Consider the following in respect of a complex number z: 1. \(\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}\) 2. zz -1 = |z| 2 Which of the above is/are correct?
1 only
Analyzing Complex Number Properties
We are asked to evaluate two statements regarding properties of a complex number \(z\). Let's analyze each statement carefully.
Statement 1: Conjugate of Inverse Property
The first statement is \(\overline{\left(z^{-1}\right)}=(\bar{z})^{-1}\). This statement relates the conjugate and the inverse of a complex number.
Let \(z = x + iy\), where \(x\) and \(y\) are real numbers and \(z \neq 0\).
The inverse of \(z\) is \(z^{-1} = \frac{1}{z}\).
To find the inverse, we multiply the numerator and denominator by the conjugate of \(z\):
\[z^{-1} = \frac{1}{x + iy} = \frac{1}{x + iy} \times \frac{x - iy}{x - iy} = \frac{x - iy}{x^2 - (iy)^2} = \frac{x - iy}{x^2 + y^2}\]
Now let's find the conjugate of this inverse:
\[\overline{\left(z^{-1}\right)} = \overline{\left(\frac{x - iy}{x^2 + y^2}\right)} = \frac{\overline{x - iy}}{x^2 + y^2} = \frac{x + iy}{x^2 + y^2}\]
Next, let's find the conjugate of \(z\), which is \(\bar{z} = x - iy\).
Now let's find the inverse of the conjugate \(\bar{z}\):
\[(\bar{z})^{-1} = \frac{1}{\bar{z}} = \frac{1}{x - iy}\]
Similar to finding \(z^{-1}\), we multiply the numerator and denominator by the conjugate of \(\bar{z}\) (which is \(z\)):
\[(\bar{z})^{-1} = \frac{1}{x - iy} \times \frac{x + iy}{x + iy} = \frac{x + iy}{x^2 - (iy)^2} = \frac{x + iy}{x^2 + y^2}\]
Comparing the results:
\[\overline{\left(z^{-1}\right)} = \frac{x + iy}{x^2 + y^2}\] \[(\bar{z})^{-1} = \frac{x + iy}{x^2 + y^2}\]
Since the results are equal, the statement \(\overline{\left(z^{-1}\right)}=(\bar{z})^{-1}\) is correct for any non-zero complex number \(z\).
Statement 2: Product of a Complex Number and its Inverse
The second statement is \(zz^{-1} = |z|^2\).
By the definition of a multiplicative inverse, for any non-zero complex number \(z\), the product of \(z\) and its inverse \(z^{-1}\) is the multiplicative identity in the complex number system, which is 1.
So, \(zz^{-1} = 1\).
Now let's consider \(|z|^2\). If \(z = x + iy\), then the modulus of \(z\) is \(|z| = \sqrt{x^2 + y^2}\). The square of the modulus is \(|z|^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2\).
The statement \(zz^{-1} = |z|^2\) claims that \(1 = x^2 + y^2\). This is only true if \(x^2 + y^2 = 1\), which means \(|z|=1\). This is not true for all complex numbers \(z\).
For example, if \(z = 1 + i\), then \(|z|^2 = 1^2 + 1^2 = 2\). But \(zz^{-1} = (1+i) \cdot \frac{1}{1+i} = 1\). Here \(1 \neq 2\).
Therefore, the statement \(zz^{-1} = |z|^2\) is incorrect in general.
Conclusion
Based on our analysis:
- Statement 1 \(\overline{\left(z^{-1}\right)}=(\bar{z})^{-1}\) is correct.
- Statement 2 \(zz^{-1} = |z|^2\) is incorrect.
Thus, only Statement 1 is correct.
The option stating that only 1 is correct is the correct choice.
Revision Table: Key Complex Number Definitions
| Term | Definition (for \(z = x + iy\)) | Notation |
|---|---|---|
| Complex Number | A number of the form \(x + iy\), where \(x, y \in \mathbb{R}\) and \(i = \sqrt{-1}\) | \(z\) |
| Real Part | The real number \(x\) | \(Re(z)\) |
| Imaginary Part | The real number \(y\) | \(Im(z)\) |
| Conjugate | A complex number with the imaginary part negated | \(\bar{z} = x - iy\) |
| Modulus | The distance of the complex number from the origin in the complex plane | \(|z| = \sqrt{x^2 + y^2}\) |
| Inverse | A complex number \(z^{-1}\) such that \(zz^{-1} = 1\) | \(z^{-1} = \frac{1}{z}\) |
Additional Information on Complex Number Properties
Understanding the fundamental properties of complex numbers is crucial. Here are some additional points related to the concepts discussed:
- Inverse of a Complex Number: The inverse \(z^{-1}\) exists if and only if \(z \neq 0\). It can be calculated as \(z^{-1} = \frac{\bar{z}}{|z|^2}\). Let's verify this formula: \(z \cdot \frac{\bar{z}}{|z|^2} = \frac{z\bar{z}}{|z|^2}\). We know that \(z\bar{z} = (x+iy)(x-iy) = x^2+y^2 = |z|^2\). So, \(\frac{z\bar{z}}{|z|^2} = \frac{|z|^2}{|z|^2} = 1\). This confirms the formula.
- Properties of Conjugates: Conjugation has several useful properties, such as \(\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}\), \(\overline{z_1 z_2} = \bar{z_1} \bar{z_2}\), and \(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}\) (for \(z_2 \neq 0\)). The property \(\overline{\left(z^{-1}\right)}=(\bar{z})^{-1}\) is a specific case of the division property where \(z_1 = 1\). Since \(\bar{1} = 1\), \(\overline{\left(\frac{1}{z}\right)} = \frac{\bar{1}}{\bar{z}} = \frac{1}{\bar{z}}\).
- Modulus Properties: The modulus also has important properties, such as \(|z_1 z_2| = |z_1| |z_2|\) and \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\) (for \(z_2 \neq 0\)). Also, \(|z|^2 = z\bar{z}\), which we used above.
- Geometric Interpretation: Multiplication by \(z^{-1}\) is equivalent to division by \(z\). Geometrically, dividing by \(z\) involves dividing the modulus and subtracting the argument. Conjugation \(\bar{z}\) is a reflection across the real axis in the complex plane. The property \(\overline{\left(z^{-1}\right)}=(\bar{z})^{-1}\) means that reflecting the point representing \(z^{-1}\) across the real axis is the same as reflecting \(z\) across the real axis and then finding the inverse of the reflected point.
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$$ z_1 = r_1(\cos \theta_1 + i \, \sin \theta_1) $$
$$ z_2 = r_2(\cos \theta_2 + i \, \sin \theta_2) $$
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 $i = \sqrt{-1}$
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- What is the real part of the number \(15 + 2i\)
- What is the real part of the number $5i$?
- What is the value of $(6 + 3i) + (2 + 5i)$?
