Complex Numbers

What are complex numbers?

No real number squares to a negative, so we define $i = \sqrt{-1}$, which gives $\sqrt{-25} = 5i$. A complex number is $x + iy$ with real part $x$ and imaginary part $y$.

Modulus, argument and conjugate

In polar form $x + iy = r(\cos\theta + i\sin\theta)$, where $r = \sqrt{x^2+y^2}$ is the modulus and $\theta$ the argument, chosen so that $\cos\theta = \tfrac{x}{r}$ and $\sin\theta = \tfrac{y}{r}$. The conjugate of $x + iy$ is $x - iy$, and $(x+iy)(x-iy) = x^2 + y^2$.

Arithmetic on complex numbers

Add by combining real and imaginary parts: $(1+2i) + (2+i) = 3+3i$. Multiply using $i^2 = -1$:

Worked example: multiplying and dividing

To divide, multiply top and bottom by the conjugate of the denominator:

Worked example: the argument in the right quadrant

Find the modulus and argument of $-1 + \sqrt3\,i$.

Solution: $r = \sqrt{1 + 3} = 2$. A calculator's $\tan^{-1}\left(\tfrac{\sqrt3}{-1}\right)$ gives $-\tfrac{\pi}{3}$, but the point $(-1, \sqrt3)$ is in the second quadrant, so the argument is $\pi - \tfrac{\pi}{3} = \dfrac{2\pi}{3}$. Check: $2\left(\cos\tfrac{2\pi}{3} + i\sin\tfrac{2\pi}{3}\right) = 2\left(-\tfrac{1}{2} + \tfrac{\sqrt3}{2}i\right) = -1 + \sqrt3\,i$.

Powers of i

Powers of $i$ repeat every 4: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Divide the exponent by 4 and use the remainder: $45 = 4 \times 11 + 1$, so $i^{45} = i$. Negative powers cycle too: $i^{-1} = -i$, $i^{-2} = -1$, $i^{-3} = i$, $i^{-4} = 1$.

The modulus of a product

Moduli multiply: $|z_1 z_2| = |z_1||z_2|$. From the example above, $|3+2i| = \sqrt{13}$ and $|1-4i| = \sqrt{17}$, so the product should have modulus $\sqrt{221}$. Directly, $|11 - 10i| = \sqrt{121 + 100} = \sqrt{221}$, which matches. Arguments add too (up to a multiple of $2\pi$, since an argument is only fixed to within one turn), which is why multiplying by $i$ rotates a point by $90^\circ$ on the Argand plane.

Common mistakes

• Taking the argument straight from $\tan^{-1}(y/x)$. Check the quadrant, as in the example above.
• Writing $i^2 = 1$. It's $-1$.
• Dividing without the conjugate. Multiply numerator and denominator by the denominator's conjugate to make it real.
• Treating $\sqrt{a}\sqrt{b} = \sqrt{ab}$ as always true. With two negatives it fails: $\sqrt{-1}\sqrt{-1} = -1$, not $1$.

Where this comes up in exams

JEE Main 2026 lists complex numbers in the form a + ib, the Argand diagram, their algebra, and modulus and argument. JEE Advanced 2026 lists the algebra of complex numbers, conjugation, polar representation and the modulus and principal argument, and the 2025-26 CBSE Class XI syllabus lists the algebraic properties of complex numbers and the Argand plane.

Checked against JEE (Main) 2026 syllabus (NTA), JEE (Advanced) 2026 syllabus and CBSE Mathematics curriculum, Classes XI-XII (2025-26). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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