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Q3:
Express the complex number $i^{-39}$ in the form $a + ib$.
Solution :
Step 1: Theoretical Foundation of Imaginary Units
The powers of the imaginary unit $i$ (where $i = \sqrt{-1}$) follow a cyclic pattern of four:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
Step 2: Decomposition of the Exponent
To evaluate $i^{-39}$, we first rewrite the negative exponent using the law of exponents: $a^{-n} = \frac{1}{a^n}$.
$$i^{-39} = \frac{1}{i^{39}}$$
To simplify $i^{39}$, we divide the exponent $39$ by $4$ to find the remainder [By Division Algorithm: $a = bq + r$]: $$39 = 4 \times 9 + 3$$ Therefore, $i^{39} = i^{4(9) + 3} = (i^4)^9 \cdot i^3$.
Step 3: Algebraic Simplification
Substitute the known values of the powers of $i$: $$i^{39} = (1)^9 \cdot i^3$$ $$i^{39} = 1 \cdot (-i) = -i$$ [Since $i^3 = -i$].
Now, substitute this result back into the expression from Step 2: $$i^{-39} = \frac{1}{-i}$$
Step 4: Rationalization of the Denominator
To express the complex number in $a + ib$ form, we eliminate $i$ from the denominator by multiplying the numerator and denominator by the conjugate or by $i$ itself: $$\frac{1}{-i} \times \frac{i}{i} = \frac{i}{-i^2}$$ Since $i^2 = -1$, the denominator becomes $-(-1) = 1$: $$\frac{i}{-(-1)} = \frac{i}{1} = i$$
Step 5: Representation in $a + ib$ Form
The resulting value is $i$. Expressing this in the standard form $a + ib$: $$0 + 1i$$ Where $a = 0$ and $b = 1$.
Final Solution: 0 + 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 $a + ib$.
- 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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