Why are some precalc topics taught but not used in calc?

[annoyinggirl]

There are quite a few pre calc topics that are not used in calc. For example, complex and imaginary numbers. Also, systems of equations
why?

 

[e.bar.goum]

Who says that complex numbers aren't used in calculus? Or systems of equations? They're such basic topics in mathematics that get used everywhere.

 

[symbolipoint]

BUT those ARE used in Calculus. If you try to ignore complex numbers, then Calculus 2 will clobber you. Systems of Equations will be used in your Calculus 2 or Calculus 3 or both.

 

[micromass]

For example, complex and imaginary numbers.

Two examples:
1) Finding the integral ##\int e^x \sin(x)dx## can be done with conventional methods by applying integration by parts twice (and by recognizing a little trick). But it can also be done quite easily by complex numbers, and then you don't need integration by parts at all. The trick is to write ##\sin(x)## as complex exponentials.

2) We have an equality: ##\frac{1}{1-x^2} = \sum_{n=0}^{+\infty} x^{2n}##, but this is only true if ##|x|<1##. It is easy to see why this cannot be extended beyond this, as ##\frac{1}{1-x^2}## is not defined in ##x=1##. So the convergence of series is tied to singularities of the function. But now consider ##\frac{1}{1+x^2}= \sum_{n=0}^{+\infty} (-1)^n x^{2n}##. This is also valid only for ##|x|<1##. But now it is not at all clear why it breaks down beyond ##|x|<1##, as the function has no singularities. Complex numbers solve this neatly by saying the function has a singularity as ##\pm i##.

Also, systems of equations

Consider (for example) partial fraction decomposition in calc 2. This naturally yields a system of equation. There are many many examples of systems of equations throughout physics and math.

 

[Stephen Tashi]

There are quite a few pre calc topics that are not used in calc. For example, complex and imaginary numbers. Also, systems of equations
why?

I think a pre calculus course is supposed to prepare you for doing calculus and beyond . So if you take courses more advanced that the first semesters of calculus, you'll probably encounter complex numbers and systems of equations. (e.g. complex numbers are encountered in courses on differential equations, system of equations are encountered in a course on linear algebra).