idea from rob
#Writing proofs in Maths
crystal herald
crystal herald
here I have to prove that for every positive a, there's a+1/a>=2, and I have to answer when there's equality
I have a hint that I should firstly prove a^2 + 1 >= 2a
so to prove that a^2 + 1 >= 2a I'd have to seperate them into a^2 + 1 = 2a and a^2 + 1 > a , I guess?
from my understanding a^2 + 1 = 2a when a=0
and the other is when a is above 0
white compass
So you understand how you get from the given statement to a^2 + 1 >= 2a, correct?
crystal herald
So you understand how you get from the given statement to a^2 + 1 >= 2a, correct...
how I get from a+1/a>=2 to a^2 + 1 = 2a?
white compass
Yes
crystal herald
I don't really get it unfortunately
white compass
But since you're applying the same operation (*a) on both sides, proving that a^2 + 1 = 2a is also proving that a + 1/a = 2
white compass
and it'd be <=> right
Yeah, it's the same equation
crystal herald
so by doing *a I proved that they're both equal to eachother
white compass
No that's just reshaping the equation
doing the *a doesn't prove anything yet
But it brings the equation into a nicer form that's easier to work with
crystal herald
Ok
crystal herald
since it's both >= then I think it is true for all real numbers
actually it's not true for negative numbers is it?
white compass
Oh sorry I mixed things up, didn't see the >=
In the task it should tell you what numbers you are supposed to prove the equation for
crystal herald
for every positive a I need to prove that a+1/a>=2
white compass
So it's only for positive a
crystal herald
a+1>=2a would be true for all positive a
white compass
So we have a^2 + 1 >= 2a
We can rearrange that to be
a^2 -2a + 1 >= 0
Do you remember any funny rearrangements for the left side here?
crystal herald
1 sec
crystal herald
So we have a^2 + 1 >= 2a
can you explain how we got a^2?
white compass
a+1/a>=2
(a + 1/a) * a >= 2 * a
a^2 + 1 >= 2a
shouldn't this just get rid of /a?
crystal herald
that's okay, I make worse mistakes all the time lol
white compass
at least you caught my mistake, shows youre thinking about it
white compass
You don't
idk if I made a mistake again somewhere lmao
from a+1/a>=2
to a^2+1 >= 2a
by multiplying both sides with a
crystal herald
is it just by logic that a^2 + 1 >= 2a is the same as a + 1 >= 2a because a^2 >= a?
crystal herald
a+1/a>=2
is the same as
a^2 + 1 >= 2a
sorry I don't really get where the ^2 comes from
white compass
```
a+1/a>=2
(a + 1/a) * a >= 2 * a
a^2 + 1 >= 2a
```
this
we multiply both sides by a
marsh nestBOT
crystal herald
ohhhhhhhhhhh
white compass
maybe im blind or something
crystal herald
no you're right
white compass
but i think thats what you sent
crystal herald
I got confused, I read a + 1/a as (a+1)/a
ok I get you now
so at this point by multiplying by a, I proved that a^2 + 1 >= 2a is a thing right?
white compass
It doesn't prove anything, we're just rearranging
Now with this modified equation we can do the actual proof
crystal herald
Now with this modified equation we can do the actual proof
I can see that a^2-2a+1 >= 0 is possible am I in the right direction?
white compass
Yes, we can rearrange it like this
Now we can maybe rearrange the left side some more
crystal herald
Now we can maybe rearrange the left side some more
this looks like what I think is called a quadratic equation in english
would it make sense to use the quadratic formula?
I guess it wouldn't? As I'm not trying to find a
crystal herald
ohhh
I always get caught off guard by the binomial formulas
so I'd get (a-1)^2 >= 0
my first guess would be to use square root on both sides to get a-1 >= 0
crystal herald
Or you can use the square property again
I don't really get what square property means
white compass
Anything squared >= 0
crystal herald
Anything squared >= 0
but that's just (a-1)^2 >= 0
crystal herald
so I should somehow use the information that a^2 >= 0 I'm guessing
so if I reverse the order of a^2 >= 0 in the same way I'd get a+1/a>=2 wouldn't I
crystal herald
so if I reverse the order of a^2 >= 0 in the same way I'd get a+1/a>=2 wouldn't ...
am I right here?
white compass
idk what you mean by reversing the order
crystal herald
idk what you mean by reversing the order
as in doing the opposite of the steps I did to get to this point
white compass
You don't need to
You only rearranged it up to here so it's still the same equation
If you prove (a-1)^2 >= 0 then you have also shown that the original equation holds
crystal herald
If you prove (a-1)^2 >= 0 then you have also shown that the original equation ho...
wouldn't I have to expand it back to a^2-2a+1 >= 0 in order to prove that?
white compass
No, because you only did rearrangements until then
crystal herald
No, because you only did rearrangements until then
(a-1)^2 > 0 would be true for every positive number of a above 1, and (a-1)^2 = 0 is true if a = 1
as a^2 always leads to a positive number if a is a positive number
white compass
✅
crystal herald
✅
is it formal enough? Or do I have to write it in a similar fashion:
Given:
To prove:
Proof:
Case 1:
...```What does the full proof look like?
crystal herald
What does the full proof look like?
in the schoolbook answer?
white compass
Yours
white compass
a+1/a>=2
(a + 1/a) * a >= 2 * a
a^2 + 1 >= 2a
a^2 -2a + 1 >= 0
(a - 1)^2 >= 0
yea I wrote it in a similar fashion
crystal herald
I have to go to sleep now, your help was incredible and I honestly really appreciate that you took time out of your day to help me @white compass
I'm really thankful for that
white compass
Good night 🥺
heady crater
yea I wrote it in a similar fashion
I would do it the other way
white compass
Now once we understand how we can prove it in text we can thincc about the formal notation
heady crater
I would do it the other way
Should probably explain what I mean by this. If what you want to prove is that a + 1/a >= 2 then you cannot start your proof by assuming that it's true and getting to another true statement
Instead you start with what you know is true, then eventually using a chain of logical implications, you get to showing that what you want to prove must also be true at the end
crystal herald
Instead you start with what you know is true, then eventually using a chain of l...
so I was right about doing the opposite of the steps I did to get from (a - 1)^2 >= 0 to a+1/a>=2?
heady crater
Yes, it's more correct
Of course you have to state that (a - 1)^2 >= 0 is true but that's easy
crystal herald
is it formal enough? Or do I have to write it in a similar fashion:
```
Given:
...
to state that it is true I have to use this format? @heady crater
heady crater
What
heady crater
You just say it in the proof?
and justify it
Ez though cause any real number x, will have x^2 be positive
Not sure what's confusing you
crystal herald
Ez though cause any real number x, will have x^2 be positive
from what I understand that justifies (a - 1)^2 > 0 and then I'd have to also justify (a - 1)^2 = 0?
sorry if I sound a bit lost, proofs is a foreign thing for me
also just started learning about inequalities
heady crater
Ez though cause any real number x, will have x^2 be positive
Sorry, my mistake
The square of any real number is non-negative