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

In summary, simplifying an expression involves finding the simplest possible form, which may not always be unambiguous. This can be achieved by combining like terms and cancelling terms in a fraction. However, there may be different ways to simplify an expression and it often depends on individual preference or the context in which it is being used. Some computer programs like Mathematica have a "Simplify" operation that helps determine the simplest form of an expression. There may also be optimizations in compilers for programming languages that can improve the efficiency of evaluating expressions. However, it is still a subjective and complex process, similar to appreciating art.
  • #1
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?
 
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  • #2
It is not always unambiguous which expression is the easiest, but typically there are not many options.
 
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  • #3
Fascheue said:
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.
 
  • #4
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.
 
  • #5
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.
 
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  • #6
eys_physics said:
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.
 
  • #7
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.
 
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  • #8
mfb said:
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.
 

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

What does it mean to simplify an expression?

Simplifying an expression means to manipulate it in such a way that it is written in its most compact and simplest form. This typically involves combining like terms, removing parentheses, and reducing fractions to their lowest terms.

Why is it important to simplify expressions?

Simplifying expressions can make them easier to understand and work with. It also allows us to find equivalent expressions that may be more useful in solving a problem or simplifying a larger equation.

What are the steps for simplifying an expression?

The steps for simplifying an expression may vary depending on the specific expression, but generally involve removing parentheses, combining like terms, and reducing fractions. It is also important to follow the order of operations, simplifying expressions within parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Can all expressions be simplified?

No, not all expressions can be simplified. Some expressions may already be in their simplest form, while others may not have any like terms to combine or any other simplification rules to apply.

Are there any strategies for simplifying complex expressions?

Yes, there are several strategies for simplifying complex expressions. One is to break the expression down into smaller parts and simplify each part individually before combining them back together. Another is to look for patterns or common factors that can be factored out. Practice and familiarity with algebraic rules and techniques can also help in simplifying complex expressions.

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