaginary unit in the denominator [By the Property of Division in Complex Numbers]:
$z^{-1} = \frac{1}{-i} \times \frac{i}{i}$
Step 4: Algebraic Manipulation
Performing the multiplication: $z^{-1} = \frac{i}{-i^2}$
Recall the fundamental definition of the imaginary unit: $i^2 = -1$. Substituting this into the equation:
$z^{-1} = \frac{i}{-(-1)}$ $z^{-1} = \frac{i}{1}$ $z^{-1} = i$
Visual Representation of the Complex Inverse
Step 5: Verification
Check the result using the product identity $z \cdot z^{-1} = 1$: $(-i) \cdot (i) = -(i^2)$ $= -(-1)$ $= 1$ The condition $z \cdot z^{-1} = 1$ is satisfied. [By Identity Property of Multiplication].
Final Solution: The multiplicative inverse of $-i$ is $i$.
More Questions from Class 11 Mathematics Complex Numbers EXERCISE 4.1
- Q1: Express the complex number in the form $a + ib$.
- Q2: Express the complex number $i^9 + i^{19}$ in the form $a + ib$.
- Q3: Express the complex number $i^{-39}$ in the form $a + ib$.
- Q4: Express the complex number $3(7 + i7) + i(7 + i7)$ in the form $a + ib$.
- Q5: Express the complex number $(1 - i) - (-1 + i6)$ in the form $a + ib$.
- Q6: Express the complex number in the form $a + ib$.
- Q7: Express the complex number in the form $a + ib$.
- Q8: Express the complex number $(1 - i)^4$ in the form $a + ib$.
- Q9: Express the complex number in the form $a + ib$.
- Q10: Express the complex number in the form $a + ib$.
- Q11: Find the multiplicative inverse of the complex number $4 - 3i$.
- Q12: Find the multiplicative inverse of the complex number .
- Q14: Express the following expression in the form of $a + ib$:
CBSE Solutions for Class 11 Mathematics Complex Numbers
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Complex Numbers
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Ashok Nagar, Delhi
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Mulund East, Mumbai
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I have been teaching students for the last 20 years. Many of my students had passed with more than 95% marks or higher grade. I always love interactive classes so always motivate my students to ask as many questions as they can.Once the teaching process becomes dialogue from monologue then I understand that half of the task has been done and thereafter concept building process becomes very easy as I constantly guided by the questions from my students. Keeping class regularly on time, solving questions everyday, clearing doubts on a regular basis and lastly keeping weekly test make it very simple to make a good score.
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Sector-18, Faridabad
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I have approximately 3 years of experience of teaching Mathematics across different boards including CBSE, IGCSE and state board in reputed schools like DPS International. I am post graduate in statistics and honors degree in mathematics from reputed universities like DU. I have helped many students to overcome their fear of Mathematics and Statistics. My aim is to develop a genuine love for the subject in students.
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Maan, Pimpri-Chinchwad
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I have been teaching students for the past 6 years and have guided many to students in getting excellent results, showcasing their best performance. Concept-based teaching and an extra emphasis on every student needs has helped me provide a quality teaching environment to my students.
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Shahdara Chitrakoot, Delhi
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With years of experience teaching Class 11 Mathematics, I have honed my skills to provide students with a deep understanding of complex concepts. My teaching approach is student-centered, focusing on clarity, problem-solving techniques, and real-world applications. I have a strong command over the subject, making intricate topics accessible and engaging. My classes are interactive and designed to foster a genuine interest in mathematics. By incorporating innovative methods and advanced tools, I ensure that learning is both effective and enjoyable. Join my classes to experience a supportive learning environment where your academic success is the ultimate goal!
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