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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?

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Not all math is computation. – Henry Swanson Jul 29 '14 at 16:24
Not all mathematicians only care about calculations. I never do any calculations. The only time in recent memory I wrote an explicit number beside 0, 1, or 2 was to pay my bills. – William Jul 29 '14 at 16:27
Even to formulate the algorithms that perform numerical computations, you need symbolic notation to develop/explain/understand them in the first place. – Hagen von Eitzen Jul 29 '14 at 16:30
@William, when I was a grad student taking a course in analytic number theory, a fellow student stopped the professor in mid-proof to ask about a funny-looking symbol on the blackboard. The answer: "It's a 5." – Barry Cipra Jul 29 '14 at 16:34
If you couldn't compute symbolically you'd never know what to compute numerically. – Christian Blatter Jul 29 '14 at 18:38

24 Answers 24

up vote 110 down vote accepted

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.

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One could go so far and say that there is no such thing as a "numerical solution", only a numerical approximation. – Turion Jul 30 '14 at 8:42
@Turion I don't think that's entirely accurate. Some problems (e.g. in number theory) may have exact solutions that can be computed numerically. – Carlos Jul 31 '14 at 19:43

$x^2 = 2$ 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.

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IF in a computation there will not be a square operation later on THEN a rational approximation of the required precision is AT LEAST as good – totalnoob Jul 29 '14 at 20:06
@totalnoob There might well be other operations than simple squaring that work with the real number $\sqrt2$ but not with any of its rational approximations. For example, one can easily compute $1/(\sqrt2-1)$ as $\sqrt2+1$. That doesn't work for any rational approximations. – Andreas Blass Jul 29 '14 at 21:52
@totalnoob: The key word is "approximation": you have a notion that you are approximating something. What is the thing that you're approximating? It's a real number. So you need to study real numbers anyways to understand what you're doing, even if you only ever intend to work with rational approximations. – Hurkyl Jul 30 '14 at 8:39
Many of the concrete constructions of real numbers can be viewed as defining a real number as a way to approximate things. e.g. the construction by Dedekind cuts identifies each irrational real number with a way to label every rational number as being "too big" or "too small". (of course, people rarely think about a real number in this fashion, even if they opt to construct them that way) – Hurkyl Jul 30 '14 at 8:43

If there was no symbolic math, we would still be living in caves and eating roots.

Formulas have meaning:

$$F=m\cdot a$$ is enough to explain a great deal of the universe.

Numbers are uninformative and unstructured: $$41.04=7.54\cdot5.443$$ tells you about nothing.

Real numbers have been invented because they model very well our intuition of a continuum, like points in space, and they give rise to billions of interesting topics.

Numerical values are essentially integers, desperately trying to mimic the former.

Symbolic solutions to a problem are compact and show patterns that can be handled by the human mind.

Numerical solutions are huge and as dumb as computers.

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"still be living in caves and eating roots" all history professors would disagree – totalnoob Jul 29 '14 at 20:09
This is a perfect example: Newton's method would just not exist if we didn't know what are derivatives and how you compute them symbolically. This said, the closed formulas for the cubic and quartic case can be faster than root searches. – Yves Daoust Jul 29 '14 at 22:07
and you can solve the general quintic using elliptic functions. – Francis Davey Jul 30 '14 at 11:37
@YvesDaoust If you compare with Newton, sure, I agree. The standard numerical method for polynomial solution is using the Francis QR iteration for eigenvalues on the companion matrix though, and that needs no guessing for initial values. – Federico Poloni Jul 31 '14 at 19:44
@YvesDaoust What? You mean you went to Leopold Kronecker's grave and gave his tombstone a lecture on "finite expressions involving integers constants only"? – James47 Sep 18 '14 at 21:07

In addition to what others are saying: 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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Well, there is "computational group theory", but your point is still valid. – naslundx Jul 31 '14 at 8:04
+1 and many of those branches of math (i.e., Graph Theory) are absurdly practical. – Seth Aug 3 '14 at 3:58

This is a great question, and I like it especially since I'm a student of computational science and so spend my days around numerical software and approximations.

Symbolic math exists for the same reason that not all physics, chemistry, biology, economy etc. can be reduced to one large computer program. We still need to think and reason about problems. Even for pure math, this is no exception.

In a way, your queston answers itself. Numerical methods can be great for lots of problems, but none of them are perfect. There is continuing ongoing development for new ones, and attempts to develop efficient numerical algorithms for problems that can not yet be solved numerically. However, numerical software cannot develop new numerical software. And you cannot numerically prove that certain numerical algorithms are stable, or show how efficient they are, or why one is better than the other in certain instances. This entire research in numerical methods themselves has to be done symbolically.

(Oh, and of course, there is a lot of math which is not based around computation and cannot be done numerically in any way.)

Also: As another answer points out, some problems can be solved more easily symbolically than with a complicated numerical method. The mathematical approach can often yield more information, more insight, than just the answer.

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I doubt we would have come up with the highly useful imaginary and complex numbers if we insisted on numerical computations only. – James47 Aug 5 '14 at 21:44

The "point" of maths is not to be "practical" or "useful" in my opinion. The point of maths is maths itself. Sure it can be used to help modelling the universe and the laws it obeys but I think maths are beyond that.

And although applied mathematics, physics, engineering and such deal with "close enough" numerical aproximations of a solution of a problem, the math it's based upon is way more abstract. Pure maths is required to be a solid base if you are to use it, otherwise you can't be certain of wether a numerical solution is a good enough aproximation or not.

In my experience, pure mathematics are way more beautiful and elegant that the aplied stuff. I don't do it to "use" it as a means to an end. Pure maths is both a mean and an end...

(But that's just my point of view)

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The major problem I have with your question is that the premise is false.

I just read that most practical problems (algebraic equations, differential equations) do not have symbolic solution, but only numerical.

This is patently incorrect--it is very rare for me to find an exact, numerical solution to any of the problems I have encountered in physics, engineering, chemistry, or computer science. The real numbers are exceedingly useful in the sciences.

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"Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dynamical systems." from the Preface of Structure and Interpretation of Classical Mechanics – totalnoob Jul 29 '14 at 16:46
in physics/engineering/chemistry, as far as can I see, everyone is always approximating (the models used for setting the equations) so there are NO exact solutions anyway, in computer science there are "only integers" – totalnoob Jul 29 '14 at 16:48
@totalnoob: In response to your quotation: that doesn't say "most problems cannot be solved with symbols" rather, it says "we focus on a small class of problems that just so happen to be symbolically solvable." In response to your other comments: I see exactly the opposite with regards to approximation--we approximate our experimental results, but the models are exact. (Example: Maxwell's Equations. Extremely basic example: Circumference of a circle) Re CS: You have clearly never read TAOCP by Knuth. ;) – apnorton Jul 29 '14 at 17:06
@totalnoob Every equation is write is exact, mathematically, and approximate, realistically. You may compute its solutions numerically, but that does not mean it is not exact. It is quite arrogant to say you have all details of reality captured, for all instances, in a set of equations. – Mark Fantini Jul 29 '14 at 17:14
In any field, you solve as much as you can symbolically (or rely on known solutions), then do the work to solve the few remaining very difficult problems numerically. This makes it seems like practical problems are numerical, but it only means the part of those problems that takes a long time is numerical. – farnsy Aug 1 '14 at 20:57

Perhaps an analogy is in order. Consider the field of statistics: we have applied statistics, in which the primary interest is in performing statistical tests of significance to make inferences or predictions about real-world phenomena. In this area, we concern ourselves with appropriate methods of experimental design, data collection, and data analysis. The correct choice of design and analysis is critical for arriving at a justifiable conclusion.

By contrast, we also have mathematical statistics. This is concerned with discovering and justifying the theoretical foundations upon which various concepts that are commonly employed in applied statistics we rely. Here, one is interested in such things as proving that a particular statistic might have desirable properties that make it amenable for using in a hypothesis test. Without this foundation, which is itself reliant on pure mathematical concepts such as measure theory, one could not rigorously justify the use of the wide variety of methods available to us in applied statistics. We could not, for example, defend the use of random forests as a classification method, or principal component analysis. We would not have a solid basis for Bayesian decision theory. All of these things rely on a well-developed theoretical exposition.

That is not to say one aspect of statistical practice is superior to any other, or that the only meaningful purpose of statistics is to be applied. The same is true for mathematics and applied mathematics in general: the nature of the subject--and indeed, the nature of the process of discovering and expanding human knowledge--is such that it is generally not possible to anticipate the utility of something at the time it is researched; consequently, utility alone cannot be the goal of academic research. Knowledge for knowledge's sake is a good philosophy, one that we should have faith will repay its investment in the long run, as this has overwhelmingly been the case throughout the course of human civilization.

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Much of the theory behind numerical estimates is based on 'symbolic' math. For instance if you use a certain numerical method to approximate something, symbolic math can help explain why the method works and how accurate the approximation is.

I also agree with the idea that math has wider interest than just 'real world' problems that may use approximation techniques.

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Page 24, section 1.8 of this book:

It is lucky that computers had not yet been invented in Jacobi's time. It is possible that they would have prevented the discovery of one of the most beautiful theories in the whole of mathematics: the theory of elliptic functions, which leads naturally to the theory of modular forms, and which, besides being gorgeous for its own sake [Knop93], has been applied all over mathematics (e.g., [Sarn93]), and was crucial in Wiles's proof of Fermat's last theorem.

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If you tried to examine stability of the solutions of differential equations, you wouldn't get anywhere without symbolic maths. With purely numerical solutions, all you would figure out that sometimes your solutions don't work, and you wouldn't have a clue why. And you wouldn't have a clue which solutions to trust, and which solutions not to trust.

And of course there are tons of problems where the solution isn't numerical. A nice set of problems is the design of floating-point arithmetic where numerical mathematics is actually of no help whatsoever! (Try proving that whenever x/2 < y < 2x, the result of the floating-point subtraction y - x is exact). The whole world of combinatorics. The four colour theorem (numerical version: Every map can be coloured using between 3.21 and 4.61 colours). Try finding gamma (100.7) without using symbolic maths.

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It seems to me that this question is a consequence of the awful way that math is taught these days. If you see math as "some methods to calculate solutions to practical problems", then you might be tempted to think the way you do, although some of the answers above clarify that even in this mindset the theory is important.

But mathematics is so much more than that. Theoretical physics is based in abstract mathematics, and it has provided the framework through which we attempt to understand the world we live in, and has produced many concrete applications (both quantum mechanics and relativity have a profound influence in today's technology, and they are phrased in terms of very abstract mathematical concepts).

Group theory and other algebraic concepts have applications both within other branches of mathematics and also real-world problems (cryptography, for example). Others have mentioned how important it is to understand differential equations, not just attempt to solve them numerically. And the list goes on and on.

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You ask:

Why do we still do symbolic math?

because you read that

(...) most practical problems (algebraic equations, differential equations) do not have symbolic solution, but only numerical.

Well, I would say you would NOT even be able to HAVE your 'most practical problems, like algebraic or differential equations', WITHOUT symbolic math. When you denote a variable value with a symbol like $x$ and say it's proportional to a logarithm of some other variable $y$, you already used 'symbolic math' with its notions of continuum, functions, real numbers and others.

If you want to solve a simple algebraic equation with Newton's method or minimize a real function of more than one variable, you need a real analysis to formulate an algorithm. Of course the algorithm then works on some integer and rational approximations, but you probably will fail to define and prove it without real numbers, limits, derivatives ets.

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Even in numerical analysis, if you can find a symbolic, closed form solution for your problem, you can use it as a regression test. And any numerical software without regression tests is probably wrong.

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Symbolic algebra is far newer than numerical approximations, and developed in Europe during the Enlightenment (ref. Florian Cajori, A History of Mathematical Notation). Even the Eighth-century Arabic Algebra of Al-Khowarizmi described its operations in words (it didn't even use numerals). This older "algebra" is in many ways more of a strategy guide for performing arithmetic.

So in a way, the question is moot. Since the symbolic technique is newer, there is no natural metaphor for it to automatically fall out of favor, and the question "why still do it?" loses its sense. Whatever it was that made us start!

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Without symbolic math it would be impossible to solve for instance Einstein equations for some more complex metrics that Minkowski's. Chaos theory is all about nonlinear equations. Mostly all of the physics laws are too complex to be solved exactly. Therefore we need approximations (I don't mean numerical approximations) and symbolic math to solve them.

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Suppose you run a numerical analysis program, and it outputs


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.

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The table on this page: cleared some of my misunderstandings.

I suppose numeric "solution" means using only arithmetic expressions, which in many cases can only be approximations and always result in rational numbers.

I suppose "computation" is also a colloquial term for using only arithmetic expressions, simply because computer hardware has only +,-,*,/ built-in.

I am still wondering what "symbolically tractable" means in this quote:

Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dynamical systems.

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If you are the same person who asked the question, note that your accounts got split. You should register your account (set up a password) and then request merge using contact us link at the bottom. – user147263 Jul 30 '14 at 7:35
did not find the quote you state, so i think symbolically tractable may be related to symbolic dynamics. Check also my answer to this question if you like (for this question) – Nikos M. Aug 6 '14 at 17:06

Symbolic mathematics has helped in discovering new physics. For example, Dirac (I think) was looking at the symbolic solutions for electrons & couldn't just ignore a sqrt(-1) issue. His solution was to postulate that electrons can have both positive & negative charge & hence we end up with a description for the positron. i.e. antimatter.

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There's already a ton of useful answers here, but I figured I throw in another stimulus for reflection.

From the way you ask your question (which is very legitimate), it looks to me as you consider numerical (discrete) solutions as if they had nothing to do with the real (continuous) problem. Or, in other words, that there's no need of real numbers and symbolic calculus if all you will do is use numerical approximations on a computer. Well, this could make sense once the methods are already available and safe to use, but that would limit the sets of problem that we can (numerically) solve to those that we already know how to solve. Let me explain.

What is a numerical method? In few words, it is a procedure that, under certain conditions, provides us with an approximate solution to the problem at hand. But what does it mean that the solution is "approximate". Well, sounds pretty clear: it is "close" to the true solution of the true problem. But how can you measure the distance between the approximate (discrete) problem and the real (perhaps continuous) problem if you don't position yourself in the setting of the real/continuous model?

Numerical methods, in order to be safe to use, must come with a "certificate" (a proof, in math language) that guarantees that, if some assumption are satisfied, then the method will work. This is crucial, otherwise we're just pressing a button and hoping that the number that the computer spits out is actually trustworthy. But again, in order to produce this certificate, you must be able to compare the discrete and the real problem. Since the discrete problem is formulated in a setting that is (usually) a strict subset of the setting of the continuous problem, your only hope is to compare the two problems in the continuous setting. Therefore, you need to be able to work (at least symbolically) in the continuous world, which is a superset of the discrete one.

Once you know (i.e., prove) that the procedure that you came up with makes sense in the continuous world, only then you can temporarily forget about the continuous space structure and only work in the discrete one. But even then, philologically, an approximation can be called as such only if there is something else to which the approximation is related (and hopefully close). So, even though you don't need to be able to manipulate the object in the continuous setting, you still need to be able to formulate the problem in the continuous setting before you start talking about the approximation.

Hope this made some sort of sense.

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The question of why we need real numbers is a good question. The basic answer is that real numbers are vital to the theoretical foundation of analysis (calculus) - "completeness property", is the key. It is true that people of an engineering or physics background who rely on calculus techniques do not require the theory of real numbers, since they can just pretend they approximated it with rational numbers. But to do calculus in a mathematical way one requires the real number system. Analysis never took off as a rigorous branch of mathematics until the real number system was developed.

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Without symbolic mathematics there would definitely be less to calculate. And without symbolic mathematics there would be no or very little development.

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There are too many sub-topics in math but it is split with two main topic theorical and applied mathematics. You can't prove 2n is an even number with numbers it goes to infinity and a simple question like ' a car try to reach a target. There is a road between target and car about 100 km. car is reaching half of way every time. 50,25,12.5,... when it is end ? Can car reach the target ? You cant answer questions like this without symbolic math.As a mathematician I can say numbers is just seen side of iceberg

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All nice and well, but people tend to forget that eventually symbols are numbers or numbers are symbols (what is called? ahh Goedel encoding). So numbers/symbols are interchangeable tokens. One can very well do math using just the "symbols" from $0$ to $9$ plus the "inference rules" of numerical analysis.

Now the fact remains that either with "real symbols" or "fake symbols" (numbers), the computations remain the same difficult (but the difficulty manifests in different way).

Someone said that one cannot (absolutely) solve $x^{1/3}=0$ numerically. But this is convention, like $1+1=2$, this is also convention, it is not about an inherently higher power of the symbol $x$ over the symbol $1$, it is a convention used.

My $.1234567$ cents :)

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