The number that satisfies the equation is
[tex]x^2=-1[/tex]Doing the square root on both sides
[tex]x=\pm\sqrt{-1}[/tex]On complex number we use the definition:
[tex]i=\sqrt[]{-1}[/tex]Therefore, the number that satisfies the equation is
[tex]x=\pm i[/tex]That means that a solution is a complex number.
Complex numbers have two parts, the complex part and the real part:
[tex]z=a+bi,\quad a,b\in\mathbb{R}[/tex]Look that a and b are real coefficients, but z is a complex number because we have b multiplying i, the complex unit.
But why complex numbers are so important?
• Complex numbers can be used to solve polynomial equations, that we couldn't solve before
,• Complex numbers can connect exponential function with trigonometric functions
[tex]e^{ix}=\cos (x)+i\sin (x)[/tex]Complex numbers are a powerful set that we can use to solve many complex problems in physics and mathematical, it may seem hard but they actually make things easier, sometimes working with complex numbers is way easier than working with real numbers.
