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Q7:
Express the complex number
in the form $a + ib$.
Express the complex number in the form $a + ib$.
Solution :
Step 1: Given Expression and Theoretical Framework
The expression to be simplified is:
$\left[ \left( 3 + 7i \right) + \left( \frac{1}{3} + \frac{7}{3}i \right) \right] - \left[ \left( 4 + \frac{1}{3}i \right) - \left( -\frac{4}{3} + i \right) \right]$
To express this in the form $a + ib$, we apply the Algebra of Complex Numbers [Per the Closure Property under addition and subtraction of complex numbers].
Step 2: Simplifying the First Bracketed Term
Grouping the real parts and imaginary parts of $\left( 3 + 7i \right) + \left( \frac{1}{3} + \frac{7}{3}i \right)$:
$= \left( 3 + \frac{1}{3} \right) + i\left( 7 + \frac{7}{3} \right)$
$= \left( \frac{9 + 1}{3} \right) + i\left( \frac{21 + 7}{3} \right)$
$= \frac{10}{3} + \frac{28}{3}i$
Step 3: Simplifying the Second Bracketed Term
Distributing the negative sign in $\left[ \left( 4 + \frac{1}{3}i \right) - \left( -\frac{4}{3} + i \right) \right]$:
$= \left( 4 + \frac{1}{3}i + \frac{4}{3} - i \right)$
$= \left( 4 + \frac{4}{3} \right) + i\left( \frac{1}{3} - 1 \right)$
$= \left( \frac{12 + 4}{3} \right) + i\left( \frac{1 - 3}{3} \right)$
$= \frac{16}{3} - \frac{2}{3}i$
Step 4: Combining the Results
Now, perform the final subtraction between the two resulting complex numbers:
$\left( \frac{10}{3} + \frac{28}{3}i \right) - \left( \frac{16}{3} - \frac{2}{3}i \right)$
$= \left( \frac{10}{3} - \frac{16}{3} \right) + i\left( \frac{28}{3} - (-\frac{2}{3}) \right)$
$= \left( \frac{10 - 16}{3} \right) + i\left( \frac{28 + 2}{3} \right)$
$= \left( -\frac{6}{3} \right) + i\left( \frac{30}{3} \right)$
$= -2 + 10i$
Visual Representation of Complex Addition
Final Solution: -2 + 10i
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 $a + ib$.
- 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 .
- Q13: Find the multiplicative inverse of the complex number $-i$.
- 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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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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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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