Complex Numbers and Quadratic Equations
Kuch equations real numbers me solve nahi hote — jaise \(x^{2} + 1 = 0\), kyunki kisi bhi real number ka square negative nahi hota. Isi problem ko solve karne ke liye complex numbers aaye. Is chapter me complex numbers ki algebra aur quadratic equations ke complex roots dekhenge.
1. Imaginary Unit aur Complex Number
Imaginary unit \(i\) ko aise define karte hain: \(i = \sqrt{-1}\), yaani \(i^{2} = -1\).
Powers of \(i\) cycle me chalte hain:
\[ i^{1} = i, \quad i^{2} = -1, \quad i^{3} = -i, \quad i^{4} = 1, \quad i^{5} = i, \dots \]
Ek complex number \(z = a + ib\) hota hai, jahan \(a, b \in \mathbb{R}\). Yahan \(a = \operatorname{Re}(z)\) (real part) aur \(b = \operatorname{Im}(z)\) (imaginary part).
Do complex numbers equal tabhi jab unke real aur imaginary parts dono barabar hon.
2. Algebra of Complex Numbers
\(z_1 = a + ib\) aur \(z_2 = c + id\) lo.
- Addition: \(z_1 + z_2 = (a + c) + i(b + d)\)
- Subtraction: \(z_1 - z_2 = (a - c) + i(b - d)\)
- Multiplication: \(z_1 z_2 = (ac - bd) + i(ad + bc)\)
\((3 + 2i)(1 - 4i)\) nikaalo. \[ = 3 - 12i + 2i - 8i^{2} = 3 - 10i - 8(-1) = 11 - 10i. \]
3. Conjugate aur Modulus
\(z = a + ib\) ka conjugate \(\bar{z} = a - ib\) hota hai, aur modulus \(|z| = \sqrt{a^{2} + b^{2}}\).
Useful properties:
\[ z\,\bar{z} = a^{2} + b^{2} = |z|^{2}, \qquad |z_1 z_2| = |z_1|\,|z_2|. \]
Division
Divide karne ke liye numerator aur denominator dono ko denominator ke conjugate se multiply karo:
\[ \frac{z_1}{z_2} = \frac{z_1 \,\bar{z_2}}{z_2 \,\bar{z_2}} = \frac{z_1 \,\bar{z_2}}{|z_2|^{2}}. \]
\(\dfrac{1}{2 + 3i}\) ko \(a + ib\) form me likho. \[ \frac{1}{2+3i} \times \frac{2-3i}{2-3i} = \frac{2-3i}{4+9} = \frac{2}{13} - \frac{3}{13}i. \]
4. Argand Plane aur Polar Form
Complex number \(z = a + ib\) ko plane me point \((a, b)\) ki tarah dikhate hain — ise Argand plane kehte hain. \(x\)-axis real axis, \(y\)-axis imaginary axis.
Polar form: \(z = r(\cos\theta + i\sin\theta)\), jahan \(r = |z| = \sqrt{a^{2}+b^{2}}\) modulus hai aur \(\theta\) argument (\(\tan\theta = \frac{b}{a}\)).
5. Quadratic Equations (Complex Roots)
Quadratic \(ax^{2} + bx + c = 0\) (\(a \neq 0\)) ke roots:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}. \]
Yahan discriminant \(D = b^{2} - 4ac\) kahani batata hai:
- \(D > 0\): do alag real roots.
- \(D = 0\): do barabar real roots.
- \(D < 0\): do complex (conjugate) roots — yahan \(\sqrt{D} = \sqrt{-|D|} = i\sqrt{|D|}\).
\(x^{2} + x + 1 = 0\) solve karo. \[ x = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}. \] Dono roots complex conjugate hain.
Key Takeaways
- \(i = \sqrt{-1}\), \(i^{2} = -1\); powers of \(i\) 4 ke cycle me repeat hote hain.
- \(z = a + ib\); conjugate \(\bar{z} = a - ib\), modulus \(|z| = \sqrt{a^{2}+b^{2}}\).
- Division me conjugate se multiply karo.
- Polar form: \(z = r(\cos\theta + i\sin\theta)\).
- \(D < 0\) ho to quadratic ke roots complex conjugate hote hain.