Definition of "simplify"

Question

In mathematics the word "simplify" is used a lot. In a lot of cases it is obvious what actually makes an expression simpler, but not always. Is there a measurable definition of simplicity or is it subjective? (I realize this could also depend on the specific question/topic/etc.)

For example, which is simpler: $(2x-3y)(4x+y)$ or $8x^2-10xy-3y^2$ ?

• The first is simpler because it involves linear expressions as opposed to quadratics?
• The second is simpler because it has no parentheses?

If possible I'd love some references to answers.

Answer 1

The word simplify is used a lot in drill books, primarily school books. This is bad practice since, as you seem to understand quite well, simplify is not an objective term. You can simplify toward achieving something, and then the simplification can be measured somehow. But, without the aim of the operation, it is meaningless to claim simplification was achieved. It is equally meaningless to ask to simplify without being specific about what is to be done and how difficulty is measured. For instance, "simplify $\frac{50}{100}$" is a common exercise in books, and the expected answer is $\frac{1}{2}$. However, when asked to simplify $\frac{1}{2}+\frac{1}{100}$ the answer goes like this: $...=\frac{50}{100}+\frac{1}{100}=\frac{51}{100}$. But hey, did we just 'unsimplify' $\frac{1}{2}$ on the way to get $\frac{50}{100}$??? Well, no, we actually simplified it, since the simplification of $\frac{1}{2}$ toward adding it to $\frac{1}{100}$ results in $\frac{50}{100}$.

It is unfortunate that so many textbooks ask for context-less simplifications - something that simply does not exist. Any question of the form "simplify blah" can (and should) be answered quite literally: blah.

Answer 2

I agree that is it very bad practice to ask students to "simplify." It usually means "Do that thing I've just been telling you how to do ".

One possible criterion for simplification would be the number of operations required to evaluate the expression

$(2x-3y)(4x+y)$ requires 4 multiplications and 2 additions

$8x^2-10xy-3y^2$ requires 6 multiplications and 2 additions

So based on that critereon I would say the factored form is simpler.

Answer 3

I don't know if anyone's ever tried to define "simple" in any objective way, and I believe the nature of simplicity in a broader context than mathematics is one of the great questions on which philosophers have spilled a lot of ink.

I predict that two or three or five or thirty centuries from now mathematicians will say those in our day didn't have a very good understanding either of simplicity or of the mathematical logic of motivation, just as we say Euclid and Euler didn't have our present-day standards of rigor in the deductive logic of mathematics.

Some things seem undisputably simpler than others: $3x^2 - 9x + 14x + 2$ is not as simple as $3x^2+5x+2$. Quite a lot of "simplification" in mathematics is of that kind. If you can express the same thing with less expression, then that's simpler.

Then there are things like "simplest radical form", which requires no radical in denominators, no denominators in radicals, only square-free expressions under radicals of index $2$, etc. Why that is "simpler" than other forms often goes unexplained in introductory courses, but is readily explainable: it's because when you reduce things to canonical form then you can tell whether two things are equal by seeing whether they have the same canonical form.

Only a couple of years ago in about 1992 or '94, Susan Landau made a notable contribution to "simplification" when she discovered an algorithm for de-nesting radicals.

An essay of the late Gian-Carlo Rota, Professor of Applied Mathematics and Philosophy at M.I.T., said that after the prime number theorem was proved in the late 19th century, several hundred papers over about seven decades were devoted to simplifying the proof, and the ultimate simplification of it was a short paper of Norman Levinson in the mid-1960s.

Answer 4

The reason we "simplify" is that two expressions can look much different from each other, but still represent the same number. By simplifying we attempt to put expressions in a form where the relationships between them become obvious.

How do $3\over16$ and $7936\over40704$ compare? No clue. But when simplified, it is easy to see that ${3\over16} < {31\over159}$.

What is simplified? That depends on the context. In grade school, a fraction like $7\over3$ is simplified when it is in mixed form: $2{1\over3}$. This is because that makes it easier to see where it falls on the number line in comparison to other fractions. In higher grades, we give that up in favor of the "top-heavy" form because it is easier to manipulate. What "simplified" really means is "put in whatever form is most convenient for the current task".

Answer 5

If you had asked this over at cs.stackexchange.com, you would have gotten a different sort of answer. Someone would have asserted that your first form, requiring only $4$ multiplications and $2$ additions would be simpler than your second form, requiring $6$ multiplications and $2$ additions. This comes from their focus on the time/space expense of the computation.

Ultimately, "simplification" is context dependent. If I want to find the $0$-level set of your expression, your first form is simpler. If I want to determine the genus a binary quadratic form, I prefer your second form. If I have to evaluate by hand (or pocket calculator) your forms for several values of the variables (and the values are "unsimple" real numbers), I prefer your first form because I don't have to type each variable value two or three times (as I do in your second form) (and whether it's two or three depends on the pocket calculator). If I want to take a derivative, I prefer your second form. I.e., "simplified" = "that thing which creates the least hassle for me in my next step of work".

Answer 6

I do not disagree with your question. However, independent of all Byzantinism, rather wishing philosophy, the word "simplify" in mathematics is directed to common sense. For example if you ask simplify the following expression $$E=\sqrt[3]\frac{x^3-3x+(x^2-1)\sqrt{x^2-4}}{2}+\sqrt[3]\frac{x^3-3x-(x^2-1)\sqrt{x^2-4}}{2}$$ when you solve the question you find $E=x$.

Which of the two forms of E is "less complicated", the first or the second? Everyone would answer that $E = x$