What is a complex number?
Considere the following equation: $x^2=-4$. This equation has no real number solution, because any real number, when squared, results in a positive number. The solutions to this equation are the complex numbers "2i" and "-2i", where "i" is called the Imaginary Unit, and is defined as the square root of -1 ($i=\sqrt{-1}$). Thus,
$(2i)^2=2^2(i^2)=4(-1)=-4$ and
$(-2i)^2=(-2)^2(i^2)=4(-1)=-4$.
Complex number notations
1) Algebraic Form: $a+bi$, where "a" and "b" are real numbers, "a" is the real part, "b" is the imaginary part, and "i" is the imaginary unit.
2) Ordered Pair Form: (a, b), where "a" and "b" are real numbers, "a" is the real part, and "b" is the imaginary part. That is, the ordered pair $(a, b)$ correponds to the algebraic form $a+bi$.
The ordered pair form can be visualized in a Cartesian system of coordinates called complex plane or Argand-Gauss diagram, in which the horizontal axis displays the real part, and the vertical axis displays the imaginary part of the complex number (following figure).
3) Trigonometric Form or Polar Coordinates. $z=|z|.(cos{\theta}+i.sen{\theta})$, where $|z|$ is called the Modulus of the complex number, and $\theta$ is its Argument.
In the complex plane (following figure), $|z|$ is the distance from point "z" to the origin of the system, and $\theta$ is the angle between the line that connects the origin of the system to point z, and the positive real axis ($|z|.cos{\theta}=a$ and $|z|.sen{\theta}=b$).
Complex conjugate
The complex conjugate of a complex number $a+bi$ is the complex number $a-bi$.
The product of a complex number and its conjugate is a real number:
$(a+bi)(a-bi)=(a)^2-(bi)^2=a^2+b^2$
Solved SAT Practice Tests
Find Practice Tests in the following link:
SAT Practice Tests - Complex Numbers
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