Complex Numbers
The expression \( \displaystyle \frac{1}{2-11i} \) can be rewritten with a real denominator (using the complex conjugate) in the form \[ \displaystyle \frac{a+bi}{c}, \]
where \( a \), \( b \), and \( c \) are positive, co-prime integers. What is the sum of \( a \), \( b \), and \( c \)?
Details and assumptions
\( i \) is the the imaginary unit, defined by \( i^2 = -1 \).
Consider the complex numbers \[A=12+i \mbox{ and } B=11-3i.\] If the value of \(2A\overline{A}+2\overline{A}B+A\overline{B}+B\overline{B}\) is \(X+Yi\), what is the value of \(X-Y\)?
Details and assumptions
\(\overline{A}\) denotes the complex conjugate of the complex number \(A\).
\(i\) is the imaginary number that satisfies \(i^2 = -1\).
For complex numbers \(\alpha=5+7i\) and \(\beta=3-i,\) what is the value of \[\alpha \overline{\alpha} +\overline{\alpha} \beta +\alpha \overline{\beta} +\beta \overline{\beta}?\]
Details and assumptions
\(\overline{\alpha}\) denotes the complex conjugate of the complex number \(\alpha\).
\(i\) is the imaginary number that satisfies \(i^2 = -1\).
Consider the complex numbers \[A=8-2i \mbox{ and } B=4-i.\] What is the value of \((A+B)^2+\left(\overline{A+B}\right)^2\)?
Details and assumptions
\(\overline{A}\) denotes the complex conjugate of the complex number \(A\).
\(i\) is the imaginary number that satisfies \(i^2 = -1\).
Let \(\omega\) and \(z\) be the complex numbers \[ \omega =2-i, \quad z =\frac{\omega+4}{4\omega-1}. \]
If \(z\overline{z}\) can be expressed as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, what is \(a+b?\)
Details and assumptions
\(\overline{z}\) denotes the complex conjugate of the complex number \(z\).
\(i\) is the imaginary number that satisfies \(i^2 = -1\).