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 $1,948.50
Silver $23.68
Platinum $1,015.00
Palladium $1,459.00
Rhodium $7,450.00

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

Gold $1,961.70
Silver $24.38
Platinum $1,018.00
Palladium $1,356.00
Rhodium $7,100.00

Silver up a decent amount. Gold a bit less so (and it went down a few bucks on Friday). Palladium and rhodium continue to slide which either means, someone is finally mining the stuff and meeting the demand, or…the economy is in a world of hurt. (The problem is they tend to piggyback on platinum mining, and no one is bothering with platinum because it’s been a glut on the market for years.)

Arbitrary Powers

No, this is not political, even though our would-be overlords are exercising arbitrary powers.

The last two posts on e and π have probably bent a few minds as people read them. I talked about some pretty wacky arithmetic…and it is arithmetic. Some few people here might not know what fractional and negative powers mean, or that they can even exist, so I’m going to take a break and bring ’em up to speed. Then I’ll bring in the Taylor series.

It seems like it would be a bit tricky multiplying an irrational number by itself, or even by some other irrational number. If it’s a rational number, even a repeating decimal, you can do the multiplication with fractions (where you get an exact answer), then if necessary convert to a decimal. But when the number doesn’t end as a decimal and can’t be represented as a fraction what do you do? You can just take it out to about twice as many digits as you need, do the multiplication, then throw away the digits you don’t need….only some of them at the end were wrong anyway. But that’s not really satisfying to mathematicians (engineers settle for it, because this is the real world and computers can only go so far for rendering numbers–there is a whole branch of mathematics devoted to estimating the errors in approximations).

And especially in the post on e, I talked about raising numbers to fractional powers. OK…at first blush, that is just plain screwy. e you might have a grip on. e²…well you multiply e x e. And e¹⁵…well it’s a lot of work and even the most sadistic math teacher wouldn’t expect that done longhand, but you just multiply 1.0 times e, then repeat..a total of fifteen times. But what in the heck does it mean to raise e to the 2.4th power? e^2.4 is 1.0 times e…2.4 times. But multiplication is a single operation, you can’t do 4/10ths of a multiplication. It’s either done or it’s not done. And that’s not because e is irrational and transcendental, you have the same issue with 2^x.

A similar question arises when someone talks about raising a number to a negative power…what does it mean to multiply 1 by x…negative three times? That one’s a bit easier, and I’ll give you a hint up front…division is the opposite of multiplication. Read on for the answer.

Yet with all these questions about what powers other than 1, 2, 3, etc. mean, you see a nice smooth graph of e^x, like this one, with values drawn in between the integer powers:

You’ll note that the line goes through x=1, x=2 and so forth as if it’s meaningful to raise e to any power…fractions, compound numbers, irrational numbers, transcendental numbers, you-name-it. How do we know that line is smooth between e¹ and e², e² and e³, and so forth, instead of jiggling all over the place in some wacky way that only just has to cross through the integer powers?

Well, it seems fairly intuitive that e^2.5, whatever the heck it means, should be somewhere between e² and e³. And that e^2.25 really ought to be somewhere between e² and e^2.5. Follow that one far enough and you have a plausible argument that whatever that line should do, it should climb the whole way between e squared and e cubed, and not jiggle around. That’s not a solid proof, but a fairly strong suggestion. And of course that still leaves the possibility of the line being straight or even convex upward between the knowable, sensible points.

But we can actually go there, in a much more rigorous way–and the graph has it right. Nice smooth curve.

Consider what it means to raise some number, like, say, two, to an integer power…like, say, ten. In other words, 2¹⁰. That;s simply 2·2·2·2·2·2·2·2·2·2 = 1024 (and that’s where You-Know-Who’s fraction of Indian (feather, not dot) ancestry comes from).

What happens if you multiply 2⁴ by 2³? Let’s see…that’s 2·2·2·2 multiplied by 2·2·2…which is to say, 2·2·2·2·2·2·2 or 2⁷. And 7 happens to be 4 + 3. That’s not a coincidence. Provided you’re working from the same base, here 2 in both cases, (rather than doing 2⁴ times 3⁵, where they are different) you simply add the exponents. That’s a quick shortcut in algebra, where you might have to multiply x⁴ by x³. You just write x⁷. (But if you’re faced with x⁴ multiplied by y³, you’re stuck with x⁴y³.)

You can similarly divide by subtracting exponents, because dividing is the opposite of multiplying, and subtracting is the opposite of addition. So: x⁴/ x³ is x^4-3 = x¹ = x. Or written out, x·x·x·x/x·x·x. The three xs on the bottom cancel out three xs on top leaving you with x. And if the latter number is bigger than the first number, you end up with a negative power. That means more xs on the bottom, and (if you’re dealing with x^3-4, you end up with x^–1 = 1/x.

So a negative power is just the reciprocal of the same positive power. Thus x^-3 is 1/x³, etc.

And you now know that any number raised to the zeroth power is 1. In other words, x⁰ = 1, no matter what x is. Don’t believe me? Do some algebra. Divide x² by x² to get 1, then do the same thing by subtracting the exponents.

OK, now for the next trick. There is a number–let’s call it s, which, when you square it, gives you 2. So s² is 2. Or, s·s = 2. Or strictly speaking, s·s = 2¹. But we’ve already seen that you can get a number, raised to a power, by raising that number to other powers and adding the powers. So, you could figure s is two raised to some power, and multiply them together, like this 2^p·2^p=2¹. So what is p then? What is the one number, which when added to itself equals one? That’s ½.

So we have: 2^½·2^½=2¹.

Of course, s, which is equal to 2^½, is simply the square root of 2. So…raising a number to the 1/2 power gives you its square root. And raising a number to its 1/3 power is the same as taking its cube root. And so on. That means to take the nth root of a number…you can divide the exponent, whatever it is, by n. The cube root of some number, x, is symbolized by first writing x as x¹, so that it has an exponent…then dividing the exponent by 3.

So do you want to know what 2^2.4 is? Now we can figure it out. It’s 2² x 2^1/5 x 2^1/5 because 2.4 = 2+1/5+1/5. Or you can treat it as 2^12/5. To compute that, multiply 1 by 2…repeating 11 times for a total of 12 multiplications. Then take the fifth root.

No, I am not suggesting you do this, but at least now you can see what a calculator might be doing behind the scenes when you enter 2, then x^y, then 2.4.

How about raising the power of some number, to yet another number? Like, say, raising 2 to the third power, then raising that to the fourth power? In other words, (2³)⁴? Well, let’s see, that would be (2³)·(2³)·(2³)·(2³) which we already know you can figure out by adding 3+3+3+3 to get 2¹². But it’s also multiplying 3 by 4 to get 12. So when you raise a power to another power, you multiply the two exponents. (And again the base…the number you’re raising…has to match–in this case it will match.)

I said a couple of paragraphs ago that your calculator might be raising a number to a power, then taking some root of it, to compute a fractional power. But it’s probably not. Taking some weird root of a number is hard. (It’s easier using certain tricks, but still…) There is instead something called the Taylor Series. And unfortunately, deriving it requires calculus, but it can be used for any “infinitely differentiable” (i.e., “smooth-curve”) function.

You use it to approximate the value of some function that might be otherwise very hard to compute. Just run it out as far as you like; eventually you’ll see your answer isn’t changing enough to change the last digit on the calculator’s display, and you stop. (That’s called “converging on the value.”)

Here’s the full Taylor series. Don’t worry too much about it; it’s about to get much simpler.

(For calculus geeks and the brave: A is some number you want to start from; most of the time that just gets set to zero and when you do that, it’s called the Maclaurin series. The fs with all the tick marks are derivatives of the function f. And the ! notation means to multiply a number by all the smaller numbers, so 6! is 6x5x4x3x2x1=720.)

As I said, you can use any smooth function here. And it just so happens that raising a positive number to the power x gives you a smooth function. So we can use the Taylor/Maclaurin series.

If that number you’re raising happens to be e then it gets very simple. The Taylor series for e^x simplifies as follows if a=0…because the derivatives of e^x are all themselves e^x. So you’re taking e^x of zero…over and over again. But that’s just one. And with a equals zero all of those (x-a)s become simply x. So you’re left with:

(Sorry if you don’t know calculus…you can take my word for it…or dig, somewhere. If you can’t dig, ask other people who know calculus to confirm I’m not just some guy on the internet bullshitting you…rather, I’m just some guy on the internet actually giving you the straight dope. Will miracles never cease?)

Notice One Very Cool Thing: You’re computing e^x without actually using the number e. And you’re doing it without having to take some weird root of the number for a fractional case. And x can be a fraction, a compound number, the square root of 2, some negative version of any of these…you name it! It’s all multiplication, division, and addition. You just have to keep going until it converges. Which, granted, might be a long time.

Yes, it’s a lot of arithmetic…but arithmetic is what calculators do.

I just set this up in a spreadsheet. Each term is simply x/n times the previous term, then you add them all up. So if I raise e¹, it converges on 2.718281828 after twelve steps (and you’re just adding 1+1+1/2+1/6+1/24+1/120…). So you can compute e this way, as far out as you like. However, if I raise e¹⁰, it takes 35 steps to converge on 22,026.46579. The farther away from zero you get, the longer it will take to converge. Your calculator would probably take some shortcuts; I could guess what they are (and bore you, if I haven’t already) and probably be wrong anyway.

OK, well that’s bloody nice, you might be saying. But what if I want to raise some other number to a fractional exponent? Like, say, ten? Surely ten is a more useful base than friggin’ e is! So….just for an example, what’s 10^3.7?

This series can help you do that!

There is some number, call it q, that if you raise e to that power, you get 10. In other words, e^q=10. Or in yet other words, q is the natural logarithm of 10, q = ln(10). (Apparently ln was chosen as the symbol for this by the French…or some other language that would want to put “natural” after “logarithm.”)

[Logarithms deserve their own post…and they’re not going to get one, at least not in this little miniseries. Suffice it to say you can figure out what two numbers multiplied together is…by adding their logarithms. Calculators probably use them a lot, too, inside and behind the scenes.]

How does that help us? Well, your original problem, “what’s 10^3.7?” is equivalent to “what’s (e^q)^3.7? But we already know that that is the same as: e^q·3.7. Remember how you multiply exponents, to take the exponent of an exponent? And we know how to raise e to any power; use the series!

You just need to know what the natural log of your desired base is. And your desired base can be anything…a fraction, an irrational number…you name it, as long as you can get there by raising e to some notional power, you can use it. You just need to know what that notional power actually is.

Well, guess what, there’s a Taylor series for that too, a series for ln(x). I’m not going to hit you with that, though, because it won’t matter down the road. Suffice it to say natural logs are readily obtainable…naturally.

OK…so where are we?

We went from wondering how the heck you can raise numbers to negative and fractional values, and wondering how to raise an irrational, even transcendental, number like e to a power…to realizing that fractional and negative powers actually do mean something, and being able to raise e to any fractional or negative value without even having to use the number e to do it, using nothing but multiplication, division and addition. Pretty slick. And going from there to being able to raise any real number to any real power.

You might not be impressed but your calculator loves it.

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 !!!