Simplifying expressions -- What exactly does it mean to simplify an expression?

[Fascheue]

What exactly does it mean to simplify an expression. I understand that an expression becomes more simple when you combine like terms but what other conditions are there? Is x(5x+1) more or less simple than 5x^2 +x? Do square roots need to be moved to the numerator?

 

[mfb]

It is not always unambiguous which expression is the easiest, but typically there are not many options.

 

[WWGD]

What exactly does it mean to simplify an expression. I understand that an expression becomes more simple when you combine like terms but what other conditions are there? Is x(5x+1) more or less simple than 5x^2 +x? Do square roots need to be moved to the numerator?

You also often cancel terms in a fraction when you know ( or assume) they are not zero.

 

[phyzguy]

One way to look at it is to see how people have implemented it in a computer program. When programming a computer, one generally needs to be unambiguous. Mathematica has a "Simplify" operation, and here's what the Wolfram website has to say about this:

"Simplifying Algebraic Expressions
There are many situations where you want to write a particular algebraic expression in the simplest possible form. Although it is difficult to know exactly what one means in all cases by the "simplest form", a worthwhile practical procedure is to look at many different forms of an expression, and pick out the one that involves the smallest number of parts."

Mathematica appears to regard your two cases of x(5x+1) and 5x^2 +x as equally simple (they each contain 3 "parts"), so it is a matter of taste.

 

[eys_physics]

As a side note, from the point of view of evaluating the expression numerically (on a computer) the first expression might be better. The reason is that the first one requires 4 operations (2 multiplications and 1 addition) but the second one 5 (3 multiplications and 1 addition). Many compilers are doing this kind of optimizations but it could have importance for script languages, if this sort of expressions are evaluated many times.

 

[WWGD]

As a side note, from the point of view of evaluating the expression numerically (on a computer) the first expression might be better. The reason is that the first one requires 4 operations (2 multiplications and 1 addition) but the second one 5 (3 multiplications and 1 addition). Many compilers are doing this kind of optimizations but it could have importance for script languages, if this sort of expressions are evaluated many times.

Do these compiler contain optimizers in the way SQL Server does? Optimizer in SQL Server looks at query format, usage statistics, indexes and decides on best way of running query. Would be nice to have similar for other software.

 

[eys_physics]

The compilers I referred were the ones for programming languages such as e.g. C/C++, Fortran, etc. SQL Server on the other hand more dedicated mainly for selecting entries in Tables (in a database). The ones I referred to are for more general purpose. I don't think they contain the sort of the optimizations you mentioned and if even is possible. But, I'm not an expert on compilers.

 

[FactChecker]

It is not always unambiguous which expression is the easiest, but typically there are not many options.

It's like art -- you know it when you see it.