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I was wondering, when we consider several functions at once in the same graph, is it ok if this whole is not a function itself ??? Do we care about whether this whole is function or not ?
Ok, my problem to why I was not understanding is because I thought that the limit inequality needed to hold ! So if the limit of f(x)= L and g(x)=M
I could have L>M
right ?
Hi, I'm having trouble understanding the following fact about limits :
If f(x)<=g(x) for all x on (a,b) (except possibly at c) and a<c<b then,
lim f(x) <= lim g(x)
x -> c x->c
Here's how I interpret the definition : We have two functions f(x) and g(x), and the inequality f(x)<=g(x) hold true for all values that are not c. (That our interval (a,b)) If we were to evaluate the functions at c (considering that we can do it for our two functions.) then the inequality wouldn't hold anymore. (For example, f(x) would be superiro to g(x))
Please tell me if I have any errors.
THank you!
If you want to read more, go here : http://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx
Ahok, thank you for your answer
Hey, I have another question. I was reading proposition 15 http://aleph0.clarku.edu/~djoyce/java/e … opI15.html and it also uses the same method to prove that 2 * 90 = 2 * 90
Now, I was wondering, shouldn't we use the common notion 2 which says : 2. If equals are added to equals, then the wholes are equal
Because we are dealing with a sum. So, we would have 90=90 and 90=90 by postulate 4 for both cases. After, we would use CN2 to prove that 2*90=2*90
or 90+(90)=90+(90)
What do you think about it ?
Oh you mean on stack exchange, right ???
Joel's answer ?What answer ? Who's Joel?
For a certain reason, I find it weird. So, we can't see that 0.249..=0.25 with the example of the line subdivision because each subdivision point represents a finite decimal. Or am I wrong ? For example, If I have 0.24999 and 0.25000 on my line, I couldn't say that they're equal.... I would need to imagine an infinite process of subdivision of a line to establish the true equality ?
Also, must the decimal expansion of 0.249 be an infinite one so that we can say that it is equal to 0.25 ??? Couldn't we have 0.249=0.25
Well, I gave a resume of the first part, if you want, I can give you the whole.
I was reading something which I found really special. It goes like this : Imagine we have a line with unity division (0,1,2,etc.) Now, we have a point on this line. The point can be on a point of division or it can be contained between two points of division. If it isn't on a point of division, then we can continue cutting the line into 10 equal pieces. (The first piece being 0, the second 1, the third 2,...) So, if my point was between 1 and 2, I'll have 1.0,1.1,etc. We continue this cutting until our point arrives at a point of division.If this arrives, we have two choices : We can pick the interval on the left or the interval on the right. And we can continue this way at infinity.
We also have the following formulae to express our numbers : g+a/10+a2/100+a3/1000+... g being an integer and a being the number of the piece which contains our point.
Now, my question concerns this :
http://i.imgur.com/RlKuUG9.jpg
How can it be possible that we can have 1/4=0.2499... and 1/4=0.25000... Also, for this idea to be valid, must it necessarily be an infinite decimal expansion or I could have the following 1/4=0.249 and 1/4=0.250 ?
Thank you !
Hi,
Here's a definition : "For every two points in space, there is a straight line passing through them, and such a line is unique."
Now, there's just a particular point which I'm not sure of. When it says : "Such a line is unique." Does it mean that the position of the line in space makes it unique ? Or in other words, I couldn't create a distinct line that is positionned at exactly the same space as the other one. The two lines would be the same. Anyway, thank you for helping my lantern !
Ok thank you !
Hi,
I have the following to multiply :
secondroot(d)*thirdroot(ab)
I have the following answer : thirdroot(a^2*b^2*d^3)
I think that the answer is an error
Here's my answer : thirdroot(abd*d^1/2)
Thank you
Yes, it does seem weird to be stating the obvious. But Euclid was the first (I think) to insist that everything must be stated as an axiom (ie. assumed to be true) or proved from those axioms. He defines his terms carefully, states his axioms (as postulates; another word for the same thing) or as common notions (which I think means axioms that can also be used in other disciplines).
All that follows are careful proofs using those starting points. Even things that seem obvious to us; such as if two lines meet at a point and the sum of their angles with a third point is 180 => the two lines are as one line, even such things must be proved. It is the starting point for all of modern maths; which is one reason for studying the Elements.
It can teach us something else too. Euclid's geometry doesn't actually exist in the 'real' world. There is no such thing as a point with no dimensions or a line with no thickness. If you draw a triangle and measure it's angles and total them, you won't get 180. You might get 179 - 181 if you're very careful. But you cannot really achieve the absolute accuracy of Euclidean geometry. But we are still able to build useful models of reality using his ideas and successfully land a man on the Moon. All maths is like that: we can make a model of the real world and provided we understand the limitations of the model, make use of it in practical situations. Use it badly and you can get silly results. We all know 1 + 1 = 2 but if you add one pile of sand to another pile of sand, how many piles of sand have you got ? Wrong model; that's all.
Bob
Interesting read, thank you!
Then, by the same logic, you would not even need Postulate 4.
Mmmh. Good point. I think that the reason why I wasn't understanding the use of postulate 4 was because, even if I tried to search, I assumed already that the two right angles were equal to the two other right angles. In other words, I didn't see them as two separate perpendiculars but only as one, which caused me trouble. I'm not sure if you're understanding what I'm saying...
That's how I'm understanding it.
Well, I find it weird that we need to prove that two rights=two rights...I mean, isn't it obvious that we're talking of the same thing ???
Wait, to be sure I'm understanding,
We proved that the sum of two angles = 2*90 ( we don't know if these angles are right angles are not, we only know that they sum to two rights)
We assumed that the sum of two angles= 2*90
Are you saying we must prove that 2*90=2*90 (We must show that 2*90 is the same as 2*90)
So that we can after say : sum of two angles=sum of two angles (Regardless of what each angle is worth.)
So, basically, we're considering 2*90 and 2*90 like two separate ideas that needs to be connected together, right ?
We'll, I don't know how good it is but it has been translated by Heath, so I guess it's correct
hi Al-Allo
Couldn't get that page.
In the version I have that is how this is proved.
Therefore, since the straight-line AB stands on the straight-line CBE, the (sum of the) angles ABC and ABE is thus equal to two right-angles [Prop. 1.13]. But
(the sum of) ABC and ABD is also equal to two right- angles. Thus, (the sum of angles) CBA and ABE is equal to (the sum of angles) CBA and ABD [C.N. 1]. Let (angle) CBA have been subtracted from both. Thus, the remainder ABE is equal to the remainder ABD [C.N. 3], the lesser to the greater. The very thing is impossible.
Thus, BE is not straight-on with respect to CB.from:
http://farside.ph.utexas.edu/euclid/Elements.pdf
Just tried that link and it failed. I'll try again tomorrow.
Bob
Weird, I just tried the link that I had posted and everything seems to work out fine. Anyway, the text you refer to doesn't seem to use nor common notion 1 or postulate 4.
Here's the text :
If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.
With any straight line AB, and at the point B on it, let the two straight lines BC and BD not lying on the same side make the sum of the adjacent angles ABC and ABD equal to two right angles.
I say that BD is in a straight line with CB.
If BD is not in a straight line with BC, then produce BE in a straight line with CB. (I.Post.2)
Since the straight line AB stands on the straight line CBE, therefore the sum of the angles ABC and ABE equals two right angles(I.13). But the sum of the angles ABC and ABD also equals two right angles, therefore the sum of the angles CBA and ABE equals the sum of the angles CBA and ABD.(I.Post.4 and C.N.1)
C.N.3
Subtract the angle CBA from each. Then the remaining angle ABE equals the remaining angle ABD, the less equals the greater, which is impossible.(C.N.3) Therefore BE is not in a straight line with CB.
Similarly we can prove that neither is any other straight line except BD. Therefore CB is in a straight line with BD.
Therefore if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.
Q.E.D.
Hi,
I was reading proposition 14
http://aleph0.clarku.edu/~djoyce/java/e … opI14.html
of Euclid's elements and there is only one thing which I find weird:
Why do we need postulate 4 to conclude that “the sum of the angles ∠CBA and ∠ABE equals the sum of the angles ∠CBA and ∠ABD.”
Why can't we just use common notion 1? It seems useless to me to use the postulate…
Thank you!
Well, of course I'm expecting rigour and exercices. The two of them
Hi,
Could someone give me a reference to a rigorous introduction (So I understand what's going on, that doesn't just give out formulas without explanation,etc.)to analytical geometry text ?? Keep in mind that I only have knowledge of high school math!(equations of second degree,etc.)
Thank you again !
Ok, yeah, I think that's it :
We have shown that such factorization exists which can make the equation equal :
1*1/1*1-0*x/1*1=+1/1*1*1*1
which results in 1-0=1
The other one is :
0*x/1*1-1*1/1*1=-1/1*1*1*1
0-1=-1
I don't know if there exists other ways of 1 or -1, but anyway, this is out of the context of the inital problem.