Convert a real number to polar form

Expert intro: For a positive real number, polar form is one of the easiest cases: the point lies on the positive real axis, so the angle is zero.

Detailed walkthrough

A complex number in polar form is written as

$r(\cos\phi+i\sin\phi)$

For the number $3$:

• Magnitude: $r=|3|=3$
• Angle: since it lies on the positive real axis, $\phi=0$

So the polar form is

$3(\cos 0+i\sin 0)$

If your worksheet wants the blanks filled in:

• $r=3$
• $\phi=0$

💡 Pitfall guide

Do not overthink the angle for a positive real number. Some students try to force a nonzero angle, but $0$ is the standard principal value in $0\le \phi<2\pi$. Also, keep $r$ nonnegative in polar form.

🔄 Real-world variant

If the number were a negative real number like $-3$, the modulus would still be $r=3$, but the angle would be $\phi=\pi$ instead of $0$. That is the main change when moving from positive to negative real values.

🔍 Related terms

polar form, complex numbers, argument