I. \(1 + 4i > 3 + 2i\)
II. \(2 + 3i < 3 + 4i\)
III. \(4 + 3i > 3 + 4i\)
where \(i = \sqrt{-1}\)
How many of the above are valid ?
Analyzing Complex Number Inequalities Validity
The core principle is that standard ordering operators like \(>\) (greater than) and \(<\) (less than) are not defined for complex numbers if their imaginary parts are non-zero. These operators only apply to real numbers.
Let's examine each inequality based on this principle:
Inequality I: \(1 + 4i > 3 + 2i\)
- Complex numbers involve both a real part and an imaginary part.
- Standard comparison operators (\(>\), \(<\)) are undefined for complex numbers unless they are purely real (imaginary part is zero).
- Therefore, the inequality \(1 + 4i > 3 + 2i\) is not valid.
Inequality II: \(2 + 3i < 3 + 4i\)
- Similar to Inequality I, the comparison operator \(<\) is not applicable to complex numbers \(2 + 3i\) and \(3 + 4i\) as they have non-zero imaginary parts.
- Thus, the inequality \(2 + 3i < 3 + 4i\) is invalid.
Inequality III: \(4 + 3i > 3 + 4i\)
- Again, the operator \(>\) cannot be meaningfully applied to compare the complex numbers \(4 + 3i\) and \(3 + 4i\).
- Hence, the inequality \(4 + 3i > 3 + 4i\) is invalid.
Conclusion on Validity
Since none of the provided inequalities involving complex numbers are mathematically valid according to standard definitions, the correct count is zero.
- Inequality I: Invalid
- Inequality II: Invalid
- Inequality III: Invalid
Therefore, zero out of the three inequalities are valid.
Similar Questions
If
\(\left( \frac{1-i}{1+i} \right)^{2m} \left( \frac{1+i}{1-i} \right)^{2n} = 1\)
where \(i = \sqrt{-1}\), then what is the smallest positive value of \((m – n)\)?
- If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
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- What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)? - If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
If A2 + B2 + C2 = 0, then what is the value of the following?
\(\Delta = \begin{vmatrix} 1 & \cos C & \cos B \\\ \cos C & 1 & \cos A \\\ \cos B & \cos A & 1 \end{vmatrix}\)If ω is a non-real cube root of unity, then what is a root of the following equation?
Important Questions from Complex Numbers
If A + iB = tan (x + iy), then the value of tan 2x is?
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
