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Complex Number in Polar Form

Complex Numbers · Class XI

Set real and imaginary parts of z = x + iy. See modulus |z| and argument arg(z) live.

1
type a value
-55
1
type a value
-55
Live values
  • Modulus1.4142
  • Argument (rad)0.7854
  • Argument (deg)45
  • z² (real part)0
  • z² (imag part)2
xy
  • |z| as Im(z) varies

Formulas in this lab

  • Modulus
    ∣z∣=x2+y2|z| = \sqrt{x^2 + y^2}
  • Argument (rad)
    arg⁡(z)=atan2⁡(y,x)\arg(z) = \operatorname{atan2}(y, x)
  • Argument (deg)
    arg⁡(z)⋅180/π\arg(z) \cdot 180/\pi
  • z² (real part)
    Re⁡(z2)=x2−y2\operatorname{Re}(z^2) = x^2 - y^2
  • z² (imag part)
    Im⁡(z2)=2xy\operatorname{Im}(z^2) = 2xy
Tip: Modulus traces a hyperbola in the imaginary part, minimum |re| at y = 0.

Frequently asked questions

▶What does polar form of $z = x + iy$ mean?

Every complex number can be written as $z = r(\cos\theta + i\sin\theta)$, where $r = |z| = \sqrt{x^2+y^2}$ is the modulus and $\theta = \arg z = \operatorname{atan2}(y,x)$ is the argument. Polar form makes multiplication easy: moduli multiply and arguments add.

▶How do I use the complex-polar lab?

Slide the real part $\operatorname{Re}(z)$ and imaginary part $\operatorname{Im}(z)$. The lab returns $|z|$, $\arg z$ in both radians and degrees, plus $z^2 = (x^2-y^2) + 2xyi$. Try $z = 1 + i$ to see $|z| = \sqrt{2}$ and $\arg z = \pi/4 = 45^\circ$.

▶Why is $\arg(z)$ tricky in different quadrants?

$\arctan(y/x)$ alone cannot distinguish quadrant 1 from quadrant 3 — both give the same ratio. The function $\operatorname{atan2}(y,x)$ adds the correct sign and offset. Students lose marks for writing $\arg(-1-i) = \pi/4$ instead of $-3\pi/4$. The lab handles this automatically.

▶Where does polar form appear outside maths class?

AC circuit analysis represents voltage and current as complex phasors $V = V_0 e^{i\omega t}$. Signal processing, control systems used in ISRO's rockets and even quantum mechanics rely on Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$. Polar form is the language of waves.