Speaker Dungsmear? Or?

McCarthy 2.0 seems a vast improvement over Speaker Dungsmear. So here’s the question. Was he once a conservative with a little fire in the belly that got captured by the system and now is finding the bellyfire again? Or is this all completely under duress?

For the moment, it doesn’t matter which one. But some day it will matter, and we will have our answer.

RINOs an Endangered Species?
If Only!

According to Wikipoo, et. al., the Northern White Rhinoceros (Ceratotherium simum cottoni) is a critically endangered species. Apparently two females live on a wildlife preserve in Sudan, and no males are known to be alive. So basically, this species is dead as soon as the females die of old age. Presently they are watched over by armed guards 24/7.

Biologists have been trying to cross them with the other subspecies, Southern White Rhinoceroses (Rhinoceri?) without success; and some genetic analyses suggest that perhaps they aren’t two subspecies at all, but two distinct species, which would make the whole project a lot more difficult.

I should hope if the American RINO (Parasitus rectum pseudoconservativum) is ever this endangered, there will be heroic efforts not to save the species, but rather to push the remainder off a cliff. Onto punji sticks. With feces smeared on them. Failing that a good bath in red fuming nitric acid will do.

But I’m not done ranting about RINOs.

The RINOs (if they are capable of any introspection whatsoever) probably wonder why they constantly have to deal with “populist” eruptions like the Trump-led MAGA movement. That would be because the so-called populists stand for absolutely nothing except for going along to get along. That allows the Left to drive the culture and politics.

Given the results of our most recent elections, the Left will now push harder, and the RINOs will now turn even squishier than they were before.

I well remember 1989-1990 in my state when the RINO establishment started preaching the message that a conservative simply couldn’t win in Colorado. Never mind the fact that Reagan had won the state TWICE (in 1984 bringing in a veto-proof state house and senate with him) and GHWB had won after (falsely!) assuring everyone that a vote for him was a vote for Reagan’s third term.

This is how the RINOs function. They push, push, push the line that only a “moderate” can get elected. Stomp them when they pull that shit. Tell everyone in ear shot that that’s exactly what the Left wants you to think, and oh-by-the-way-Mister-RINO if you’re in this party selling the same message as the Left…well, whythefuckexactly are you in this party, you lying piece of rancid weasel shit?

In Defense of Ranked Choice Voting

One of the biggest obstacles to direly-needed change is RINOs, and one of the weapons in their arsenal is the “Wasted Vote” argument.

Periodically a third party has arisen, trying to hold RINOs to account by putting pressure on them from outside of the party, since doing so from the inside has historically done very little good. But, even if you find a third party candidate who perfectly reflects your views, you’re likely to vote for the RINO anyway. Why? Because if you don’t, the Democrat might win, and that would be even worse. So if you vote for that third party (that few will vote for), you’re throwing your vote away and increasing the likelihood of the Democrat winning. (It’s half as much a gain for the Democrat, as actually voting for the Democrat would be. Not as much, but half as much. Because although you denied the R your vote, you did not flip your vote to the Democrat.)

The Republican Party Establishment knows you don’t love them. But they know you hate the Democrats worse, and they use that to continue to herd you into supporting them. With gritted teeth you cast your vote, but your vote counts the same whether you cast it enthusiastically. And the other alternative, pissing on the voting apparatus to express your actual feelings, is probably a felony.

But what if you could vote for that third party without increasing the chances of the Dem walking away with the prize?

This is what ranked choice voting, or instant runoff voting, can do provided it is properly implemented. (And this includes the votes, and only genuine votes, being counted honestly, of course. However, I’m going to compare it to what we have today, and pretend that is honestly done too. RCV can’t work if it’s not honestly administered, just like our current system isn’t working because it isn’t honestly administered.)

The idea behind RCV is to vote by expressing your order of preference. You could vote for the Patriot Party, then for the RINO Party as your second choice (and ignore the Democrat, the Green, the Overt Socialist Schmuckmonkey Party, etc).

What does this do? It nullifies the wasted vote argument. Your vote will be counted for the Patriot party, first, then instead of it being “wasted” when the Patriot Party loses, it ends up going to the RINO. Actually, it’s just barely possible that the Patriot Party would actually beat the RINO, if people weren’t all individually afraid to vote for it.

It’s just like the famous “Prisoner’s Dilemma” where your fear of other peoples’ actions prevents you from doing the optimal thing–and vice-versa. As long as Job Lowe is afraid to vote Patriot because he’s afraid you’ll vote RINO, you’ll have to vote RINO because you fear that Job Lowe will, because he fears you will.

So on the whole I like RCV. It gives you a no-risk way to vote against the RINO scum, and in favor of someone who deserves your vote.

The problem is, as done here in the US, it comes packaged with a “jungle primary.” A bunch of candidates get to put their name out there, and the top four (or so) candidates get onto the “main” ballot. This gives party establishments their way around the threat of a good third party bumping them off. Because they know that few people bother with primaries, and third parties don’t have the resources to run in a primary…so they throw two or three establishment hacks into the primary and they will probably beat the third party. The result is the RINOs end up with two of the four slots in the general election, and the Dems get the other two. Now there’s suddenly no third party candidate on the ballot at all.

If we were to combine RCV with the present system where each party could nominate exactly one candidate to appear on the November ballot, or at the very least, ensure minor parties could get onto the ballot with at least one candidate regardless of the primary, we would be getting somewhere, but the establishment is smarter than we like to give them credit for. They will support the jungle primary + RCV “solution” rather than the more appropriate one-candidate-per-party + RCV solution.

It’s not RCV that is the problem, it’s the primary structure grafted onto it.

Justice

It says “Justice” on the picture.

And I’m sure someone will post the standard joke about what the fish thinks about the situation.

But what is it?

Here’s a take, from a different context: It’s about how you do justice, not the justice that must be done to our massively corrupt government and media. You must properly identify the nature of a person, before you can do him justice.

Ayn Rand, On Justice (speaking through her character John Galt, in Atlas Shrugged):

Justice is the recognition of the fact that you cannot fake the character of men as you cannot fake the character of nature, that you must judge all men as conscientiously as you judge inanimate objects, with the same respect for truth, with the same incorruptible vision, by as pure and as rational a process of identification—that every man must be judged for what he is and treated accordingly, that just as you do not pay a higher price for a rusty chunk of scrap than for a piece of shining metal, so you do not value a rotter above a hero—that your moral appraisal is the coin paying men for their virtues or vices, and this payment demands of you as scrupulous an honor as you bring to financial transactions—that to withhold your contempt from men’s vices is an act of moral counterfeiting, and to withhold your admiration from their virtues is an act of moral embezzlement—that to place any other concern higher than justice is to devaluate your moral currency and defraud the good in favor of the evil, since only the good can lose by a default of justice and only the evil can profit—and that the bottom of the pit at the end of that road, the act of moral bankruptcy, is to punish men for their virtues and reward them for their vices, that that is the collapse to full depravity, the Black Mass of the worship of death, the dedication of your consciousness to the destruction of existence.

Ayn Rand identified seven virtues, chief among them rationality. The other six, including justice, she considered subsidiary because they are essentially different aspects and applications of rationality.

—Ayn Rand Lexicon (aynrandlexicon.com)

Justice Must Be Done.

Trump, it is supposed, had some documents.

Biden and company stole the country.

I’m sure enough of this that I put my money where my mouth is.

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system. (This doesn’t necessarily include deposing Joe and Hoe and putting Trump where he belongs, but it would certainly be a lot easier to fix our broken electoral system with the right people in charge.)

Nothing else matters at this point. Talking about trying again in 2024 or 2026 is pointless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud in the system is not part of the plan, you have no plan.

This will necessarily be piecemeal, state by state, which is why I am encouraged by those states working to change their laws to alleviate the fraud both via computer and via bogus voters. If enough states do that we might end up with a working majority in Congress and that would be something Trump never really had.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Spot Prices

Last week:

Gold $1,943.60
Silver $23.73
Platinum $930.00
Palladium $1,290.00
Rhodium $4,750.00

This week, 3 PM MT on Friday, markets closed for the weekend

Gold $1,914.10
Silver $22.75
Platinum $921.00
Palladium $1,338.00
Rhodium $4,750.00

Everything down or flat except palladium. Interesting!

Sines and Cosines from Plain Old Arithmetic

Well after last week’s difficulty with editing I’m going to have another go at it.

I ended by stating that there was a way of computing the sine and cosine of any arbitrary angle, without knowing any other sines or cosines.

This is good because if you do need other sines and cosines as inputs, then roundoff error in the previously determined sines and cosines will start you off wrong, and you’ll be making an roundoff error here (remember you can’t specify most of these numbers precisely, they’re irrational). These errors will only accumulate. A direct method that doesn’t depend on other calculations would be much better.

We have that method. Sort of. There are a couple of caveats: This method will give you an approximation, though you can put more work into it and get it as close to the actual sine or cosine as you want. The more accurate you want, the more work you will need to do. (It becomes precisely accurate if you do infinite work.

And the other caveat is you have to give your angle in radians, because that is the “natural” way to measure an angle: as a distance along the arc of a circle.

OK, here we go:

Remember a few weeks ago (https://www.theqtree.com/2023/06/10/2023%C2%B706%C2%B710-joe-biden-didnt-win-daily-thread/) I talked about the Taylor series, and a special case of it called the MacLaurin series. These can be used, in some cases to compute values of a function that might otherwise seem intractable, and by using pure arithmetic.

The function must be infinitely differentiable, which is a calculus concept. I’m going to try to explain it, but if you don’t want to know it won’t hurt you not to know…much. You may want to skip the next few paragraphs…I’ll put bold where it’s safe to pick up so you can scan for it.

Basically, if you have a function like sinx which is shown below with x in radians.

…you can create another function that gives the slope of sin x at any given point. So what would such a function look like? Well, when x is zero, the slope of the wiggly line is just about 1. Which is to say, right there at x = 0, it’s climbing at the rate of one square vertically for every one unit to the right (in radians). Our new “slope of sin x function” has a value of 1 at x = 0. At π/2 (or 90°), the curve of sin x is level, so the slope is zero. At π, the slope is -1. At 3π/2, it’s zero again. And at 2π, it’s back to being 1 again.

We already know of a function that has these values at these points. It’s the cosine of x. Is that a coincidence? Actually, no. It turns out (after a lot more proof than I am going to show you), that the “slope of sin(x) function” is indeed cos(x). Now for some terminology; the correct way to phrase this is “the derivative of sin(x) is cos(x).”

You can repeat this exercise for cos(x), and you will get -sin(x). So the derivative of cos(x) is -sin(x). But cos(x) is already the derivative of sin(x), so the derivative of the derivative of sin(x) is -sin(x). But we don’t want to say “the derivative of the derivative of..” so we call that the second derivative.

Unsurprisingly, the derivative of -sin(x) is -cos(x). And the derivative of -cos(x) is back to sin(x). In other words, you can keep taking the derivative of the derivative of the derivative…forever. Kind of boring.

But that’s what it means to say that sin(x) is infinitely differentiable. With most functions you’ll eventually end up with a derivative that’s zero for all values of x, and then you’re done. But not with sin(x) and also (since it shows up in the chain) not for cos(x).

Which means that sine and cosine are infinitely differentiable and we can use the Maclaurin series on them.

(OK the calculus-averse are back with us now, and we can continue.)

So here is the Taylor series of any function, f. The idea is to pick a central point, a, and then compute the value of f for some nearby x. Note that all of the terms have x–a in them; you get closer if x is very close to a.

If a is zero, it becomes the Maclaurin series:

And the tick marks denote the first, second, third, and four derivatives.

The 1!, 2!, 3! etc are factorials. You get a factorial by multiplying the number by every smaller number down to 1. So 4! is 4 x 3 x 2 x 1. (Practically, of course, you don’t bother multiplying by 1.)

So if we set a equal to zero (we’re going to be interested in values for x close to zero), the formula can simplify.

I pointed out in the section you skipped over that the first derivative of sinx is cosx, the second derivative is -sinx, the third derivative is -cosx, the fourth derivative is sinx again, and so on…the cycle repeats.

So if I substitute those in we have:

sinx = sin(0) + (cos(0)/1!)x + (-sin(0)/2!)x² + (-cos(0)/3!)x³ + (sin(0)/4!)x⁴ + …

But we know that the sin(0) is zero, and so is -sin(0). So we can simply throw a bunch of these terms out (and as I do so I’ll bring in more terms from where the ellipsis was)

sinx = (cos(0)/1!)x +(-cos(0)/3!)x³ + (cos(0)/5!)x⁵ + (-cos(0)/7!)x⁷…

And the others (cos(0) and -cos(0)) can be replaced with 1 and -1 respectively, so simplifying:

sinx = x – x³/3! +x⁵/5! – x⁷/7!…

This is a nice pattern. Every odd power of x, divided by the factorial of the power. Every other term gets a minus sign. The further you carry it out, the more accurate you will be. Also, the closer to zero x is, the more accurate it will be, but you can always just carry it out further if x isn’t very close to zero, to make up for it. If you can follow this infinitely far, you can get the correct answers for numbers very far away from 0, as far away as you like.

Here is a graph of sinx, along with a graph of what you get out of the Maclaurin representation taken to the x⁷ term.

As a practical matter you do not need to go past x=π/2, because the function makes the mirror-image curve going down to π as it did climbing to π/2. So the sine of 89° is the same as the sine of 91°, and so on. And the stretch from π to 2π is also a mirror image, except flipped vertically. After that, it repeats. So even though you can see some difference at x = π, and a lot of difference at 2π, you can probably get everything you need from the range 0 to π/, where this is pretty close.

So you can get pretty close with this function, between –π/2 and π/2. I’ll repeat it with the factorials written out:

sinx is roughly equal to: x – x³/6 +x⁵/120 – x⁷/5040.

Certainly the brain inside a calculator should be able to do that very quickly! And so, that’s probably what it does. It might go out one or two more terms:

x – x³/6 +x⁵/120 – x⁷/5040 + x⁹/363,880 – x¹¹/39,916,800

But you can see if x is small you’re not going to get much change from the x¹¹ term.

So let’s do the same thing for cosines. Now when we go back to the MacLaurin series, everything shifts over, it starts with cosine, then -sine, then -cosine, then sine.

cosx = cos(0) + (-sin(0)/1!)x + (-cos(0)/2!)x² + (sin(0)/3!)x³ + (cos(0)/4!)x⁴ + …

And now we know that the cosine of zero is 1, and the sine of zero is zero, so:

cosx = 1 + –x²/2! + x⁴/4! + –x⁶/6! + x⁸/8! +…

So here we have a pattern oddly like, but oddly unlike, the sine version. Now it’s the even powers of x (including x⁰ = 1) that show up, again with alternating signs.

And we have:

cosx is roughly equal to: 1 – x²/2 +x⁴/24 – x⁶/720 + x⁸/40,320.

So we can get increasingly accurate. Because, remember sine and cosine swap at 45 degrees. The sine of 60° is the same as the cosine of 30°, so if you want the sine of 60°, compute the cosine of 30° instead; your answer will be closer. (Needless to say first, convert to radians!) So we really only need these series to be accurate enough for whatever our purposes are, to 45° or π/4 radians.

Next time: I pull a mathematical rabbit out of my hat!

Obligatory PSAs and Reminders

China is Lower than Whale Shit

To conclude: My standard Public Service Announcement. We don’t want to forget this!!!

Remember Hong Kong!!!

https://youtube.com/watch?v=L3tnH4FGbd0

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

News Flash

Today, it is still the case that Joe Biden didn’t Win.

I realize that to some readers, this might be a shock; surely at some point things must change and Biden will have actually won.

But the past cannot actually be changed.

It will always and forever be the case that Joe Biden didn’t win.

And if you, Leftist Lurker, want to dismiss it as dead white cis-male logic…well, you can call it what you want, but then please just go fuck off. No one here buys that bullshit–logic is logic and facts are facts regardless of skin color–and if you gave it a moment’s rational thought, you wouldn’t either. Of course your worthless education never included being able to actually reason–or detect problems with false reasoning–so I don’t imagine you’ll actually wake up as opposed to being woke.

As Ayn Rand would sometimes point out: Yes, you are free to evade reality. What you cannot do is evade the consequences of evading reality. Or to put it concretely: You can ignore the Mack truck bearing down on you as you play in the middle of the street, you won’t be able to ignore the consequences of ignoring the Mack truck.

And Ayn Rand also pointed out that existence (i.e., the sum total of everything that exists) precedes consciousness–our consciousnesses are a part of existence, not outside of it–therefore reality cannot be a “social construct” as so many of you fucked-up-in-the-head people seem to think.

So much for Leftist douchebag lurkers. For the rest of you, the regular readers and those lurkers who understand such things: I continue to carry the banner once also carried by Wheatie. His Fraudulency didn’t win.

Let’s Go, Brandon!!

Hey China!

Or rather, “Hey Chinese Communist Party and your entire array of servitors, ass-wipers, and fellators!”

You’re not even worth my time this week. When you decide to act like civilized people, maybe I’ll give you a lesson or two in how non-barbarians behave.

Hey BiteMe!
(Or, Whoever Has Their Hand Rammed Up That Putrefying Meat Puppet’s Ass)

[Language warning]

You and yours have caused a lot of injury. Literal injury with your war on people who don’t want to take an untested vaccine. When people die in an emergency room because a hospital won’t admit them because they haven’t had their clot shot, that’s a crime.

I’m going to address here the insult on top of the injury, because I am among the insulted. I still have my health but apparently you want me to live under the 8th Street Bridge (which actually isn’t on 8th Street, but whatever, that’s what the I-25 overpass over Cimarron is called), so maybe if you have your way that won’t be true for long. Dreadful time of year to become homeless.

No, you’re just trying to make me unemployed, because I won’t take your fucking shots.

Well, that threat is NOT going to work. I. Won’t. Take. Your. Fucking. Shots.

And it looks like enough people agree, that you’re having to back down, you worthless asswipe.

You’re LOSING.

You LOSER.

You Chinese-bought ratfucking traitor.

I would love to see you die an agonizing, humiliating death. (This isn’t a threat, because I am not threatening to cause that death. I am just announcing my intention to party if it happens.) It would be just recompense for the way you’re killing America…and millions of Americans.

His Fraudulency

Joe Biteme, properly styled His Fraudulency, continues to infest the White House, we haven’t heard much from the person who should have been declared the victor, and hopium is still being dispensed even as our military appears to have joined the political establishment in knuckling under to the fraud.

One can hope that all is not as it seems.

I’d love to feast on that crow.

(I’d like to add, I find it entirely plausible, even likely, that His Fraudulency is also His Figureheadedness. (Apparently that wasn’t a word; it got a red underline. Well it is now.) Where I differ with the hopium addicts is on the subject of who is really in charge. It ain’t anyone we like.)

Justice Must Be Done.

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system.

Nothing else matters at this point. Talking about trying again in 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Tranheuser Busch

Those asshats either get the world, all time record for obtuseness…or this is deliberate somehow. And their namby pamby non-apology that amounts to a “We’re sorry you’re too much of a knuckledragger to appreciate what we did” simply underscores the stupid. It’d be like a 400 pound golfer wearing spikes tromping on your instep and being utterly clueless why you are screaming in pain.

Me

I wrote that months ago, and they are casting about now trying to figure out how to get people to come back. However the one thing that might work (an abject admission that they were wrong to even think of doing this) is the one thing they cannot do. Because the decision makers are woke and cannot conceive that they were actually fuckheadedly wrong to do this.

Spot Prices.

Kitco Ask. Last week:

Gold $1,960.40
Silver $24.41
Platinum $945.00
Palladium $1,280.00
Rhodium $4,800.00

This week, markets closed as of 3PM MT.

Gold $1,943.60
Silver $23.73
Platinum $930.00
Palladium $1,290.00
Rhodium $4,750.00

Gold and silver still farting about in roughly the same places. Someone with a big futures presence can make some money off of these moves…some poor guy who bought physical metal a few months ago ain’t making squat (it has to rise enough to cover the buy/sell spread, which can be pretty wide). PGMs mixed.

More on Sines and Cosines

As you may have realized, I generally start to write a post by copying the one from five weeks earlier and modifying it…that way I can recycle my eagle pictures. (Right now that’s six weeks, because the Darwin Memorial Post doesn’t have an eagle on it.)

So I get to see an older science (or math) post of mine and I will sometimes read through it. When I did that this time I found multiple typos (not so bad) and also one glaring arithmetic error…that no one one called me on. (I’ve gone and fixed it.)

Makes me wonder.

Anyhow, we left off realizing that a complex number can be thought of as representing an angle, and that multiplying any complex number (including those that are pure real or imaginary) by another complex number, produces a complex number that represents the sum of the two angles. If one of the complex numbers has a magnitude of 1, the effect of the multiplication is to simply rotate the other number by the first number’s angle.

OK, those sentences were a bit complicated, so let’s show some examples: Take two complex numbers:

A = <a, α>
B = <b, β>

The first complex number A, is a units long at an angle of α radians, and similarly for B. (And a quick reminder. For clarity, I’m using angle brackets to indicate polar coordinates–a distance and an angle–but this is not any kind of standard notation.)

If you multiply those two complex numbers together, you get <ab, α+β>. The magnitudes get multiplied, the angles get added.

If b just happens to be 1, then you have an answer of <a, α+β>. In other words multiplying by B just means rotating A by the angle β.

And if both a and b are 1, you are simply adding the angles together to get another answer that has magnitude 1; it’s a pure rotation.

As I said before this is not just a geek parlor trick; it underlies much of electrical engineering.

Also discussed, a lot, last time were the sines and cosines of angles. I stated that some angles have obvious sines and cosines. For example, zero degrees measured on the unit circle has no y value, so its sine is clearly 0. And it’s x coordinate is 1, so the cosine is 1. Similarly 90° or π/2 radians is pure y coordinate with no x component, so its sine is 1 and its cosine is 0. You can easily figure out 180° and 270°, and of course 360° is the same as 0°.

But those aren’t the only obvious angles. 45° (or π/4 radians) is another obvious one. Why? Because you know the x and y coordinates of 45° on the unit circle are the same number, so it forms an isosceles right triangle…the two “legs” (the sine and cosine of 45°) are the same.

You can then get the rest of the way with Pythagoras. Squaring the two legs gives you the same number too, and the hypotenuse is 1:

1² = (sin(45°))² + (cos(45°))²

But the cosine and sine are equal so just replace cosine with sine (and 1² with 1):

1² = (sin(45°))² + (sin(45°))²
1 = 2(sin(45°))²

Dividing through by 2 we get:

½ = (sin(45°))²

Taking the square root of both sides:

√½ = sin(45°) = 1/√2

So we know that the sine of 45° or π/4 radians is one divided by the square root of 2.

But you do not want to divide by an irrational number if you can possibly avoid it, because they have an infinite number of digits, so think about writing that one out doing long division…you’re going to be multiplying long numbers a lot and subtracting them too.

So for historical reasons mathematicians never let a square root sign be in the denominator. You fix this by multiplying top and bottom by that number:

(√2 x 1) / (√2 x √2) = √2/2

Which works out to roughly .707.

So if you’ve done trig for any length of time when you see 45° (or 135°, or 225°, or 315°) you just automatically know the sine and cosine will be 0.707 (or -0.707).

You can use similar reasoning for 60°. That angle is the angle for an equilateral triangle, one where all three sides are the same length, and all three angles are 60°. So the origin (0,0), the (x,y) for 60 degrees on the unit circle, and (1,0) form an equilateral triangle. But the (x,y) are just (cos(60°), sin(60°)). The top angle of the triangle is half as far to the right as 1,0 is, so cos(60°) must be 1/2. You can then use Pythagoras on half of the equilateral triangle to show that the sine must be √3/2, or about .866.

(I must apologize for this diagram, it only shows half of the equilateral triangle. I found that any diagram that showed the whole triangle made the sides of length 2, and I thought that would be even more confusing than this diagram is.)

For 30° you have a mirror image of 60°, and you can swap the sine and cosine.

OK, what about the other angles?

Not every problem you’re going to want to solve with trig is going to involve one of these angles. In fact, if one ever actually does find oneself doing these angles, one is probably working a story problem!

It turns out that there are formulas for computing the sine (or cosine) of half of a given angle. So you could find the sine and cosine of 15° by running the formulas on 30°, and you could get the sine and cosine of 7.5° by running the formulas on that. You can get as small an angle as you like by repeatedly doing half angles. And then you can add those angles together in some combination with other formulae. For instance to get the sine and cosine of 11 1/4 degrees, run the addition formulae on the sines and cosines of 7.5° and 3.75°.

Actually you know how to do the addition formula: just turn the angles into complex numbers and multiply them together. The real part is the cosine of the sum of the angles, the imaginary part is the sine of that same sum.

Many people spent countless hours building up tables of sines and cosines this way, and back in those days they had to work longhand multiplying six digit numbers together just to get halfway decent results after rounding, and reusing the rounded numbers to do the next computation.

But there IS another way. It may not be easier but you can go directly to the sine and cosine of the angle you want instead of having to build off of other, known numbgers. So maybe it is easier because you could compute the sine and cosine of 11.25° directly instead of first computing the sine and cosine of 7.5° and 3.75°.

And I want to tell you what it is, and I was planning to do so tonight, but for some reason the Wordpus editor is slowing down on me more and more, and I’ve already restarted Firefox four times. It’s now slow enough that backspacing over one letter takes a second or two. And I have to wait for it to catch up as I type.

So I’ve run out of patience. Next time, the direct way to get a sine or a cosine. A hint: the sine and cosine are “infinitely differentiable.” (Not much of a hint, for people who haven’t taken calculus.)

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

A Political Statement

One of my stock jokes is to say, as I head off to the bathroom or washroom or restroom, “I’m going to go make a political statement.”

It certainly seems like urinating is a good way to make a statement about the YSM, and defecating works for politicians. Or the other way around is good too, though “#1” just never seems emphatic enough.

Maybe “#1” is just being polite, and “#2” is telling the fixtures what you really think.

Anyhow, I pretty much do plan to be polite to RINOs that electioneer or fundraise. I’ll tell them, Sorry, but I need to GO Pee…

The Twenty Stand-Up Shitbags (Plus Five More Who Were Just Cowardly Shitbags)

Twenty were stand-up shitbags, who actually voted against punishing Adam Schitt, leaving no doubt where they stood. Five more voted present or were sure to be elsewhere…effectively helping shut down justice, but without being overt about it.

At least the first 20 weren’t weasely about it; we know how the insides of those bags stink.

But it won’t work, you Cowardly Five. We’re onto you.

But then…

I read something in the YSM today that suggested that the 20 who voted against censuring, ejecting. and fining Adam #2 (see above if you’re wondering who Adam #1 is–there isn’t one) were mostly concerned about the constitutional propriety of the fine.

OK, fair enough. I can actually see that being an issue; one I’d want to hear arguments on, pro- and con-.

We’ll get some clarity as to the real reason soon. Next week will be a simple censure motion. I’d like to see the little pencilneck turd kicked out as well, but let’s see how “The Twenty” vote this time around.

Language Warning

In the next piece I had to discuss a particular topic. Unfortunately, I couldn’t discuss it without naming it. Therefore I apologize in advance for having to do so, and apologize to anyone offended by the sight of the name.

RINO McDaniel

RINO McDaniel continues to infest the GOP. But RINO McDaniel isn’t the problem.

Let me be crystal clear on this, RINO McDaniel is a lower-than-whale-shit, piss guzzling ratfucking shit eating traitorous rancid syphillitic cunt. Her worth as a human being is substantially less than zero, any oxygen sucked into her lungs is wasted, and it would be, no matter what job she had.

I fear I haven’t been clear enough, but that will have to suffice.

But she is not the problem…or rather, she would not be a problem were it not for others. She’d still be as I have described, but we wouldn’t know who she is and would not care, because she could do no damage. She’d just be anonymous human refuse.

No, the real problem is the fact that a majority of the 168 top GOP people voted for her. And now that has happened five times so they cannot claim they didn’t know what she was.

In spite of the fact that under her “leadership” the party has deliberately sabotaged the will of its base, has deliberately refused to challenge blatant election fraud, had gone out of its way to ensure certain candidates do not get nominated, has diverted donor money to namby-pamby candidates who have all the electoral appeal of a puddle of dog vomit…and in general has done nothing whatsoever to help fix the problems that plague America.

However that last is to be expected; I cannot expect anyone who IS the problem to help FIX the problem.

RINO McDaniel would be powerless without an entire party leadership of the same mind as her. They want this dismal performance; they want to ignore the party base.

If she were to drop dead this instant, it would solve nothing as someone just like her would be elected by those same pustulous people.

According to Charlie Kirk, about 55 people voted against her, 10-12 wanted something different but were too chickenshit to do the right thing, and roughly 100 people voted for her enthusiastically, and even had the unmitigated gall to complain to Kirk about US. Fuck ’em. Rusty 12 gauge bore brushes would be too good for these arrogant pricks and cunts.

Every single one of those hundred is just as bad as she is. In other words, they are all worse than I described at the beginning of this piece. And no doubt those people in turn have people who supported them to be state party chairs and whateveritis they call the other two people from each state and territory who were voting.

It’s time to face up to the fact that the Republican party is effectively owned by the shit-eating RINOs. We’ve got more work to do, a lot more work, to make the GOP an instrument for the restoration of the United States of America. And that’s in addition to cleaning up our elections.

There’s no point in cleaning up elections just to elect ratfucking RINOs.

OK, hopefully now you will have some inkling of my true attitude towards RINOs. Sorry that words were inadequate to give you the full picture.

The Real Fascist is His Fraudulency Joe Biden*

*Or whoever has his hand rammed up that meat puppet’s ass.

Brandon (which I will use as a term for whoever is the power behind the Porcelain Throne) has thrown down the gauntlet…but in a way where most of America will never see it. The networks didn’t carry his tirade. CNN air brushed it (or whatever you call editing the red background) for its five viewers (who aren’t trapped in airports).

Luckily for me I live in Colorado, and therefore, despite my best efforts, I probably didn’t vote for Donald Trump.

Of course, for this purpose who I actually did try to vote for will be essential, and they undoubtedly know.

Come and get us, asswipes!

https://www.youtube.com/watch?v=3h6ZZ28QtX4

Justice Must Be Done.

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system.

Nothing else matters at this point. Talking about trying again in 2022 or 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

Small Government?

Many times conservatives (real and fake) speak of “small government” being the goal.

This sounds good, and mostly is good, but it misses the essential point. The important thing here isn’t the size, but rather the purpose, of government. We could have a cheap, small tyranny. After all our government spends most of its revenue on payments to individuals and foreign aid, neither of which is part of the tyrannical apparatus trying to keep us locked down and censored. What parts of the government would be necessary for a tyranny? It’d be a lot smaller than what we have now. We could shrink the government and nevertheless find it more tyrannical than it is today.

No, what we want is a limited government, limited not in size, but rather in scope. Limited, that is, in what it’s allowed to do. Under current circumstances, such a government would also be much smaller, but that’s a side effect. If we were in a World War II sort of war, an existential fight against nasty dictatorships on the brink of world conquest, that would be very expensive and would require a gargantuan government, but that would be what the government should be doing. That would be a large, but still limited government, since it’d be working to protect our rights.

World War II would have been the wrong time to squawk about “small government,” but it wasn’t (and never is) a bad time to demand limited government. Today would be a better time to ask for a small government–at least the job it should be doing is small today–but it misses the essential point; we want government to not do certain things. Many of those things we don’t want it doing are expensive but many of them are quite eminently doable by a smaller government than the one we have today. Small, but still exceeding proper limits.

So be careful what you ask for. You might get it and find you asked for the wrong thing.

Political Science In Summation

It’s really just a matter of people who can’t be happy unless they control others…versus those who want to be left alone. The oldest conflict within mankind. Government is necessary, but government attracts the assholes (a highly technical term for the control freaks).

A Few Things We Cannot Blame on His Fraudulency

I am pretty sure Joe Biden had nothing whatsoever to do with the 30 Years War that ran from 1618-1648 and probably killed about a third of the people then living in what is now Germany.

Nor did he cause the collapse of either Roman empire (Western, 476 CE, Eastern 1453 CE). Nor the ignominious failure of most of the Crusades. Nor the collapse of Bronze Age civilization around 1200 BCE (including the collapse of the Minoans and the blowup of Santorini).

However, my utter lack of ability to imagine how he could possibly be responsible for these things is not a valid argument against them, so I await correction if appropriate.

His Truth?

Again we saw an instance of “It might be true for Billy, but it’s not true for Bob” logic this week.

I hear this often, and it’s usually harmless. As when it’s describing differing circumstances, not different facts. “Housing is unaffordable” can be true for one person, but not for another who makes ten times as much.

But sometimes the speaker means it literally. Something like 2+2=4 is asserted to be true for Billy but not for Bob. (And when it’s literal, it’s usually Bob saying it.) And in that sense, it’s nonsense, dangerous nonsense. There is ONE reality, and it exists independent of our desires and our perceptions. It would go on existing if we weren’t here. We exist in it. It does not exist in our heads. It’s not a personal construct, and it isn’t a social construct. If there were no society, reality would continue to be what it is, it wouldn’t vanish…which it would have to do, if it were a social construct.

Now what can change from person to person is the perception of reality. We see that all the time. And people will, of course, act on those perceptions. They will vote for Trump (or try to) if their perception is close to mine, and vote against Trump (and certainly succeed at doing so) if their perception is distant from mine (and therefore, if I do say so, wrong). I have heard people say “perception is reality” and usually, that’s what they’re trying to say–your perception of reality is, as far as you know, an accurate representation of reality, or you’d change it.

But I really wish they’d say it differently. And sometimes, to get back to Billy and Bob, the person who says they have different truths is really saying they have different perceptions of reality–different worldviews. I can’t argue with the latter. But I sure wish they’d say it better. That way I’d know that someone who blabbers about two different truths is delusional and not worth my time, at least not until he passes kindergarten-level metaphysics on his umpteenth attempt.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

(Paper) Spot Prices

Kitco “Ask” prices. Last week:

Gold $1,961.90
Silver $24.66
Platinum $972.00
Palladium $1,322.00
Rhodium $4,750.00

This week, 3PM Mountain Time, markets have closed for the weekend.

Gold $1,960.40
Silver $24.41
Platinum $945.00
Palladium $1,280.00
Rhodium $4,800.00

Gold barely moved at all, everything else is down except rhodium which moved up by what appears to be the minimum increment (I don’t believe I’ve ever seen a price for it which wasn’t a mulitiple of $50.

So it seems to have been a quiet week, but then again I rarely look during the week.

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

https://youtube.com/watch?v=L3tnH4FGbd0

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

Joining The Herd Of Lemmings

I’ve had cause to consider a few things. Maybe we’re going about it the wrong way, and we need to ditch Trump

Yeah, NO

Trump all the way! Why? Because being hated by the people who hate him is a sign of impeccable character, that’s why.

The haters can go fuck themselves with rusty twelve gauge bore brushes. I’d prefer ten gauge but that’s kind of scarce, so…I’m willing to compromise.

The RINO’s Dilemma

The RINOs who who have burrowed in and taken over most GOP organizations, from the state down to local organizations, have quite a dilemma on their hands, and most of them have their heads too far up their asses to realize it.

OK, I’m not talking about the liberal in a Republican area, who knows they’re in the wrong party, but is there because it’s the only game in their town; they hope to capture a nomination someday, at which point they’re guaranteed to be elected…otherwise, they never will be. These people are a hazard in any heavily conservative area.

No, I’m talking about the guys who are a little bit conservative and want to do some good by going into politics, and they’re in a closely matched area, closely enough that they can join the party they are most aligned with and still have a chance. They think the Democrats…particularly the ones who end up running for office…are nuts.

They don’t think much better of the Deplorable types, either. A bunch of bumpkins whose hearts are in the right place, mostly…OK a bit extreme. But they think Deplorables can’t understand that first you have to get elected, then work within the system to change things…a slow process. They genuinely want many of the things Deplorables want…just not as much. The government is spending too much. Or they need to spend money on highways instead of welfare for illegal immigrants. But they want to work within the system to get these things done.

Or maybe they think things are pretty close to ideal right now, and they want to nail it in place.

The problem is, that means they don’t stand for anything in particular. And it shows. They’re about as unappetizing to the electorate as a puddle of dog vomit. The folks in the middle, who they think they are appealing to because they themselves are not extreme, would honestly prefer a clear-spoken radical to someone who qualifies everything they say to the point where they sound like they don’t believe anything at all.

The problem these “Mild RINOs” have, is they just can’t see that. And the reason they just can’t see that, is their entire sense of self-worth is tied up in not seeing that. In their minds, they’ve worked tirelessly for their party, to keep those crazy Democrats out…only to have to constantly fight with a small number of crazy Republicans–who are only liabilities if they end up as candidates. They’ve fought the good fight, and if they can just find the right candidate, someone with some charisma, they might stop the crazies…without being too beholden to the OTHER crazies. In the meantime it’s not working. What’s a responsible guy in politics to do?

They simply cannot understand that the Republicans can’t succeed as the party of nothing in particular. Not really in the past, and certainly not today when people are starting to realize that no matter what they do in the voting booth, the country is still about to fly off a precipice. If they did see it, they’d suddenly have two choices: Go away and let the GOP succeed, or stay and fight. But “go away” isn’t really an option, because what’s the point of having a party now owned by the crazies, win?

Well, they have a dilemma…and WE, therefore have a problem. And we would have that problem even IF they realized that they had a problem…that they were the problem.

No one ever thinks they are the bad guy. Even Epstein probably thought he was the good guy. Right up to the moment where he didn’t kill himself.

So if you ever wonder why these unappetizing dufuses cling on even when their fingernails are being left behind…that’s why. They don’t understand no one wants them, and can’t imagine that no one should want them. And oftentimes their greatest pride is in all the hard work they’ve done for the party. They’re not going to give that up; it’d be psychological suicide.

If you’ve worked with these people, there’s a good chance you like them and consider some of them your friends. But even if so…we’re going to have to give them a good, hard shove. Because America is more important than those milquetoasts’ egos.

Justice Must Be Done

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system.

Nothing else matters at this point. Talking about trying again in 2022 or 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Spot (i.e., paper) Prices

Last week:

Gold $1,955.30
Silver $24.96
Platinum $982.00
Palladium $1,304.00
Rhodium $5,450.00

This week, 3PM Mountain Time, markets have closed for the weekend.

Gold $1,961.90
Silver $24.66
Platinum $972.00
Palladium $1,322.00
Rhodium $4,750.00

Gold creeping upward. Other things decidedly mixed, with little movement, prercentage wise. Except of course rhodium which continues its collapse.

Comparing Cartesian And Polar Coordinates

Last week I introduced polar (circular grid) coordinates as an alternative to (and equivalent of) Cartesian (square grid) coordinates.

I also stated that because they’re equivalent either one could be used at any time, so you could pick whichever one was most convenient at the time.

What did I mean by that? Well, hold that thought.

I was talking about x, y pairs last week, but I could just as easily have been talking about complex numbers. Remember they can be represented on a plane that looks a lot like the Cartesian plane, with the real part taking the place of x, and the imaginary part standing in for y. So you can convert complex numbers to polar coordinates, too…but there’s a hitch. The resulting r and θ do not map nicely to real and imaginary, the pair as an inseparable whole is part imaginary and part real. You just have to remember that the thing represents a complex number, and if you want to split it up, you have to convert to Cartesian.

In polar coordinates, r is often called the “absolute value” of the complex number (or you can think of it as the magnitude or length), while θ can be called the “argument.”

Actually…this is not standard notation, but for clarity I am going to surround polar-style complex numbers with angle brackets: < and >

OK, now let’s consider the complex number plane, in both Cartesian and polar forms. And let’s look at some examples.

We’re going to represent a couple of complex numbers in both ways and do two different things to them in order to see which system is more convenient for each of those things.

The first example number we will call A, and in Cartesian form, it is 1+i. Converting to polar coordinates, it’s <√2, 45°> or <√2, π/4> depending on whether you’re thinking in degrees or radians.

The second pair is B and in Cartesian coordinates, it is √3 + i. Converting to polar coordinates, it’s <2, 30°> or <2, π/6>, depending.

[Angles like 30°, 45° and 60° degrees are the “easy” angles in trigonometry because their sines and cosines are well known and fairly simple; the sine of 30° is 1/2, the cosine is √3/2. For 45° both the sine and cosine are √2/2. For 60°, swap the sine and cosine of 30 degrees. Other easy angles are 0 and 90°, plus of course any corresponding angles in the other quadrants of the plane, e.g. 120° is simply the mirror image of 60° so its sine and cosine are going to be the same except for the cosine changing sign to -1/2. People who have been doing trigonometry for very long simply know these angles, and their matches in other quadrants, off the tops of their heads. So (returning to our two example complex numbers), A is at 45° but its length is clearly not 1 since both the x and y (real and imaginary) values are 1. We can get the length from Pythagoras, though, x² = 1 and y² = 1, so the hypotenuse (the length of A) squared is 1+1=2, so the length of A is √2. For B, I simply doubled the well-known sine and cosine of 30°, so I’ve got a 30° angle and a length of 2.]

OK, let’s add A + B. That’s easy to do in Cartesian coordinates, it will be (1+√3) + (1+1)i. (If you remember vector addition from the physics series, this works exactly the same; you can even treat the two numbers as if they were vectors and do a tail-to-head visual addition.) So what do we get in polar coordinates? Well, it’s not obvious how to add things in polar coordinates…but let’s convert the sum into polar coordinates just to see if some sort of pattern suggests itself. Doing so, we find that 1+√3 + 2i converts to <3.3859, 36.206°>…approximately.

Yuck.

There’s no obvious way to get from <√2, 45°> + <2, 30°> to get <~3.3859, ~36.206°>. Not even if you write 1.414… for √2. You can add 1.414… and 2 and get 3.414…but that’s different from 3.3859. It’s close but not close enough. In fact if you actually did do a tail-to-head vector addition diagram, you saw why. The two “vectors” are in almost, not quite the same direction, so the sums of their lengths is not that much greater than the length of their sum. But it’s not the same. And of course the angle 36.206° has no obvious derivation from 45° and 30°. It’s not halfway in between or anything like that.

So it seems that, if you want to add x, y pairs, or complex numbers in Cartesian form…you’d better leave them that way. Adding those is easy. Adding r, θ pairs looks a lot more complicated. So, for addition, the Cartesian form is most convenient.

How about multiplication? Multiplication is defined for complex numbers (but defining it for vectors gets a bit dicey). Okay, so let’s do the multiplication.

(1 + i) x (√3 + i)

Remember for complex numbers, you use “FOIL” (first, outside, inside, last), a generalization of the rule that you multiply everything in the first vector, complex number, or whatever, by everything in the second, then combine like terms.

1 x √3 = √3 (first)
1 x i = i (outside)
i x √3 = √3i (inside)
i x i = -1 (last)

Gathering like terms together, we have (I’ll put parentheses around the real and imaginary parts for clarity):

(√3 – 1) + (√3+1)i.

This seems kind of almost-symmetrical, but it’s a mess, actually.

OK. It’s not obvious here how we’d use polar coordinates to multiply, so let’s just convert the answer and see what pops out.

The first component, r, is going to be the square root of the sum of the squares of the real and imaginary parts. That’s basically Pythagoras, r² = (real part)² + (imaginary part without the i)²

Squaring both pieces (again using FOIL) we get:

(3 – 2√3 + 1) [first part, squared] + (3 + 2√3 + 1) [second part, squared]

(Note that we don’t include the i in the squaring because we are trying to assess the sizes of the sides of the triangle, and those are all real numbers.)

And lo and behold the 2√3’s cancel and we’re left with r² = 8 = 4×2, so r = 2√2. Which is sort-of tidy. And the angle turns out to be (I won’t show the work, but I pushed it into a calculator even though I know the answer already because…well, you’ll see): 75°.

Exactly 75°.

OK now set <2√2, 75°> next to the polar version of the two pairs we were multiplying and see if anything strikes you as a pattern.

<√2, 45°> x <2, 30°> = <2√2, 75°>

Now if this were a video, I’d either wait a while or ask you to pause it and think about it.

But really…look. The rs multiplied, √2 x 2 = 2√2…and the angles added.

Geez, that is a lot less tedious than FOIL. In fact it’s almost as easy as adding the numbers is in Cartesian mode!

As it turns out, multiplying any complex number, by some other complex number, gives you another complex number with magnitude equal to the product of the two numbers, and angle the sum of the two angles. Or you can think of it as rotating the first number by the angle in the second, and then multiplying the two magnitudes.

So which mode is more convenient for multiplication? You’ve got it: the polar mode.

And as an extra bonus, we have a way to rotate a complex number. (And similar ways to rotate vectors.) Multiply it by another complex number with the desired angle. And unless we want the magnitude to change, make sure the “rotator” complex number has magnitude, r, equal to 1.

A “rotator” complex number like that will have, as its real component, the cosine of the rotation angle, and as its imaginary component, the sine of the rotation angle. Or in other words, <1, θ> a/k/a cosθ + isinθ will, by multiplication, rotate another complex number on the complex number plane by the angle θ.

This may seem like nothing more than a parlor trick for geeks, but it turns out to be very, very important in the technical realm, particularly with electrical and electronic engineering. Whatever it is you’re reading this on, ultimately depends on this fact.

OK that’s enough for now!

Fuck Joe B*d*n

Due to complaints about foul language, I’ve censored the most objectionable word in the title of this section.

B*d*n, you don’t even get ONE scoop of ice cream today.

(Please post this somewhere permanent, as it will continue to be true; the SOB will never deserve a scoop.)

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!