Algebra

Complex Numbers

Complex Conjugates - Arithmetic

         

The expression 1211i \displaystyle \frac{1}{2-11i} can be rewritten with a real denominator (using the complex conjugate) in the form a+bic, \displaystyle \frac{a+bi}{c},

where a a , b b , and c c are positive, co-prime integers. What is the sum of a a , b b , and c c ?

Details and assumptions

i i is the the imaginary unit, defined by i2=1 i^2 = -1 .

Consider the complex numbers A=12+i and B=113i.A=12+i \mbox{ and } B=11-3i. If the value of 2AA+2AB+AB+BB2A\overline{A}+2\overline{A}B+A\overline{B}+B\overline{B} is X+YiX+Yi, what is the value of XYX-Y?

Details and assumptions

A\overline{A} denotes the complex conjugate of the complex number AA.

ii is the imaginary number that satisfies i2=1i^2 = -1.

For complex numbers α=5+7i\alpha=5+7i and β=3i,\beta=3-i, what is the value of αα+αβ+αβ+ββ?\alpha \overline{\alpha} +\overline{\alpha} \beta +\alpha \overline{\beta} +\beta \overline{\beta}?

Details and assumptions

α\overline{\alpha} denotes the complex conjugate of the complex number α\alpha.

ii is the imaginary number that satisfies i2=1i^2 = -1.

Consider the complex numbers A=82i and B=4i.A=8-2i \mbox{ and } B=4-i. What is the value of (A+B)2+(A+B)2(A+B)^2+\left(\overline{A+B}\right)^2?

Details and assumptions

A\overline{A} denotes the complex conjugate of the complex number AA.

ii is the imaginary number that satisfies i2=1i^2 = -1.

Let ω\omega and zz be the complex numbers ω=2i,z=ω+44ω1. \omega =2-i, \quad z =\frac{\omega+4}{4\omega-1}.

If zzz\overline{z} can be expressed as ab,\frac{a}{b}, where aa and bb are coprime positive integers, what is a+b?a+b?

Details and assumptions

z\overline{z} denotes the complex conjugate of the complex number zz.

ii is the imaginary number that satisfies i2=1i^2 = -1.

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