What is the principal argument of (-1 –i), where i = \(\sqrt { - 1}\)
-3π/4
Finding the Principal Argument of a Complex Number
The question asks for the principal argument of the complex number \((-1 - i)\). The principal argument of a complex number \(z\) is the unique angle \(\theta\) such that \(z = r(\cos \theta + i \sin \theta)\) and \(\theta \in (-\pi, \pi]\).
Let the given complex number be \(z = -1 - i\). This number is in the standard form \(x + iy\), where \(x = -1\) and \(y = -1\).
Steps to Find the Principal Argument
To find the principal argument, we first determine the modulus and the quadrant where the complex number lies.
1. Calculate the Modulus
The modulus \(r\) of a complex number \(z = x + iy\) is given by the formula \(r = \sqrt{x^2 + y^2}\).
For \(z = -1 - i\):
$$ r = \sqrt{(-1)^2 + (-1)^2} $$
$$ r = \sqrt{1 + 1} $$
$$ r = \sqrt{2} $$
The modulus of \(z = -1 - i\) is \(\sqrt{2}\).
2. Determine the Quadrant
The complex number \(z = x + iy\) with \(x = -1\) and \(y = -1\) has both its real part (\(x\)) and imaginary part (\(y\)) negative. In the complex plane, the real axis corresponds to the x-axis and the imaginary axis corresponds to the y-axis. A point \((x, y) = (-1, -1)\) lies in the third quadrant.
3. Calculate the Reference Angle
The reference angle \(\alpha\) is the acute angle made by the line segment connecting the origin to the point \((x, y)\) with the positive x-axis. It is typically calculated using \(\tan \alpha = |\frac{y}{x}|\).
For \(z = -1 - i\):
$$ \tan \alpha = \left|\frac{-1}{-1}\right| $$
$$ \tan \alpha = |1| $$
$$ \tan \alpha = 1 $$
The value of \(\alpha\) for which \(\tan \alpha = 1\) in the first quadrant is \(\frac{\pi}{4}\).
$$ \alpha = \frac{\pi}{4} $$
4. Find the Principal Argument based on Quadrant
The principal argument \(\theta\) depends on the quadrant in which the complex number lies. Since \(z = -1 - i\) is in the third quadrant, the principal argument \(\theta\) is given by:
$$ \theta = -\pi + \alpha $$
Substituting the value of \(\alpha\):
$$ \theta = -\pi + \frac{\pi}{4} $$
$$ \theta = \frac{-4\pi + \pi}{4} $$
$$ \theta = -\frac{3\pi}{4} $$
This value \(-\frac{3\pi}{4}\) lies in the range \((-\pi, \pi]\), so it is the principal argument.
Conclusion
The principal argument of the complex number \((-1 - i)\) is \(-\frac{3\pi}{4}\).
Let's compare this result with the given options:
- \(\pi/4\)
- \(-\pi/4\)
- \(-3\pi/4\)
- \(3\pi/4\)
Our calculated principal argument, \(-\frac{3\pi}{4}\), matches the third option.
| Complex Number \(z=x+iy\) | Quadrant | Principal Argument (\(\theta\)) | Condition |
|---|---|---|---|
| \(x>0, y>0\) | First | \(\arctan(y/x)\) | |
| \(x<0, y>0\) | Second | \(\pi + \arctan(y/x)\) or \(\pi - |\arctan(y/x)|\) | |
| \(x<0, y<0\) | Third | \(-\pi + \arctan(y/x)\) or \(-\pi + |\arctan(y/x)|\) | |
| \(x>0, y<0\) | Fourth | \(\arctan(y/x)\) | |
| \(x>0, y=0\) | Positive Real Axis | \(0\) | |
| \(x<0, y=0\) | Negative Real Axis | \(\pi\) | |
| \(x=0, y>0\) | Positive Imaginary Axis | \(\pi/2\) | |
| \(x=0, y<0\) | Negative Imaginary Axis | \(-\pi/2\) | |
| \(x=0, y=0\) | Origin | Undefined |
Revision Table: Complex Number Argument
| Concept | Description | Formula/Notation |
|---|---|---|
| Complex Number | A number of the form \(x + iy\), where \(x\) and \(y\) are real numbers and \(i = \sqrt{-1}\). | \(z = x + iy\) |
| Modulus | The distance of the complex number from the origin in the complex plane. | \(|z| = r = \sqrt{x^2 + y^2}\) |
| Argument | The angle \(\theta\) between the positive real axis and the line segment connecting the origin to the complex number in the complex plane. | \(\arg(z)\) or \(\theta\) |
| Principal Argument | The unique argument \(\theta\) that lies in the interval \((-\pi, \pi]\). | \(\text{Arg}(z)\) or \(\theta\) where \(-\pi < \theta \le \pi\) |
Additional Information: Complex Plane and Arguments
The complex plane (also called the Argand plane) is a graphical representation of complex numbers. It has a horizontal real axis and a vertical imaginary axis. A complex number \(z = x + iy\) is represented by the point \((x, y)\).
The argument of a complex number is not unique. If \(\theta\) is an argument, then \(\theta + 2n\pi\) for any integer \(n\) is also an argument. For example, for \(z = -1 - i\), \(-\frac{3\pi}{4}\) is an argument, but so is \(-\frac{3\pi}{4} + 2\pi = \frac{5\pi}{4}\), \(-\frac{3\pi}{4} - 2\pi = -\frac{11\pi}{4}\), and so on.
The principal argument is specifically defined to have a unique value within the interval \((-\pi, \pi]\). This is why we chose \(-\frac{3\pi}{4}\) instead of \(\frac{5\pi}{4}\), as \(-\frac{3\pi}{4} \approx -2.35\) radians (which is between \(-\pi \approx -3.14\) and \(\pi \approx 3.14\)), while \(\frac{5\pi}{4} \approx 3.92\) radians (which is outside this interval).
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