You really should post new questions in a new thread.
here, there is not much to do after the hint the book gave.
For every positive irrational number b, there exists an irrational number a with 0<a<b
Proof:
Let b be any positive irrational number. Consider a = b/2. Note that a<b, and since a is obtained by dividing an irrational number by a rational number, a is irrational. And since a is half of a positive number, we have a>0. Thus we have 0<a<b, where a and b are irrational, as desired
QED
That does not sound familar. We have a foreign lady for our teacher so understanding her is difficult at times. I know she feels that she can't express some thoughts and ideas as well as she needs to. So, she may have tried to tell/teach us that but it sounds jiberish to me.
Let me show you an example of what I mean. remember this proof?
Let's try to do it by directly manipulating the mod equations.
Let n be an integer. Prove that if n is not congruent to 0 (mod 3), then n^2 is congruent to 1 (mod 3).
We employ a direct proof. We define "n = x (mod 3)" to be n is congruent to x (mod 3). We define "n != x (mod 3)" to be n is not congruent to x (mod 3).
Now assume n != 0 (mod 3). Then n = 1 (mod 3) or n = 2 (mod 3).
Case 1: n = 1 (mod 3)
if n = 1 (mod 3) then by squaring both sides we obtain n^2 = 1^2 (mod 3), that is n^2 = 1 (mod 3)
Case 2: n = 2 (mod 3)
if n = 2 (mod 3) then by squaring both sides we obtain n^2 = 2^2 (mod 3), that is n^2 = 4 (mod 3). Since in a modulus equation, we can add or subtract the modulus as much as we want, this means that n^2 = 4 - 3 (mod 3), that is n^2 = 1 (mod 3).
So we see in both possible cases, n^2 = 1 (mod 3), as desired.
QED
well, remember modulus equations deal with a lot with remainders.
remember, when we say n = 2 (mod 3) for instance we mean, 3|n-2 what that actually means is that if we divide n by 3 the remainder is 2. (do you see why?)
so for mod 3 there are three possibilities:
n = 0 (mod 3)
n = 1 (mod 3)
n = 2 (mod 3)
since when dividing by 3, there can only be three possible remainders, either you have a remainder of 0 (in this case, n is divisible by 3) or you have a remainder of 1 or 2 (in these cases, n is not divisible by 3)
so if i assume n is not congruent to 0 mod 3, that is, it is not divisible by 3, then n has to be one of the other 2
got it?
Whey you said, We define "n = x (mod 3)" to be n is congruent to x (mod 3). We define "n != x (mod 3)" to be n is not congruent to x (mod 3). Did you mean n! is not congruent to x? If so that makes sense other wise I don't understand that part. Why did you pick n! = x (mod 3)?
I understand the remainder concept that mod 3 has 3 possibilities.
no what i said there was not a part of the proof, i did that for you. when i said n != x (mod 3), i did not mean n!, notice that the "!" was attached to the equal sign, not the n. i was just making you aware of my notation. is used "!=" to mean "not congruent" since i don't have the symbol for not congruent (which is the equivalent sign with a strike through it". and i used "=" for congruent since i don't have the symbol for congruent (which is the equivalent sign).
well, as for the rigorous mathematical proof that shows we can do this in all cases, i don't have (or know) that. so let's just try to use common sense to figure this out--since math is a lot about common sense well, not all of it, but a lot of it
okay,
let's use mod 3 as an example again.
let's say you got n = 4 (mod 3). i claim that is the same as saying n = 4 - 3 (mod 3) which is n = 1 (mod 3)
so now let's think about it. n = 4 (mod 3) means that when we divide n by 3 we get a remainder of 4. but wait, if there is a remeainder of 4, that means we could have gotten another 3 out of it, and be left with a remainder of 1. ah, well if that's true, then n = 1 (mod 3) since we can "take out" a modulus out of 4, it is the same as subtracting the modulus.
so basically, once we go in excess to the point where we can take the modulus out of the remainder, it means we might as well have divided into the remainder more, leaving a smaller remainder. so that's the common sense answer