Class 11: Complex Numbers
Algebra, Modulus, Conjugate & Argand Plane
1. Definition and Powers of i
| Complex Number ($$z$$) | A number of the form $$z = a + ib$$, where $$a, b \in R$$. $$a$$ = Real part ($$Re(z)$$) $$b$$ = Imaginary part ($$Im(z)$$). |
| Equality | Two complex numbers $$z_1 = a + ib$$ and $$z_2 = c + id$$ are equal if and only if $$a = c$$ and $$b = d$$. |
| Powers of $$i$$ | $$i = \sqrt{-1}$$ $$i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$$ General Rule: For integer $$k$$: $$i^{4k} = 1, \quad i^{4k+1} = i, \quad i^{4k+2} = -1, \quad i^{4k+3} = -i$$ |
| Square Root of Negative Real | If $$a > 0$$, then $$\sqrt{-a} = \sqrt{a}i$$. Note: $$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$$ is false if both $$a$$ and $$b$$ are negative. |
2. Algebra of Complex Numbers
| Addition | $$z_1 + z_2 = (a + c) + i(b + d)$$ Properties: Commutative, Associative. Additive Identity: $$0 + i0$$ (Zero complex number). Additive Inverse: $$-z = -a – ib$$. |
| Multiplication | $$z_1 z_2 = (ac – bd) + i(ad + bc)$$ Properties: Commutative, Associative, Distributive. Multiplicative Identity: $$1 + i0$$. Multiplicative Inverse ($$z^{-1}$$): See section below. |
| Identities | $$(z_1 + z_2)^2 = z_1^2 + z_2^2 + 2z_1z_2$$ $$z_1^2 – z_2^2 = (z_1 + z_2)(z_1 – z_2)$$ |
3. Modulus and Conjugate
Let $$z = a + ib$$.
| Modulus ($$|z|$$) | $$|z| = \sqrt{a^2 + b^2}$$ Represents the distance from origin in Argand plane. |
| Conjugate ($$\bar{z}$$) | $$\bar{z} = a – ib$$ Represents the mirror image of $$z$$ on the real axis. |
| Multiplicative Inverse ($$z^{-1}$$) | $$z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2}$$ |
| Important Properties | 1. $$z \bar{z} = |z|^2$$
2. $$|z_1 z_2| = |z_1| |z_2|$$ |
4. Argand Plane
The plane having a complex number assigned to each of its point.
- Coordinate System: Corresponds to ordered pair $$(x, y)$$ for $$z = x + iy$$.
- X-axis: Real axis.
- Y-axis: Imaginary axis.