What does it mean to be rigorous?

[Couperin]

It's a word I hear a lot and I've looked it up but I don't understand. What does it mean to be rigorous in one's mathematics?

 

[robert Ihnot]

I suppose there are various meanings attached to this. But, I think, the one influencing education is the idea that students do better in college and on jobs if they spend more time on Algebra and Geometry rather than Arithmetic and Business Math.

Now the books on math my grandfather had used on the farm, generally stressed Arithmetic, students learned the tables to 12 x 12, (which is a gross) and learned how to multiply 2x2x2...in their heads. The better students were faster at doing these things.

 

[ice109]

it means that all of your conclusions are deduced using proper mathematical rules.

for example

$$5x^2=x$$
$$x=1/5$$ only if $$x \neq 0$$

because in deriving that conclusion you divided by x and x can't be zero because you can't divide by zero.

 

[eyehategod]

it means hard. so hard that you want to blow your brains out b/c you don't understand the problem.

 

[pivoxa15]

It means it's understandable to a mathematician.

However life is not perfect. There are different levels of rigour which have changed through history and it evolving.

 

[symbolipoint]

eyehategod in message #4 was close. Rigorous is difficult, stressing critical thinking in learning and applying fundamental notions. (Or substitute properties, knowledge, concepts, and skills, instead of "notions".)

 

[morphism]

eyehategod in message #4 was close. Rigorous is difficult, stressing critical thinking in learning and applying fundamental notions. (Or substitute properties, knowledge, concepts, and skills, instead of "notions".)

Rigorous doesn't mean difficult...

If anyone's response was close, it's ice109; in fact he pretty much nailed it.

 

[ice109]

all you people need a dictionary. rigorous doesn't mean hard. arduous means hard

 

[symbolipoint]

all you people need a dictionary. rigorous doesn't mean hard. arduous means hard

The included characterization of "hard" or "difficult" comes from finding those words associated with "rigorous courses" in website articles about college preparatory and other advanced high school course; among other sources.

 

[Gib Z]

it means that all of your conclusions are deduced using proper mathematical rules.

for example

$$5x^2=x$$
$$x=1/5$$ only if $$x \neq 0$$

because in deriving that conclusion you divided by x and x can't be zero because you can't divide by zero.

We all obviously get your point, but I thought I would point it out that it is quite an obvious statement that $$x= b$$ only if $$x \neq a , a \neq b$$ :P

Perhaps a better example would have been $$\frac{x^2}{x} = x$$ only if x is not equal to 0.

 

[Galileo]

I'd say being rigorous in mathematics means only accepting mathematical statements, not because it is 'intuitively obvious' but because you have proven it using sheer logic. Constantly check all assumptions you make and if/how they apply.
You should be able to provide the reasoning from everything you assume to be true back all the way to the axioms.

 

[Math Jeans]

My teacher's definition of being rigorous with a problem is basically using a complex proof to get an answer through a problem.

Suppose there is a simple way to define a certain limit (other then solving it), but your teacher wants you to be rigorous, so you need to do a completely correct epsilon-delta proof.

Generally, I interpret rigorous (in math) to be the act of leaving NO holes in the proof of a mathematical problem.

P.S. I hated that course.

 

[disregardthat]

Well, it's important to be precice in both your explanations and conclusion. Any proof that may be claimed unrigid ( is that a word? ) may have faults, that ultimately may mean that it doesn't hold.

 

[ZioX]

The misconceptions in this thread are disheartening.

 

[DeadWolfe]

Rigorous means, precise, no steps left out, no assumptions left unstated, not definitions vague, etc... While presenting everything in logicalese is sufficient to be rigorous, it is usually overkill.

 

[matt grime]

Rigorous means, precise, no steps left out, no assumptions left unstated

No it doesn't: one presumes a certain level of sophistication in an audience, different at different times. As long as you are correct, then leaving out steps is fine if the intended reader can be expected to obtain them.

not definitions vague, etc...

Now that is, after re-ordering words, absolutely true.