What is the principal argument of \(\frac{1}{1 + i}\) where \(i = \sqrt{-1}?\)
Finding the Principal Argument of a Complex Number
The problem asks for the principal argument of the complex number given by the expression \( \frac{1}{1 + i} \), where \( i = \sqrt{-1} \). The principal argument of a complex number is the angle \( \theta \) it makes with the positive real axis, such that \( -\pi < \theta \leq \pi \). To find this, we first need to simplify the complex number into the standard form \( x + yi \).
Simplifying the Complex Number \( \frac{1}{1+i} \)
We are given the complex number \( z = \frac{1}{1 + i} \). To simplify this fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1 + i \) is \( 1 - i \).
So, we have:
\( z = \frac{1}{1 + i} \times \frac{1 - i}{1 - i} \)
Now, we perform the multiplication:
- Numerator: \( 1 \times (1 - i) = 1 - i \)
- Denominator: \( (1 + i)(1 - i) \). This is in the form \( (a+b)(a-b) = a^2 - b^2 \). Here, \( a = 1 \) and \( b = i \). So, \( (1 + i)(1 - i) = 1^2 - i^2 \).
Since \( i = \sqrt{-1} \), we know that \( i^2 = -1 \). Substituting this into the denominator:
\( \text{Denominator} = 1^2 - (-1) = 1 + 1 = 2 \)
So, the simplified complex number is:
\( z = \frac{1 - i}{2} = \frac{1}{2} - \frac{1}{2}i \)
This is now in the form \( x + yi \), where \( x = \frac{1}{2} \) and \( y = -\frac{1}{2} \).
Finding the Argument of the Complex Number
The argument \( \theta \) of a complex number \( x + yi \) is the angle satisfying \( \tan \theta = \frac{y}{x} \). For \( z = \frac{1}{2} - \frac{1}{2}i \), we have \( x = \frac{1}{2} \) and \( y = -\frac{1}{2} \).
\( \tan \theta = \frac{-\frac{1}{2}}{\frac{1}{2}} = -1 \)
Now, we need to determine the correct angle \( \theta \) based on the quadrant the complex number lies in. The real part \( x = \frac{1}{2} \) is positive, and the imaginary part \( y = -\frac{1}{2} \) is negative. A complex number with a positive real part and a negative imaginary part lies in the fourth quadrant.
In the fourth quadrant, the angle \( \theta \) for which \( \tan \theta = -1 \) is \( -\frac{\pi}{4} \) (or \( \frac{7\pi}{4} \), but we need the principal argument).
Determining the Principal Argument
The principal argument is the value of \( \theta \) in the range \( (-\pi, \pi] \). The angle \( -\frac{\pi}{4} \) is in this range because \( -\pi < -\frac{\pi}{4} \leq \pi \).
Therefore, the principal argument of \( \frac{1}{1 + i} \) is \( -\frac{\pi}{4} \).
Summary of Steps:
- Simplify the complex number \( \frac{1}{1+i} \) to the form \( x+yi \). We got \( \frac{1}{2} - \frac{1}{2}i \).
- Identify the real part \( x = \frac{1}{2} \) and the imaginary part \( y = -\frac{1}{2} \).
- Determine the quadrant based on the signs of \( x \) and \( y \). Positive \( x \) and negative \( y \) means the fourth quadrant.
- Find the angle \( \theta \) such that \( \tan \theta = \frac{y}{x} \). \( \tan \theta = \frac{-1/2}{1/2} = -1 \).
- For the fourth quadrant, the angle with \( \tan \theta = -1 \) in the principal range is \( -\frac{\pi}{4} \).
The principal argument is \( -\frac{\pi}{4} \).
Revision Table: Principal Argument Calculation
| Step | Action | Result |
|---|---|---|
| 1 | Simplify \( \frac{1}{1+i} \) | \( \frac{1}{2} - \frac{1}{2}i \) |
| 2 | Identify \( x \) and \( y \) | \( x = \frac{1}{2}, y = -\frac{1}{2} \) |
| 3 | Determine Quadrant | Fourth Quadrant (x > 0, y < 0) |
| 4 | Find \( \theta \) from \( \tan \theta = y/x \) | \( \tan \theta = -1 \) |
| 5 | Principal Argument in Quadrant 4 | \( -\frac{\pi}{4} \) |
Additional Information on Complex Number Argument
The argument of a complex number \( z = x + yi \) represents the angle \( \theta \) in the complex plane from the positive real axis to the vector representing \( z \). The general argument is given by \( \text{arg}(z) = \theta + 2n\pi \), where \( n \) is an integer.
The principal argument, denoted as \( \text{Arg}(z) \), is the unique value of the argument that lies in the interval \( (-\pi, \pi] \). This ensures a standard representation for the angle.
To find the principal argument \( \theta = \text{Arg}(x+yi) \):
- Calculate a reference angle \( \alpha = \arctan\left|\frac{y}{x}\right| \) (where \( \arctan \) gives a value in \( [0, \frac{\pi}{2}] \)).
- Adjust the angle based on the quadrant:
| Quadrant | Sign of x, y | Principal Argument \( \theta \) |
|---|---|---|
| 1 | x > 0, y > 0 | \( \alpha \) |
| 2 | x < 0, y > 0 | \( \pi - \alpha \) |
| 3 | x < 0, y < 0 | \( -\pi + \alpha \) |
| 4 | x > 0, y < 0 | \( -\alpha \) |
Note: For points on the axes:
- Positive real axis (x > 0, y = 0): \( \theta = 0 \)
- Positive imaginary axis (x = 0, y > 0): \( \theta = \frac{\pi}{2} \)
- Negative real axis (x < 0, y = 0): \( \theta = \pi \)
- Negative imaginary axis (x = 0, y < 0): \( \theta = -\frac{\pi}{2} \)
- Origin (x = 0, y = 0): Argument is undefined.
In our case, \( x = \frac{1}{2} \) and \( y = -\frac{1}{2} \). \( \left|\frac{y}{x}\right| = \left|\frac{-1/2}{1/2}\right| = |-1| = 1 \). The reference angle \( \alpha = \arctan(1) = \frac{\pi}{4} \). Since the point is in the fourth quadrant (x > 0, y < 0), the principal argument is \( -\alpha = -\frac{\pi}{4} \).
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