Why do we still do symbolic math? I just read that most practical problems (algebraic equations, di

kreamykraka80

kreamykraka80

Answered question

2022-07-04

Why do we still do symbolic math?
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?

Answer & Explanation

iskakanjulc

iskakanjulc

Beginner2022-07-05Added 18 answers

Try to solve x 1 / 3 = 0 with a "symbolic calculation". Now try to solve x 1 / 3 = 0 with Newton's Method. The point is numerical methods can fail and anyone using numerical methods should have an understanding of mathematics so that they can detect when a numerical method is not working and perhaps even fix the problem. Also as this case shows sometimes the symbolic calculation is easier.
All these computational methods need to be designed and implemented. It takes an understanding of the underlying mathematics to do this.
civilnogwu

civilnogwu

Beginner2022-07-06Added 6 answers

x 2 = 2 doesn't have a solution in the rationals. So what could you possibly mean by it having a numerical solution?
Do you mean that you can produce a rational numbers whose squares are arbitrarily close to 2? Congratulations: your notion of "having a numerical solution" is simply reinventing the notion of "real number", just in a form that is much more cumbersome to think about and manipulate than usual.
in addition to what has already been said: Many mathematical results, and even entire branches of math, can't be done numerically because they're about objects that are much more general than numbers. Group theory and topology spring to mind as examples.

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