Math Principles and Misconceptions

[Sandmaster]

That won't help, but I think there is an equation for how large an object would appear using the altitude (distance) and the leg (size, in a way)

[General Veers]

That's what I was doing.

w=base= double the short leg

a=altitude=distance from back of eye

An isosceles triangle can be split into two back-to-back right triangles.

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I GOT IT!

Assuming that t is the angle from the altitude to h (the hypotenuse) of the right triangle that makes up an isosceles triangle,

sint=w/h
cost=a/h
tant=w/a

There it is! That is what I need!

Now my limit will be in terms of t (I will substitute w/a for tant).

lim_{t ► infinity}tant

The above expression, the limit of the tangent of t as t approaches infinity, will provide the value that w'/a' will approach.

I might need a graphing calculator for this...hold on.

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Darn.

lim_{t ► infinity}tant is undefined.

If you look at a graph of f(x)=tanx, the graph shows oscillating values. That graph does not approach any single value as it goes on to infinity.

The world will never know what an infinitely large mass an infinite distance away will look like or seem to look like...

[Sandmaster]

this could take a while...

[General Veers]

The limit is undefined. See above post.

[Sandmaster]

So i was right!

[General Veers]

The question is, what exactly does that mean if the apparent size of an object is undefined?

[Sandmaster]

It means that infinity has the attributes of 0 (try dividing infinity by infinity)

[The Dark Master]

So infinitiy's partner is 0? That just came as final proof as what the image of perfection is.

[General Veers]

The limit of x as x approaches infinity does not exist (because it approaches infinity, which is not any fixed value).

The limit of x^-1 as x approaches infinity is 0.

Yes, they are pretty much a good pair, 0 and the concept of infinity...

[General Veers]

Edit: Post removed. Thank you.

DO NOT SPAM OR POST USELESS COMMENTS

Please ensure that you actually ask questions concerning math principles, postulates, theorems, misconceptions, etc., or else that you answer such questions.

You may also bring up new principles or neat observations, such as what vaconcovat did with the palindrome numbers, or request discoveries such as what Sandmaster did with the "infinite size - infinite distance" problem.

You can also correct any mistakes that you see.

Thank you.

[General Veers]

I see that you edited the post above my reminder. Thank you, but it still isn't contributing to the thread. I am trying to keep out as much spam as possible. Please cooperate. Thank you for your patience and understanding.

Edit: Post removed. Thank you.

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Sequences (Condensed)

A sequence is a list of terms whose value depends upon where it is in the list. The first term of a sequence is referred to as a₁, the second term as a₂, and so on. Some n^th term, somewhere in the list, is called a_n. A sequence can have a finite amount of terms or an infinite amount of terms.

For example...

1,2,3,4,5,6,7,8,9,10,11

That is a sequence of numbers that follow the rule a_n=1+1(n-1)=1-1+n=n.
a₁=1
a₂=2
a₃=3
a₄=4
a₅=5
a₆=6
a₇=7
a₈=8
a₉=9
a₁₀=10
a₁₁=11

1,2,4,8,16,32,64,128,256,512,1024

That is a sequence of numbers that follow the rule a_n=1*2^n-1=2^n-1
a₁=1
a₂=2
a₃=4
a₄=8
a₅=16
a₆=32
a₇=64
a₈=128
a₉=256
a₁₀=512
a₁₁=1024

The first sequence was an arithmetic sequence: subtracting any one term by the term before it would provide the same difference no matter which two consecutive terms you choose. Arithmetic series follow the pattern a_n=a₁+(n-1)d, where n is which term in the sequence you want to find and d is the common difference found by subtracting any two consecutive terms.

The second sequence was a geometric sequence: dividing any one term by the term before it would provide the same ratio no matter which two consecutive terms you choose. Geometric series follow the pattern a_n=a₁r^n-1, where n is which term in the sequence you want to find and r is the common ratio found by dividing any two consecutive terms.

NOT ALL SEQUENCES ARE ARITHMETIC OR GEOMETRIC!

For example, the sequence whose n^th term is 1/n! is neither an arithmetic or geometric sequence.

Series (Condensed)

A series is the sum of the terms in a sequence.

For example...

S₁₁=1+2+3+4+5+6+7+8+9+10+11=66

This is the sum of the arithmetic sequence example from a₁ to a₁₁. Another way to solve for the sum is with the formula S_n=(n/2)(a₁+a_n).

S₁₁=1+2+4+8+16+32+64+128+256+512+1024=2047

This is the sum of the geometric sequence example from a₁ to a₁₁. Another way to solve for the sum is with the formula S_n=a₁(1-rⁿ)/(1-r).

NOT ALL SERIES ARE ARITHMETIC OR GEOMETRIC!

[Qwerty]

Okay...

Any way to say it in English?

Indeed...

[General Veers]

I edited it some, although not by much. I hope this helps?

Sequences: list of terms that follow a pattern

Series: sum of terms in a sequence.

[Qwerty]

Yup, that clears it up.

Indeed, it does...

[The Dark Master]

0x0=infinity
Infinity x infinity=infinity

infinity-infinty=0
0 x infinity= 0

Likeness? Could this mean 0=infinity? You may think: 'The man's mad, 0 is nothing and infinity is, well...Infitity!'
Howver, that is in physics, as maths thingymabobies, they are very similar....
I'm not really a dab hand at maths, but it's worth a shot.....

[General Veers]

Any value that is not undefined times zero is zero. E.g. 0n=0, when n is a defined value.

Zero times an undefined number technically cannot be determined. (0/x)(x/0)=0x/0x=0/0, which is an indeterminate form.

Zero times infinity...well, the limit of 0n as n approaches infinity would indeed be zero, but you couldn't perform that operation without limits.

Practically any operation with the concept of infinity used as a value would be indeterminate, although limits can be used to find one of several possible values for any of the indeterminate forms. For example, (5*infinity⁵-2*infinity⁴+10*infinity³+infinity²-7*infinity+8)/(infinity⁶+6*infinity⁵+15*infinity⁴+20*infinity³+15*infinity²+6*infinity+1) would be indeterminate as is, but by taking the limit of that function (with x's instead of infinities) as x approached infinity and using L'Hopital's Rule, you would get the following:

limit_{x ► infinity}(5x⁵-2x⁴+10x³+x²-7x+8)/(x⁶+6x⁵+15x⁴+20x³+15x²+6x+1)
limit_{x ► infinity}(25x⁴-8x³+30x²+2x-7)/(6x⁵+30x⁴+60x³+60x²+30x+6)
limit_{x ► infinity}(100x³-24x²+60x+2)/(30x⁴+120x³+180x²+120x+30)
limit_{x ► infinity}(300x²-48x+60)/(120x³+360x²+360x+120)
limit_{x ► infinity}(600x-48)/(360x²+720x+360)
limit_{x ► infinity}(600)/(720x+720)
0 (The limit of a constant over a variable as that variable approaches infinity is equivalent to zero.)

You might not understand L'Hopital's Rule until you take differential calculus, since L'Hopital's Rule uses derivatives to make one of two specific indeterminate forms (infinity/infinity and 0/0) determinable.

[Sandmaster]

What about phi (1.168...or something)?

It's awsome

Also you get closer to it every time you divide the larger by the smaller of two consecutive numbers in the Fibbonacci sequence!

[General Veers]

Phi, the golden ratio?

It's exact value is (1 + 5^1/2)/2

It is approximately 1.61803398875...

[Sandmaster]

yeah, phi is awesome

[General Veers]

I never did realize that lim_{n ► infinity}(a_n/a_n-1)=phi when a_n is a term in the Fibonacci sequence.

Thank you.

Did you know that sum(1/n!,n,0,infinity)=e?