To determine whether the polar form of a complex number is unique when its argument is restricted to the range [tex]\([0, 2\pi)\)[/tex], let's understand what the polar form of a complex number is and the role of the argument.
1. Complex Number in Polar Form:
A complex number [tex]\( z \)[/tex] can be represented in polar form as:
[tex]\[
z = r e^{i\theta}
\][/tex]
where:
- [tex]\( r \)[/tex] is the magnitude (or modulus) of the complex number.
- [tex]\( \theta \)[/tex] is the argument (or angle) of the complex number.
2. Argument Restrictions:
The argument [tex]\( \theta \)[/tex] of a complex number can take any real value. However, in order to have a unique representation, we often restrict [tex]\( \theta \)[/tex] to a principal value range. A common choice is [tex]\( [0, 2\pi) \)[/tex].
3. Uniqueness in Polar Form:
When the argument [tex]\( \theta \)[/tex] is restricted to the range [tex]\( [0, 2\pi) \)[/tex]:
- Each unique complex number corresponds to a unique pair [tex]\((r, \theta)\)[/tex].
- No two different values of [tex]\( \theta \)[/tex] within this range can represent the same complex direction, ensuring that each complex number's polar form is unique within this range.
Thus, given the restriction [tex]\( \theta \in [0, 2\pi) \)[/tex], the polar form of a complex number is indeed unique.
Therefore, the statement "As long as its argument is restricted to [tex]\([0,2\pi)\)[/tex], the polar form of a complex number is unique" is:
True.
