###### Waste less time on Facebook — follow Brilliant.
×
Back to all chapters

# Complex Numbers

"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

# Complex Conjugates - Arithmetic

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 $$i^2 = -1$$.

×