The modulus and principal argument of the complex number \(\frac{{1 + 2{\rm{i}}}}{{1 - {{\left( {1 - {\rm{i}}} \right)}^2}}}\) are respectively
1, 0
Calculating Modulus and Principal Argument of a Complex Number
To find the modulus and principal argument of the given complex number, we first need to simplify the expression.
The given complex number is:
\( z = \frac{{1 + 2{\rm{i}}}}{{1 - {{\left( {1 - {\rm{i}}명이right)}^2}}} \)
Let's simplify the denominator:
\( 1 - (1 - {\rm{i}})^2 \)
We expand the term \( (1 - {\rm{i}})^2 \):
\( (1 - {\rm{i}})^2 = 1^2 - 2(1)({\rm{i}}) + ({\rm{i}})^2 \)
Since \( {\rm{i}}^2 = -1 \), this becomes:
\( 1 - 2{\rm{i}} + (-1) = 1 - 2{\rm{i}} - 1 = -2{\rm{i}} \)
Now substitute this back into the denominator:
\( 1 - (-2{\rm{i}}) = 1 + 2{\rm{i}} \)
So, the complex number simplifies to:
\( z = \frac{{1 + 2{\rm{i}}}}{{1 + 2{\rm{i}}}} \)
Since the numerator and the denominator are the same and non-zero, the fraction simplifies to 1.
\( z = 1 \)
Now we need to find the modulus and principal argument of \( z = 1 \).
Finding the Modulus
A complex number \( z = x + y{\rm{i}} \) has a modulus given by \( |z| = \sqrt{x^2 + y^2} \). For \( z = 1 \), we can write it as \( z = 1 + 0{\rm{i}} \). Here, \( x = 1 \) and \( y = 0 \).
The modulus is:
\( |z| = \sqrt{1^2 + 0^2} = \sqrt{1 + 0} = \sqrt{1} = 1 \)
The modulus of the complex number is 1.
Determining the Principal Argument
The principal argument of a complex number \( z = x + y{\rm{i}} \) is the angle \( \theta \) such that \( x = |z|\cos(\theta) \) and \( y = |z|\sin(\theta) \), and \( \theta \) lies in the interval \( (-\pi, \pi] \). For \( z = 1 \), we have \( x = 1 \), \( y = 0 \), and we found \( |z| = 1 \).
We need to find \( \theta \) such that:
- \( \cos(\theta) = \frac{x}{|z|} = \frac{1}{1} = 1 \)
- \( \sin(\theta) = \frac{y}{|z|} = \frac{0}{1} = 0 \)
The angle \( \theta \) that satisfies both \( \cos(\theta) = 1 \) and \( \sin(\theta) = 0 \) is \( 0 \) radians. This angle \( 0 \) falls within the principal argument range \( (-\pi, \pi] \).
Therefore, the principal argument of the complex number is 0.
Summary of Results
The modulus of the complex number is 1 and the principal argument is 0.
| Property | Value |
|---|---|
| Complex Number (Simplified) | \( z = 1 \) |
| Modulus \( |z| \) | 1 |
| Principal Argument \( \arg(z) \) | 0 |
Revision Table: Complex Numbers
| Concept | Description | Formula/Example |
|---|---|---|
| Complex Number | A number of the form \(x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)). | \(3 + 4i\), \(-2i\), \(5\) |
| Modulus | The distance of the complex number from the origin in the complex plane. | For \(z = x + yi\), \(|z| = \sqrt{x^2 + y^2}\) |
| Argument | The angle made by the line connecting the origin to the complex number point with the positive real axis. | For \(z = x + yi\), \( \tan(\theta) = \frac{y}{x} \) (consider the quadrant) |
| Principal Argument | The unique argument value in the interval \( (-\pi, \pi] \) or \( [0, 2\pi) \), depending on convention (usually \( (-\pi, \pi] \) in many contexts). | If \(z = 1\), \( \arg(z) = 0 \). If \(z = -1\), \( \arg(z) = \pi \). |
Additional Information: Complex Number Operations
Simplifying complex fractions often involves multiplying the numerator and denominator by the conjugate of the denominator. However, in this specific problem, the denominator simplified directly to the numerator, making the calculation easier.
The principal argument is crucial because the argument of a complex number is multi-valued (adding \(2n\pi\), where \(n\) is an integer). The principal argument provides a standard unique value.
The complex number \(z=1\) lies on the positive real axis in the complex plane. Points on the positive real axis have an argument of \(0\) radians.
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Important Questions from Complex Numbers
- Consider the following complex numbers
$$ z_1 = r_1(\cos \theta_1 + i \, \sin \theta_1) $$
$$ z_2 = r_2(\cos \theta_2 + i \, \sin \theta_2) $$
where $r_1, r_2$ are real numbers, $0 \leq \theta_1 \leq \frac{\pi}{2}$, $0 \leq \theta_2 \leq \frac{\pi}{2}$, and $i = \sqrt{-1}$
If $|z_1 + z_2| = |z_1| + |z_2|$, which one of the following conditions is necessarily CORRECT? - What is the value of \((2 - 3i) + (3 + 5i)\)
- What is the real part of the number \(15 + 2i\)
- What is the real part of the number $5i$?
- What is the value of $(6 + 3i) + (2 + 5i)$?
