scales: sharps and flats

so i just found a new teacher and i like this kid more than the last guy. he's younger so i can relate to him, he's younger than me so i'm not as embarrased when i suck around him, and he's definitely good (a guitar major) and has a more organized lesson plan.

anyway he has me starting to get into the circle of 5ths and going though the scales in the order C, G, D, A, E etc. but he only gave me the "sharps", ...E, B, F#, C#. so i went and found the whole order on my own (admittedly not using the tool on this site)

F#, C#, Ab, Eb, Bb, F, then back to C.

question is, are F# and C# considered sharp? or flat? when writing out the scales it looks like they could go either way.

secondly, what is the purpose between "sharp" and "flat" scales? from my memory of music years ago i always thought that it made the scale more "convenient" i.e. instead of writing G# A# C C# you could write Ab Bb C Db, keeping the ABCD order instead of skipping B and repeating C. or is there another reason for sharps/flats?

how would YOU write out the C#/Db scale? F#/Gb?

just pondering...

A scale is defined as follows:

7 relative notes all of different tonality following a certain pattern.

In theory, we will always have a different letter for each tonality. Therefore, writing a scale with two "C's" in it is wrong.

i.e. G#, A#, C, C#, .... -> WRONG

Ab, Bb, C, Db, .... -> RIGHT

so in other words, it would not be C# but rather Db?

C# D# F F# G# A# C wrong

Db Eb F Gb Ab Bb C correct

same with F#?

R..2..3..4.5..6..7

F# G# A# B C# D# F ?? E#?

Gb Ab Bb B Db Eb F ?? Cb instead of B?

i know that B# and E# dont' exist as unique tones, but sometimes they can be used as Cb=B, B#=C, Fb=E and E#=F

In science, the notes are the same because they have the exact same frequency.
In music, they are different because notes are typically played with other notes and the rule is you don't mix sharps and flats, and cannot have repeating notes, hence,

F# G# A# B  C# D# E# F#...

Gb Ab Bb Cb Db Eb F  Gb...

i just remember from back in the day seeing B# or Fb written. the intention is to have you play a C or an E. i mean it was extremely rare but i have seen it before with mine own two eyes!!!

I think it occurs in harmonic minor keys, when you have to sharpen the E or B. In this case, you write E# or B#, because it has to stay as an "E" or "B". However, when you play it, it's a F and C.

There are actually some organs where there is one key for the Eb and one for the D# etc..
http://www.h-pi.com/eop-keyboards.html

This is because they are used in different keys and play different roles in the melodic content off the keys AND have a slightly different frequency.
However, on the guitar it's the same tone unless you're using one of these:
http://www.truetemperament.com/site/index.php?go=9&sgo=1