Mathematical Curiosities. (2 Viewers)

[seanieg89]

We can refer to Dave easdowns inquisitive interpretation of exponentials heh

Huh?

 

[asianese]

http://www.maths.usyd.edu.au/u/UG/JM/MATH1903/r/EasdownSecondPart.pdf

 

[seanieg89]

I meant why the word inquisitive? But yeah, the second method is the one I would use to rigorously develop arbitrary complex exponentiation. (Complicated by the fact that the logarithm is a multifunction on C).

 

[asianese]

I dunno the use of MCT was really weird at first since we only saw the definition of exponentiation and logarithms at high school

 

[RealiseNothing]

I think the fact that is pretty cool, which I guess extends to the idea that

edit: I know why this is the case, I just found it cool. Probably should have put it in one of the "cool things about maths" threads.

 

[seanieg89]

I dunno the use of MCT was really weird at first since we only saw the definition of exponentiation and logarithms at high school

Haha the HS definitions are so hand-wavy and non-rigorous. You have to put them out of your mind when you are learning maths for real. The only kind of useful thing about high school maths is it will teach you to do certain computations quickly and develop a partial notion of rigor.

To prove things and define operations on real numbers, we first need to define the real numbers. The fact that this definition is deemed too difficult for high school students is why anything in the high school course than involves real (or complex) numbers cannot be proven properly using high school methods.

The extension from Q to R is a lot more intricate than the extension of Z to Q.

 

[nerdasdasd]

I am so lost ...andddddd this is why I dropped 3U.

 

[asianese]

Dem reals. I wanna take Dave Easdowns algebra and logic (replacing his logic and foundations) next year... He claims to construct the reals from the empty set. Seems good to me. I'll see if I can fit it in... Doubt it though.

 

[anomalousdecay]

I am so lost ...andddddd this is why I dropped 3U.

If you did 4U you would understand more about complex numbers.
Also, this is uni level. The only HSC level was the stuff I posted earlier on this thread, which were also stretching the syllabus.

 

[hit patel]

Keep this thread going. Good thread.

 

[anomalousdecay]

Keep this thread going. Good thread.

This odd property of a series of square numbers:

1, 4, 9, 25, 36, 49, 64, ....
Why does the difference between each set of numbers increase by 2, where the difference in the above numbers are given below:
3, 5, 7, 9, 11, 13, 15, ....

I've tried to manipulate the sigma notation but I have had no luck. I also can't find a way to prove it by induction.

 

[nerdasdasd]

I wonder how integration and differentiation came about :3.

Like how did they discover the product and quotient rule?

how did they know integration finds the area under a curve?

 

[anomalousdecay]

I wonder how integration and differentiation came about :3.

Like how did they discover the product and quotient rule?

how did they know integration finds the area under a curve?

I remember going nuts trying to find this in a textbook or online. I would like to know the reason why as well.

 

[Carrotsticks]

This odd property of a series of square numbers:

1, 4, 9, 25, 36, 49, 64, ....
Why does the difference between each set of numbers increase by 2, where the difference in the above numbers are given below:
3, 5, 7, 9, 11, 13, 15, ....

I've tried to manipulate the sigma notation but I have had no luck. I also can't find a way to prove it by induction.

 

[Carrotsticks]

I wonder how integration and differentiation came about :3.

Like how did they discover the product and quotient rule?

how did they know integration finds the area under a curve?

That is really hard to answer in a post. There are entire BOOKS dedicated to this, so it really is hard to summarise because there are SO many Mathematicians who contributed to the development of Calculus (then you have those who contributed to 'solidifying it' such as Cauchy, Riemann, Weierstrass), but the 'fathers' are considered to be Newton and Leibniz (though they were definitely fighting for the position as founder!)

Newton was one one of the first to raise the idea in his book 'De Analysi' or 'Method of Fluxions' ('fluxions' was the term he had coined for our modern 'differentiation'), and there is a very heavy study of . Integration came about the same time as differentiation when Newton discovered (in modern parlance) that the 'derivative' (fluxion) of the 'integral' (fluent) was the function itself (now known as the 'First Fundamental Theorem of Calculus'), then assumed the converse (which is our modern integration) held true. If you look carefully at the image below, you can see a curve which looks like a sideways parabola. this IS the curve that he was studying so closely.

In more 'mathy' language, he made the claim (again if you look carefully at the cover above, you can see traces of below)



You might like to read this

http://www.nctm.org/uploadedFiles/Articles_and_Journals/Mathematics_Teacher/Humanizing Calculus.pdf

As it definitely will be able to provide more information about the product and quotient rule than I can! It also has some cool info regarding the competition of Newton vs Leibniz.

 

[Carrotsticks]

Here is the English translation if you're interested.

 

[anomalousdecay]

Thanks. You are a legend.

 

[RealiseNothing]

This odd property of a series of square numbers:

1, 4, 9, 25, 36, 49, 64, ....
Why does the difference between each set of numbers increase by 2, where the difference in the above numbers are given below:
3, 5, 7, 9, 11, 13, 15, ....

I've tried to manipulate the sigma notation but I have had no luck. I also can't find a way to prove it by induction.

The way I like to think about it is in terms of actual squares. For example:

A

AB
BB

ABC
BBC
CCC

ABCD
BBCD
CCCD
DDDD

As you can see, the difference between the value of each square is represented by a unique letter. When we go to the next square we add another row and column of the next letter. Since the amount of letters in each row=column (it's a square), then we are adding on an even number value (i.e. to go from C to D we add 2 times 3). So we have something like this:

ABCD
BBCD
CCCD
DDD

However to complete the square we must add one final letter in the bottom right corner, which turns the even number value we added on into an odd number value:

ABCD
BBCD
CCCD
DDDD

Since we are adding on an extra letter to each row/column each time, the difference between the squares always goes up by consecutive odd number values.

 

[anomalousdecay]

The way I like to think about it is in terms of actual squares. For example:

A

AB
BB

ABC
BBC
CCC

ABCD
BBCD
CCCD
DDDD

As you can see, the difference between the value of each square is represented by a unique letter. When we go to the next square we add another row and column of the next letter. Since the amount of letters in each row=column (it's a square), then we are adding on an even number value (i.e. to go from C to D we add 2 times 3). So we have something like this:

ABCD
BBCD
CCCD
DDD

However to complete the square we must add one final letter in the bottom right corner, which turns the even number value we added on into an odd number value:

ABCD
BBCD
CCCD
DDDD

Since we are adding on an extra letter to each row/column each time, the difference between the squares always goes up by consecutive odd number values.

Nice thought. Tried to rep but have to wait :/

 

[ayecee]

Why the area under the curve of 1/x^2 towards infinity is finite, but towards zero it is not. I mean, they both go on for ever, so to me it would make more sense if they were both infinite, or at least both finite, but to be different?

Is there a way to find the length of a curve, and if so, does it have any real world significance? For example the derivative of a function of speed gives instantaneous acceleration, and the area under the curve would be distance travelled. Would the length of the curve on a velocity function mean something useful?

Also, a bit of a strange question my teacher wasn't even able to comprehend: what if pi =/= 3.141..., or e =/= 2.71828 etc. Like, would the world blow up ?