Weird Question about Majoring in Math

[PhysicsHelp12]

Im in 2nd year now...and I've been worrying about really trivial things

but its been driving me crazy ...

and I've been getting really good marks with no trouble

but I've been thinking about How do I know what to do

Like When I substitute into an equation ...

SOmetimes I use brackets...sometimes I don't use brackets at all

and just see the answer...

Like when I am shifting the index of a powe series ---I don't bracket

(k+1)-1
if I am subbing in k+1 for n-1 ...I just do it ...

or if this were like

1-x replace x with (t-1)^3 ...I wouldn't use brackets...

Id just go 1-(t-1)^3 ...

but the people on Yahoo Answers said to always use brackets


but now I'm worried that if I continue to just not think...

that one day itll get too complicated and I won't know what to do

DO you all use brackets in your head...

or do you skip steps when youre doing trivial things


I just find it really discomforting that I can do the questions --but its

become subconscious so I really don't know how I do it ...it just


happens in a flash..

throwing brackets on things happens ...and I don't know why ...

and then when I reason it out ---It slows me down considerably

so either I go back to not thinking about it --and just doing it

but I don't know if this is normal ...

Im worried that Ill forget it and fail ...



Anyone please help ...

 

[CompuChip]

Well, it all depends (ok, that's not the helpful answer, I know, but I think it's the truth). You don't write brackets to sooth your conscience. You write brackets to make it clear what you mean. This can either be to avoid ambiguity; for example, if you mean: 3 - 4 + 1 is not the same as 3 - (4 + 1). Or you can use them in a derivation, for example (as you said) in an expression like
\sum_{k = a}^b k - 1 = \sum_{p = a - 1}^{b - 1} (p + 1) - 1
it makes it more obvious what it is you did (in this case, replacing k by p + 1). However, if the derivation went on like
\cdots = \sum_{p = a - 1}^{b - 1} p + (1 - 1) = \sum_{p = a - 1}^{b - 1} p = F(x)
I would skip the former steps and just write down: \cdots = F(x) immediately.

Finally, it also depends on your audience. When writing for high school students, you will want to include more steps as not to leave them puzzled; when writing for mathematicians who have graduated 10 years ago you will want to be more brief so that they will not get utterly bored. (Of course, as long as you are a student, your teacher will be an exception - especially they don't usually appreciate stuff like "the calculation is lengthy, the reader can easily check that it gives ...").

 

[Pere Callahan]

\sum_{k = a}^b k - 1

Well, this is extremely ambigious...

 

[CompuChip]

I would usually take that to mean
\sum (k - 1)
however clearly, "usually" is not very good either (in practice, I wonder how much confusion arises because of such notational sloppiness).

To be completely clear, I agree with you one would have to write either that or
\left( \sum k \right) - 1
though.

Putting brackets is also not always ideal though
I once read an expression like
H(t_f - t_i)
and it took me some conscious thought to realize that H was just a constant multiplied by some difference, rather than a function of it (especially since the text went on to explain that we could introduce some function f(t_i, t_f) which actually only depended on that difference as well, and hence could be written as f(t_f - t_i)).

 

[HallsofIvy]

Im in 2nd year now...and I've been worrying about really trivial things

but its been driving me crazy ...

and I've been getting really good marks with no trouble

but I've been thinking about How do I know what to do

Like When I substitute into an equation ...

SOmetimes I use brackets...sometimes I don't use brackets at all

and just see the answer...

Like when I am shifting the index of a powe series ---I don't bracket

(k+1)-1
if I am subbing in k+1 for n-1 ...I just do it ...

or if this were like

1-x replace x with (t-1)^3 ...I wouldn't use brackets...

Id just go 1-(t-1)^3 ...

Then what do you mean by "brackets"? You have used parentheses and that's all the "brackets" you need.

but the people on Yahoo Answers said to always use brackets


but now I'm worried that if I continue to just not think...

that one day itll get too complicated and I won't know what to do

DO you all use brackets in your head...

or do you skip steps when youre doing trivial things


I just find it really discomforting that I can do the questions --but its

become subconscious so I really don't know how I do it ...it just


happens in a flash..

throwing brackets on things happens ...and I don't know why ...

and then when I reason it out ---It slows me down considerably

so either I go back to not thinking about it --and just doing it

but I don't know if this is normal ...

Im worried that Ill forget it and fail ...



Anyone please help ...