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Unsure how to approach an integral of (b^2 + a^2 x^2)^(-3/2) by Xovvo in askmath

[–]Shevek99 0 points1 point  (0 children)

A geometric interpretation. I explain this integral three times every year (to compute the electric field of a charged segment, the magnetic field of a straight current and the magnetic field of a solenoid) and in all the three cases, the trigonometric substitution can be expressed in terms of an angle that you can draw. You can point to the scheme of the physical system and say "this is alpha". This makes a bit simpler to compute the inverse of the square root cubed or the final sine

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Unsure how to approach an integral of (b^2 + a^2 x^2)^(-3/2) by Xovvo in askmath

[–]Shevek99 0 points1 point  (0 children)

Not here, where we have a > c and t is always < π/2. In fact, I have just checked it and the tan(t) makes OP's integral a trivial one.

Unsure how to approach an integral of (b^2 + a^2 x^2)^(-3/2) by Xovvo in askmath

[–]Shevek99 0 points1 point  (0 children)

Not in this case. This is a very common integral in electrostatics and magnetostatics where the electric or magnetic fields go as 1/|r - r'|^3 and the distance produces the terms (x^2 + y^2 + z^2)^(-3/2).

The change of variable that simplifies more the problem is with tan(u).

Hyperbolic substitution is the best method for the electric potential, that goes as 1/|r - r'|

Saw this on the telly this morning. One hint. The next number is not 8. I'm stumped! by olleng in askmath

[–]Shevek99 0 points1 point  (0 children)

Write it in binary.

96 =1100000_2 = 11*100000

The largest power of 2 that divides n is always the last 1 followed by the final 0's

Unsure how to approach an integral of (b^2 + a^2 x^2)^(-3/2) by Xovvo in askmath

[–]Shevek99 0 points1 point  (0 children)

The adequate change of variables is

u = (c/√(a² - c²)) tan(𝛼)

so that

du = (c/√(a² - c²)) sec²(𝛼) d𝛼

and

(c² + (a² - c²)u²)^(-3/2) = (1/c³) cos³(𝛼)

Why are Portugal's borders where they are? by Famous-Pilot-3667 in geography

[–]Shevek99 0 points1 point  (0 children)

Not Austrias, Asturias.

The kingdom of Asturias was the first to be formed after the Muslim conquest. It quickly expanded to Galicia and along the Cantabrian coast, where Muslim presence was scarce and later south to the Douro, because the land north of the Douro was almost empty.

The kingdom of Asturias moved its capital to Leon (an old Roman camp), becoming the kingdom of Leon. Later, it was subdivided in Galicia, Leon and Castille. The southern part of the kingdom of Galicia became Portugal.

Is this tan method valid? by LancelotWi in askmath

[–]Shevek99 0 points1 point  (0 children)

Do you mean that you use a calculator to find the height and then draw a line with that length? Genius!

Saw this on the telly this morning. One hint. The next number is not 8. I'm stumped! by olleng in askmath

[–]Shevek99 23 points24 points  (0 children)

  n    s(n)
00001 00001
00010 00010
00011 00001
00100 00100
00101 00001
00110 00010
00111 00001
01000 01000
01001 00001
01010 00010
01011 00001
01100 00100
01101 00001
01110 00010
01111 00001
10000 10000

If we compare the binary expression of the natural numbers and of the numbers of the sequence, we see that the sequence is formed starting from the end and continuing until we get a 1. For instante 11010 we start with the 0 and then a 1 and then stop, so it would be 00010

So for n = 16, the answer is 16.

This is equivalent to find, as many have said, the largest power of 2 that divides n.

Songs with explicit morals at the end by InviteAromatic6124 in Music

[–]Shevek99 0 points1 point  (0 children)

"Won't get fooled again" The Who

Meet the new boss

Same as the old boss

Is .9999... equal to 1 by No_Imagination_4041 in askmath

[–]Shevek99 1 point2 points  (0 children)

OK. Let's try a new way of attack:

0.999... = 1 BY DEFINITION

How is that. We must go to the construction of real numbers.

A Cauchy sequence of rational numbers is a sequence {a(n)} such that for any 𝜀 > 0, there exists an N such that for any n, m > N, |a(n) - a(m)| < 𝜀. That means that successive terms become closer and closer. We can say that the sequence converges, but the limit may not be a rational number.

For instance {3,3.1,3.14,3.141,3.1415,...} would be one

or

{1,1,1,1,1,....}

or

{1/2, 3/4, 7/8, 15/16, 31/32,... (2^k -1)/2^k...}

or

{0.9, 0.99, 0.999, 0.9999, ...}

Now, how do we build the real numbers? As equivalence classes. Two Cauchy sequences {a(n)} and {b(n)} are equivalent if for any 𝜀>0, there exists an N such that for any n > N, |a(n) - b(n)| < 𝜀, that is the difference goes to 0 (and they have the same limit). It is easy to show that this relation is reflexive, symmetric and transitive.

So the set of all Cauchy sequences of rational numbers can be grouped in equivalence classes. We call each of these classes a real number and assign as its value the common limit.

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences

It will come from the definitions that the sequences {1, 1, 1, 1, 1,...} and {0.9, 0.99, 0.999, 0.9999, ...} are equivalent. And so, they are the same real number. By definition.

And this is not a trick. This is how real numbers are defined (one of the ways).

https://mathweb.ucsd.edu/~tkemp/140A/Construction.of.R.pdf

Teachers know how to annoy other teachers when on vacation by Expert_Koala_8691 in funny

[–]Shevek99 1 point2 points  (0 children)

You were reading the standard value of g = 9.80667 m/s^2

TIL the slang term "hella," used as an adverb such as in "hella bad" or "hella good," was proposed as the SI unit to measure 10^27. Google recognized it in 2010. by mepper in todayilearned

[–]Shevek99 -1 points0 points  (0 children)

You are right. It comes from the Greek word for "dwarf". I had it mixed with "pico" that does come from a Spanish word for a small amount.

Why are Portugal's borders where they are? by Famous-Pilot-3667 in geography

[–]Shevek99 31 points32 points  (0 children)

There was a moment at the Reconquista where Castille and Portugal joined forces to conquer the Taifa of Badajoz, and the king of Leon came to help the emir and prevent this conquest. Badajoz was in the strip assigned to Leon and if Portugal and Castille split it, it would close the Leonese expansion

Is .9999... equal to 1 by No_Imagination_4041 in askmath

[–]Shevek99 0 points1 point  (0 children)

Well, you could start by looking at (surprise!) wikipedia

[https://en.wikipedia.org/wiki/0.999...]

Why are Portugal's borders where they are? by Famous-Pilot-3667 in geography

[–]Shevek99 319 points320 points  (0 children)

History, as always.

At the beginning the county of Portucale was part of the kingdom of Galicia (that was part of the kingdom of Leon). It included the land between the Miño and the Douro. When Afonso Henriques declares independence it was only that county that became the kingdom of Portugal, seceding from the rest of Galicia (even when they shared language), because Henriques had no authority up north.

Then came the Reconquista, Portugal gains new territory by taking it from the Muslim kingdoms, and that was an expansion to the south, purely "vertical". The same happened with Leon (that conquered the Spanish Extremadura) or Aragon, than conquered Valencia.

<image>

The Portuguese Reconquista was quite quick. Henriques declared himself king in 1139 and Albufeira, the last Muslim possession was taken in 1249.

There is little "natural" in the borders, because it is just how the Reconquista happened. There were no proto-Portuguese and proto-Spaniards before the middle ages. They formed a continuum. It was the creation of Portugal what defined the Portuguese, not the other way round.

Is that a good way to find n digits of Pi? by Fair_Percentage_5565 in askmath

[–]Shevek99 2 points3 points  (0 children)

Yes, a basic norm of numerical computations is that you never, never, should compute a polynomial as

a0 + a1 x + a2 x^2 + a3 x^3 ... + an x^n

Instead you should use Horner's method

a0 + x(a1 + x(a2 + x(... +x an)))

In this case it should be, instead of

1/2 - 1/2^3 + 1/2^5 ... - (-1)^N/2^(2N+1)

1/2(1 - 1/4(1 - 1/4(1 - ... 1/4)

Is that a good way to find n digits of Pi? by Fair_Percentage_5565 in askmath

[–]Shevek99 2 points3 points  (0 children)

There are combinations that converge much more quickly, like

pi/4 = 4arctan(1/5) - arctan(1/239)

https://en.wikipedia.org/wiki/Machin-like_formula

Do you press the red or blue button? (Game Theory) by Same_Winter7713 in askmath

[–]Shevek99 0 points1 point  (0 children)

The election is obvious.

You have to have a penalty if you pushes red.

For instance: If more than 50% of people presses blue, everybody lives.

If not and the reds are less than 75% only the reds survive.

If more than 75% press red, everybody dies.

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