Wright State University Lake Campus/2020-1/Intro to imaginary numbers

Students have trouble appreciating the square root of minus 1, or

${\sqrt {-1}}=i$

For this introduction, all you need is the following ideas:

Derivative of a polyomial

${\frac {d}{dt}}Ct^{n}=nCt^{n-1}$

Newton's law for an ideal spring

$F=-kx=ma=m{\frac {dv}{dt}}=m{\frac {d^{2}x}{dt^{2}}}$

The first step in understanding the solution to this is to imagine that k=m=1:

${\frac {d^{2}x}{dt^{2}}}=-x$

What function of time is the negative of it's own derivative?

The following infinite series famously equals it's own second derivative:

$f(t)=1+t+{\frac {1}{2\cdot 1}}t^{2}+{\frac {1}{2\cdot 1}}t^{2}+{\frac {1}{3\cdot 2\cdot 1}}t^{3}+{\frac {1}{4\cdot 3\cdot 2\cdot 1}}t^{4}+...$

See

• wikipedia:Exponential function
• commons:File:Serie di Taylor della funzione seno.gif
• wikipedia:Taylor_series