# What’s the difference between analytical and numerical approaches to problems?

I don’t have much (good) math education beyond some basic university-level calculus.

What do “analytical” and “numerical” mean? How are they different?

Analytical approach example:

Find the root of $$f(x)=x−5f(x)=x-5$$.

Analytical solution: $$f(x)=x−5=0f(x)=x-5=0$$, add $$+5+5$$ to both sides to get the answer $$x=5x=5$$

Numerical solution:

let’s guess $$x=1x=1$$: $$f(1)=1−5=−4f(1)=1-5=-4$$. A negative number. Let’s guess $$x=6x=6$$: $$f(6)=6−5=1f(6)=6-5=1$$. A positive number.

The answer must be between them. Let’s try $$x=6+12x=\frac{6+1}{2}$$: $$f(72)<0f(\frac{7}{2})<0$$

So it must be between $$72\frac{7}{2}$$ and $$66$$...etc.

This is called bisection method.

Numerical solutions are extremely abundant. The main reason is that sometimes we either don't have an analytical approach (try to solve $$x6−4x5+sin(x)−ex+7−1x=0x^6-4x^5+\sin (x)-e^x+7-\frac{1}{x} =0$$) or that the analytical solution is too slow and instead of computing for 15 hours and getting an exact solution, we rather compute for 15 seconds and get a good approximation.