TITLE: Why do we still do symbolic math?
QUESTION [109 upvotes]: I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one.
Numerical computations, to my understanding, never deal with irrational numbers, but only rational numbers.
Why do mathematicians construct the real numbers and go through the pain of using their complicated properties to develop real analysis, etc. to be able to solve just a few cases symbolically?
Why do non-mathematicians deal with real numbers and symbolic calculation if the majority of cases must use rationals and numerical computation only?

REPLY [8 votes]: Suppose you run a numerical analysis program, and it outputs

$$0.5323$$

What does that even mean?  Does it mean:
$$x = 0.5323$$
$$0.5323 \le x < 0.5324$$
$$0.53225 \le x < 0.53235$$
$$\text{Probability}\left(\frac{|x - 0.5323|}{0.5323} < 0.01\right) > 95 \%$$
The meaning of a numerical program still must be expressed and proven symbolically.  Mathematics isn't the study of numbers, it's the study of statements, some of which are numerical.