Math Is Fun Forum

Well, the "space" concept comes from the desire of mathematicians to do addition, substraction, multiplication, division ... freely. "Freely" means if they do such an algebra of elements in the space, they get the result from this space too. That's why they invented negative numbers and radicals and imaginary numbers-they don't wanna tell you I am stuck in telling you the answer of 2-4 or that of sqrt(3) or x^2=-2.

So a space in algebra term is that you play certain manuplications in it and you are sure the results are also in it. Integers form a good space for addition and substraction, well natuaral numbers don't. Vectors with the same degree can also form a good space for addition and substraction, and for scale product k v=  (kv1, kv2, kv3), They call it a linear space because combinately you are sure the result of a u + b v is in the vector space too.  Trivally, {(0,0,0)} is also a linear space for it satisfy the definition of a linear space.

Now they want more than a linear space. They want each two vectors from an already linear space to have a fixed number as their so called inner product, additionally a positive number if the two are the same, 0 when one is 0 vector. It does not matter how the inner product is defined, as long as the defined inner product can be applied to any two and satisfy those conditions. They call this linear space also an inner product space.

Q: Is there any inner product space not linear space?
A: No, they are defined with linear property. In other words, mathematicians don't think it is useful to define a space satisfying inner product property but not  linearity.

Post 5 is creative, how nice it would be if a textbook include this piece of proof!

I think to calculate the area will be tougher.

Polar Coordinates may be better

I have used firefox once, but it didn't display a registration tab on an important site for me, so I quit using it.

Generally I use Maxthon, which is just above IE- I mean quite similar but approporately improved. Now IE7 has adopted multiple windows, which I guess is copying Maxthon.

Tools-Internet Options-(General)-Accessibility-Select "Ignore Font Size"

I remember the negative case was proven seperately and then the conclusion of absolute value comes.

I have no idea. My explaination is only to the constant integration region (The fixed rectangle).

It doesn't matter whether you finish a map puzzle row by row or column by column (Imagine you can do that).

When s is a positive integer and cannot be expressed as a square of an integer, e.g. 2, 3, 5, 6, 7, 8, 10...,

no integer a and b can satisfy:
(a/b)[sup]2[/sup]=s

Proof:
if so,
a[sup]2[/sup]=sb[sup]2[/sup]
So a[sup]2[/sup]|s

Because s cannot be expressed as a square of a integer, such as s=1=1², s=4=2², s=9=3²..., the only way to guarantee a²|s is a|s, then
a²|s²

a²=sb²

so b²|s
so b|s and b²|s²

To Ricky's arguement.

Alright. I got it. It simply cannot compile it when it is not formally saved, where "formally saved" means saved with a name other than "untitled".

Texmaker might be one good example of splitting a task into a subtask too small.

It looks pretty, much more pretty than Microsoft Word.
It looks nice, every maths symbol ready to use.
But where the hell can I get a bunch of Latex maths symbols out instead of a string of sheer commands like
" \int e^{x}\dx=e^{x}"
????????!!!!!!!!