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Q2:
Express the complex number $i^9 + i^{19}$ in the form $a + ib$.
Solution :
Given Variables & Initial Setup
The expression provided is $Z = i^9 + i^{19}$. The goal is to simplify this expression into the standard form $a + ib$, where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$.
Step 1: Application of Power Properties of $i$
Recall that the powers of the imaginary unit $i$ follow a cyclic pattern of period 4:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
Step 2: Decomposition of $i^9$
Divide the exponent 9 by 4: $9 = 4 \times 2 + 1$ Therefore: $i^9 = i^{4(2) + 1} = (i^4)^2 \cdot i^1$ Substituting $i^4 = 1$: $i^9 = (1)^2 \cdot i = i$
Step 3: Decomposition of $i^{19}$
Divide the exponent 19 by 4: $19 = 4 \times 4 + 3$ Therefore: $i^{19} = i^{4(4) + 3} = (i^4)^4 \cdot i^3$ Substituting $i^4 = 1$ and $i^3 = -i$: $i^{19} = (1)^4 \cdot (-i) = -i$
Step 4: Summation and Final Algebraic Form
Substitute the simplified terms back into the original expression: $Z = i^9 + i^{19}$ $Z = i + (-i)$ $Z = 0$
Geometric/Visual Interpretation
To express this in the requested form $a + ib$: $a = 0$ $b = 0$ Thus, $Z = 0 + i(0)$.
Final Solution: 0 + 0i
More Questions from Class 11 Mathematics Complex Numbers EXERCISE 4.1
- Q1: Express the complex number 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 .
- 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
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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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