Today is Canada Day. Something once worth celebrating.

Oh, well.

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 2022 or 2024 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 $1,920.20
Silver $22.52
Platinum $972.00
Palladium $1,316.00
Rhodium $6,100.00

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

Gold $1,920.80
Silver $22.85
Platinum $913.00
Palladium $1,260.00
Rhodium $4,700.00

The platinum group metals are taking an absolute beating right now. Especially for palladium and rhodium.

Those metals’ primary use is industrial. Everything from catalytic converters to other uses as catalysts in chemical plants. A collapse in demand for these could indicate the actual productive economy is tanking.

A Small Science Post

I finally have a free moment to write this post…and midnight, Swamp Time, is only half an hour away.

So it will be short.

Valerie will not doubt express relief!

We left off having introduced i as the square root of minus one (√-1), pointing out that this number does not appear on the number line and therefore isn’t in the broad group of numbers that we call “real” numbers, which includes integers, fractions, irrational numbers, even the irrational numbers that are also transcendental. It’s therefore an imaginary number…but no, that doesn’t mean it’s made up. It ends up being very useful indeed not just to mathematicians (who work in a very abstract world that can sometimes be quite disconnected from reality), but to engineers, who absolutely must deal with reality.

Imaginary numbers are those which are multiples of i. Because you can do any sort of algebra or arithmetic on i that you can do on real numbers. But you cannot simply mix them with real numbers interchangeably; that’s bigtime apples-and-oranges. But nothing says you cannot make a fruit salad, where they each retain their identity but work together. When you do that, you have a “complex number.” For example, 5+3i is a complex number, as it has both a real part, 5, and an imaginary part, 3i. (But note that pure imaginaries and pure reals are also considered part of the complex numbers…as is zero. If this bothers you realize that reals can all be expressed as <real number> + 0i; we just choose to leave off the zeroed part. We do a lot of things like this, such as expressing 1x¹/1 as simply x. Similarly a pure imaginary number can be expressed as a complex number having zero for the real part.)

I mentioned earlier that i is not on the number line. I hope it’s of no surprise to you that no complex number that has an imaginary part is on the number line, either.

So when thinking about number lines, or things like number lines, aids to visualizing sets of numbers, how do complex numbers fit in? As it happens, it’s more the other way around. The real numbers, those on the number line, are part of the complex number set, not the other way around.

Think about the set of all possible complex numbers. The real part can be any number that actually is on the number line. The imaginary part can be i times any number that actually is on the number line. Any real part can be combined with any imaginary part. They are independent of each other…they are independent variables or they are orthogonal to each other.

If you have a STEM background, there’s probably something the size of a gorilla jumping up and down in your mind shrieking “Two dimensions!!! Two dimensions!!!” If you don’t, and you’re wondering why your STEM friends are acting so weirdly right now, well now you know about the jumping gorilla.

Because that’s exactly what we are going to do. The number line has one dimension, left-right. There is a complex plane with two dimensions, left-right and up-down (on the sheet of paper) or east west and north-south (if you’re imagining looking down on the plane from above).

So now we have a new mental picture, the complex number plane, where the old real number line runs side to side and is in the plane, and another line, the imaginary number line, runs vertically, and they intersect at 0 + 0i.

Here’s a picture, with a few sample complex numbers pointed out and ℝ and I axes labeled:

Next time, we’ll see how well it works. And it works very, very well, leading you to some very interesting places.

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 $1,959.20
Silver $24.29
Platinum $992.00
Palladium $1,449.00
Rhodium $7,100.00

This week, markets closed as of 3PM MT.

Gold $1,920.20
Silver $22.52
Platinum $972.00
Palladium $1,316.00
Rhodium $6,100.00

Wow, gold and silver have been taking a beating lately! Market manipulation? Or just fake good news about the dollar?

Going Out of Line

A lot of different categories of numbers have been mentioned here recently, from rational numbers, to irrational numbers, to transcendental numbers.

But those aren’t all the types. Not by a long shot.

Going the easy way first, we have a group called, variously, the counting numbers, the whole numbers, and the natural numbers. This last designation is a bit problematic because there are two competing definitions. One of them is that ℕ includes 1, 2, 3, 4, etc, and the other one starts with 0, then picks up with 1, 2, 3, 4…

I was taught that the “natural numbers” did not include zero, but that “whole numbers” were the natural numbers, with zero added in. But others got something different. Mathematicians have taken to writing ℕ₁ and ℕ₀ to distinguish the two possibilities.

One distinction that mathematicians like to make is whether a grouping like this is closed or open for some operation.

For example ℕ (either one) is closed for addition and multiplication, because you can take any two numbers from ℕ, add (or multiply) them together, and the result is a member of ℕ. However, ℕ is open (or just simply “not closed”) for subtraction and division. 5 – 19 = ?? 2 ÷ 3 = ???

Yes, there are valid answers to those problems, but they are not part of ℕ.

In early elementary school, of course, these are the only numbers you work with at all, so your teacher just told you that you couldn’t subtract a big number from a little one, and division was for later anyway.

Now there is a caveat. If you think of division as “quotient” and “remainder” (or “modulo”) then ℕ₀ is closed over division. [With of course the important caveat that you cannot divide by zero unless you are Donald Trump on a Batman day.] Because you can divide any ℕ by any other ℕ and you will get a whole-number quotient and a whole number remainder, both of them 0 or greater. But notice, ℕ₁ is not closed for division, because something like 3 ÷ 4 has an answer quotient 0, remainder 3 and 0 is not a part of ℕ₁.

Zero was the first addition to the set of numbers that pegged some people’s bullschiff meters at the time. Zero, to them, wasn’t a number, it was a lack of a number.

But arithmetic works far better with zero being considered a number than it does without it (and no I’m not just talking about the difference between working in Roman and ‘Arabic’ numerals–the Roman system didn’t have zero in it at all, much less the convenience of the way numbers can be written provided you have a zero). The fact is that people literally didn’t think of zero as a number in its own right, even after deciding it might be handier to work with “10” than “X”.

This was reflected in the way musical intervals were counted (a musical “first” is the same note, the second is the adjacent note and so forth, “first” didn’t mean the note immediately following the base note, because there wasn’t a concept of going zero steps). It even made its way into the Bible; the resurrection is depicted as happening on the third day…even though Sunday is only two days after Friday, and in fact based on times of day, it was less than 48 hours after the crucifixion. If this happened today the Gospel writers would probably have said “less than two days later…” to mean the same thing. Friday isn’t zero days after the crucifixion in the Gospels, it’s the first day. No concept of a zero.

In any case, we’re well past elementary school, and we have a way to close the number set, for subtraction at least, and that’s the negative numbers, -1, -2, -3, and so on. Add them to the set denoted by ℕ₀ and you have the integers, for which the symbol is, of course, ℤ. After all what else could the symbol for “integer” possibly be?

You can add, subtract, and multiply any integers together, and still get an integer, you can do division (again, no dividing by 0) with remainders, too.

You have two identities as well. There’s the additive identity, 0. Because you can add (or subtract) zero to anything…and it doesn’t change. And the multiplicative identity, 1. You can multiply anything by 1, and it doesn’t change.

You have inverses, too. Pick some number…oh, say 17. It has an additive inverse, -17, because when you add a number and its inverse, you get zero. And there’s also a multiplicative inverse (also called the reciprocal), the number you can multiply 17 by to get 1…and that of course is ¹/₁₇.

So this is the sort of things that mathematicians think about when talking about sets of numbers, and the operations that can be performed on them. If it sounds like this is all third grade stuff…okay, fifth grade stuff since long division is in there…that’s because we’re dealing with a simple case; these issues become a big deal when talking about vectors, matrices, and tensors.

(In fact when I was going through elementary school these “simple” rules plus ones like associativity [(a+b)+c = a+(b+c)] and commutativity [a+b = b+a] were put out there and I was wondering why they were harping on them so much. Well it’s because mathematicians need to consider this stuff for higher mathematics.)

What’s missing here? Fractions, of course! In essence any number you could get by dividing integers (again, not by zero, though you can divide zero by anything else) is some sort of fraction, or perhaps a compound number like 1¼, which you can get from 5 divided by 4. This is the set of rational numbers, symbolized with…wait for it…ℚ.

“Rational” comes from “ratio,” not our modern meaning of “rational” to describe the exact opposite of “woke”.

ℚ is closed under addition, subtraction, multiplication and division (so long as you don’t divide by zero).

You can exponentiate, i.e. raise these numbers to a power. Is ℚ closed for that operation? Well you can raise to any integer power, without a problem. Squaring, cubing, etc any of these numbers gives you another such number (and in fact that’s true even for whole numbers, like -42) and still be within ℚ. You can even use a negative number, because 5^-1 is simply ¹/₅. But that’s not all that is in ℚ. The fractions are in ℚ and if you raise a number to a fraction you will end up taking a root. The simplest most trivial case where ℚ “breaks” is 2^½, which is the square root of 2 and that is not a rational number, it’s an irrational number: 1.414… And here irrational means it cannot be expressed as a ratio between two integers.

Well. It seems like every time we introduce a new mathematical operation it “breaks” our number set and we have to extend it, doesn’t it?

We now have a set of the combined rational (ℚ) and irrational numbers. Does that have a name? Not really because there are still more numbers that cannot be written as fractions, and cannot be got at just by taking some root of a rational number. These are the transcendental numbers like e and π.

Lumping all of these together you get a set known as the real numbers, and that is symbolized by ℝ, so at least now we’re back to symbols that make sense given the name of the number set. Why “real numbers”? Well…read on, it will become clear presently.

ℝ is closed over addition, subtraction, multiplication, and division. It is still not closed over exponentiation, though. Wait, we’re no better off than we were with rational numbers ℚ??

Well we are, because as long as you are raising to an integer power, you can raise any real number to a power and you’ll still have a real number. So e²³ is a real number, so is e^-23, or -3³⁷ or even (-π)^-4. And you can raise any positive number to any real power, e.g., 27^e, 3.2^47/13. These are all in ℝ.

The problem arises when you try to take any root of a negative number (i.e., raise to a fractional power).

At the most trivial, the square root of minus one (√-1) is not in the real set. There is no real number you can multiply by itself to get minus one. Nor for any other negative number. You can try to take a crack at something like √-9, though:

√-9 = √9 ⋅ √-1 = 3 ⋅ √-1

You will end up with some answer, multiplied by this non-existent “square root of minus 1.” It’s like taking the square root of a negative number almost works like taking the root of the same number as a positive number–you can do the same arithmetic to extract the root, but then this bogus “square root of minus one” piece which just hangs around as some bit of ballast.

OK, let’s take a bit of a breather.

Another one of those things I saw in elementary school which made me think “why bother” is the number line.

Here’s one for ℕ₀:

Notice it seems to end on the left hand side at zero, but it continues after 100, implying you can get higher and higher numbers. But this is clearly not a number line you’d want to use for negative numbers. When those come up you suddenly see a whole left side of the number line with the negative numbers on the left, and it open on the left too, implying numbers can get lower and lower than that.

And fractions, of course have their places between the integers. The irrational numbers actually fit in between fractions. You can write down some fraction arbitrarily close to, say e but not quite reach it, from both above and below. You can narrow the gap with increasingly ridiculous fractions, but you can’t close it, so e, π, √2 and all the other irrational numbers are wedged in between the rational numbers.

Another way to think of ℝ is as the complete set of numbers that can be represented on the number line. Negatives, fractions, irrationals, transcendentals, you-name-it, it’s on there…somewhere.

It’s just a shame you can’t take the square root of a negative number. It’s not much consolation to know that you can take the cube root of a negative number just fine: -8 = -2 x -2 x -2, therefore ∛-8 = -2.

Actually you can do it, but you’re going to have to expand your notion of what a “number” can be, literally into another dimension.

Let’s just define √-1 to be i. (Lower case “I”.)

Why i? Because someone decided to name this number “imaginary.”

You can multiply i by a real number…like 17…to get 17i, which is still considered an imaginary number. You can square i, of course, to get -1, you can cube it to get –i. The fourth power of i is actually +1. What happens if you want the reciprocal of i? In other words, what is 1/i?

1/i = 1⋅i / i ⋅i = i / i² = i / -1 = –i.

You can basically just do algebra on imaginary numbers, and when you square i, replace with -1. If you see higher powers of i, break them down into products of squares and proceed accordingly:

i³ = –i, i⁴ = 1, i⁵= i,

and so on.

Because you can do basic algebram it turns out that anything you could do to real numbers you can do with i.

But it doesn’t show up anywhere on the number line. And because this critter doesn’t appear on the number line, it isn’t a real number. It’s not real, so it’s imaginary. (And this is why they call ℝ the real numbers, by contrast with the imaginary ones.)

Imaginary is a crappy name, honestly, because all numbers are abstractions with no concrete existence. If you want to disagree with that last statement, point to three. Not three dogs, or three amigos, but just to three…and I don’t mean a copy of the symbol ‘3’ either, it’s a symbol for the abstract concept.

So “imaginary” numbers are no more imaginary than “real” ones, nor are “real” numbers any more real than “imaginary” ones.

But that doesn’t stop people from claiming mathematicians are just engaging in mental masturbation “imagining” things.

Well, no. That’s simply ignorant.

The fact of the matter is all electronic engineering, and almost all electrical engineering as well–in other words the tech you are reading this little ramble on– absolutely relies on these so-called “imaginary numbers.” [The only thing is, since i is current in electrical engineering, when electrical engineers want the square root of minus one, they write it as j. Which drives math teachers buggy, and of course probably the biggest swath of people in many college level math classes are electrical engineering students; who end up having to flip back and forth depending on whether it’s a math course or an engineering course.]

But we’re stuck with the name “imaginary”, so I will quit putting it in quotes.

Is it possible to have combinations? A “thing” that has both real and imaginary parts?

Not just yes, but hell yes. But if we do it, we end up with a number that has two distinct pieces, not just one. And since that sounds a bit complicated, these are the complex numbers, and their symbol is ℂ.

So how do they fit together? How do you combine them?

Not to sound flippant, but you combine them by, well, combining them. You have a nice imaginary number like 4i. And a solidly real number like -2.

To combine them, just do this:

-2+4i

There’s your complex number.

If these form a valid set of numbers, then what operations is ℂ closed over?

All of the ones I’ve mentioned. Even exponentiation, but I will have to demonstrate that some other week.

But how do they work?

Addition seems straight forward. Let’s take that complex number above and add it to 3-7i.

Hmm… (-2 + 4i) + (3 – 7i). Now what? Rearrange things. Combine the real and imaginary parts like this: (-2 +3) + 4i – 7i) and you see that you get 1 – 3i. Or a bit more succicntly: Just add the real parts together, and the imaginary parts together.

What about multiplication? Remember imaginary numbers can be mangled with algebra the same as real numbers, and so can complex numbers. If you want to multiply the two numbers above:

(-2 + 4i) × (3 – 7i)

You can “foil” it just like you would do with two binomials. FOIL, of course, stands for first, outside, inside, last.

First: -2 × 3 = -6

Outside: -2 × -7i = +14i

Inside: 4i × 3 = +12i

Last: 4i × -7i = -28 i² which is to say -28 × -1 = 28

Combining bits and pieces, your answer is 22 + 26i.

And it’s now 13 minutes after 10 my time so I need to hit “post” and let you all bang your head on the wall trying to understand this! Much, much more about complex numbers…I think (unless you already know it) you’ll be a bit surprised!

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

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

Gold $1,959.20
Silver $24.29
Platinum $992.00
Palladium $1,449.00
Rhodium $7,100.00

Gold took a thumping earlier this week, but recovered almost completely. Silver, similarly. Platinum’s beating was more permanent. Palladium is actually moving up. Rhodium is flat at levels that, though quite low compared to a couple of years ago, still make me wish I had bought some in $400/ozt days.

The problem was, I would have needed to buy it a kilogram at a time, so minimum investment was north of $12K which I sure as heck didn’t have back then. But when it was at $30K I would have made nearly two orders of magnitude on it.

Speaking of orders of magnitude…

Logarithms and Slipsticks

“Slipstick” was apparently American slang…not recognized in the UK. So apparently part of the reason we fought the War of Independence was so that we could call slide rulers “slipsticks” especially when the middle bit fell completely out and clattered on the floor (and probably got a ding that would make it harder to use in the future).

I said last week I wasn’t going to cover logarithms, because they were off the path I wanted to follow in this little math “mini series” (all series are mini- compared to the 30 part physics series plus additions). But there was so much discussion. The subject was beaten until the horse was dead.

Never wanting to let a dead horse remain unpureed, I thought “maybe I should cover logarithms after all.” You didn’t ask for it…but you got it.

What a Logarithm Is

A logarithm is the reverse of raising a number to a power.

To recap: If I have some number…call it b, and want to raise it to some power, like, say, 3, I need only multiply it by itself 3 times: b✕b✕b = b³. (It’s slightly preferably to think of it as starting with 1, then multiplying by b three times.) We were able to establish that even though you can’t multiply things a fractional number of times, it does make sense to speak of a negative power, or a fractional power, or even an irrational power [remember that an irrational number is just one that can’t be expressed as a ratio of two integers, it is not ratio-nal]: e.g., b^-3, b^2.4, b^sqrt(2). Nor does b have to be a nice clean integer, it can be a fraction or an irrational number. (Though if we’re going to play with fractional/irrational powers, we should stick to b being a positive number of some kind.)

Some notational matters to take care of (and sorry to those of you who find this stuff obvious). Raising a number to the second power is often called squaring it; raising a number to the third power is often called cubing it. You’ll note I’ve used the ✕ symbol to denote a multiplication. It’s also possible to use this symbol: · and I will do so. And when dealing with symbols (as opposed to actual numbers) oftentimes you won’t see a multiplication sign at all. So: ab = a·b = a✕b. You’ll also see b^x meaning “b raised to the power x” but when superscripts are unavailable (or someone is lazy) you will see b^x to mean the same thing.

OK with notation out of the way, the key insight here is that, b raised to some power x, multiplied by b raised to some (other) power y, is the same as b raised to the power of x + y. In other words:

b^x·b^y = b^x+y

Note that b (the “base”) has to be the same for this to work. (That’s implicit in the formula, because b can’t change meaning halfway through it; that’s not necessarily implicit in the English language version if not stated (and listened to) precisely enough.)

This seems like something that might only be useful while doing algebra, but no it isn’t.

Why Anyone Ever Gave a Damn About Logarithms

Consider that, since we’re allowing any “real number” power, from minus infinity to plus infinity, including fractions and irrational numbers, any positive number can be expressed as b^x. Pick your base, b, and there is some x you can raise it to. so as to get to your positive number. More than likely x will be an irrational number, but sometimes it’s not. And some other positive number can be expressed as b^y. But remember that you can multiply two exponents by simply adding them together. And you can divide by subtracting.

So if your first number is (capital) X and is equal to b^x, and your second number is (capital) Y and is equal to b^y, then:

X·Y = b^x+y

And

X/Y = b^x-y

Now, given how much of a pain long-hand long-division is, perhaps you can see an application for this…especially more than 50 years ago when electronic calculators could cost a significant fraction of the price of a car, or 60 years ago when they could cost much more than a car.

Little x and y are logarithms. They are the number, you raise your base to, to get some other specific number. so x is the logarithm of X, and y is the logarithm of Y.

It does matter, a lot, what your base b is. So technically, one should say that x is the logarithm, base b, of X, and y is the logarithm, base b, of Y.

[In practice, only three different bases are used even somewhat commonly. The first is e, and logarithms to the base e are called natural logarithms, and are abbreviated ln, as in lnX = x, meaning that e^x = X. The other base you will see a lot is 10 (ten), and that’s usually abbreviated “log”, so logY = y meaning that 10^y = Y. However, sometimes mathematicians will write “log” when it’s the base e, so there’s now an ISO standard (a good one IMHO) that log base e is to be written ln, and log base ten is to be written lg, and any other base, like 2 (the third most commonly used base) should be written like log₂. (Logs base ten and e can also be written that way, of course.)]

So in order to multiply or divide, take the logarithm of X, and the logarithm of Y (in some base…it doesn’t matter, just so long as it’s the same base both times). Add (or subtract) those two numbers. Then raise b (the same base) to that power. The answer is your result.

That doesn’t seem to gain you much. Taking a logarithm, then raising a number to a likely-irrational power, sounds like more work than just the multiplication or division would be.

Except that you can make a table, and look up the logarithms you need in it. Then when you have the logarithm of the final answer, you can use the table backward…find that value in the table, and go back to the number it “stands for.”

But wait, you object: You can use a multiplication table as well!

Yes, you can. But that table has to be two dimensional.

A logarithm table with ten thousand entries, can be used to multiply (or divide) any one of ten thousand numbers, by any other one of those thousand numbers, because you look them both up separately. As long as your result appears in the table (and there are ways to ensure that it will, see below), you can find your answer in the table and read backwards to find your number.

To do the multiplication of any one of those ten thousand numbers, by any other one of those ten thousand numbers, you need to look up a pair of numbers both at once, and you need a table 10,000 x 10,000 in size, and that is a hundred million entries. Do you want that thing straining your bookshelf? And yes; it would be in a book…or many books in this case. Unless you want to paint the thing on your living room floor. (Sorry, no one is that hard core.)

The next objection, though, seems to be a strong one: Numbers cover an infinite range (even if they are positive), they can also be arbitrarily small. What table is going to cover all of the useful values from ¹/_{100,000,000,000,000,000,000,000} to 100,000,000,000,000,000,000,000? And what if you need to go beyond that range? (Note that this applies to multiplcation tables, too…only much worse because the table has to quadruple in size to double its range.)

The answer to that involves using logarithms base 10, also called common logarithms, and abbreviated (often) log, but now officially and pedantically, lg.

Consider: Here is the common logarithm of 15 (to five decimal places): 1.17609.

Now, here is the common logarithm of 150: 2.17609. And 1500: 3.17609. And 1.5: 0.17609.

Do you see the pattern? The logarithm base 10 has a whole number part which differs for each of these numbers, but the part after the decimal is always the same based on what the leading digits are, for digits 1 and 5, it’s always going to be .17609. The number in front of the decimal point on a log (base 10) is called the characteristic. The part after the decimal is the mantissa.

[What about 0.015? It turns out the log of that is -1.82391. What happened? But that is -1 – .82391, or -1 plus negative 0.82391. Rewrite that as -2 + .17609. In other words your characteristic goes negative, but your mantissa should remain positive. Logarithms like this would be written with a bar over the characteristic, and so you’d say the logarithm of .015 was “bar 2 point .17609” meaning the characteristic was negative but the mantissa was positive. That sounds complicated, but it actually simplifies things.]

This means all we have to do is create a table covering .1000 through .9999 (8999 entries) and it can cover all cases. Or if you want more accuracy, 0.10000 through .99999. And so on. In other words, you just need a table for the mantissa. If you can remember the characteristics and add or subtract them, you can do that in your head. (However beware of carrying and borrowing, if your logs are 1.5 and 1.5, your mantissas add to 1.0 and you’d better add that one to the sum of your characteristics; 1.5 + 1.5 = 3.0.)

If you’re wondering why that table doesn’t start at .0000, well the logarithm of zero is undefined. (Fortunately, multiplying by zero is pretty simple, dividing by zero is forbidden; maybe Trump can do it while he’s being Batman.) And a number like .0001 can simply be expressed as .1 but with a different characteristic. So no entries are needed for leading zeros.

Before the days of cheap electronic calculators, no engineer or scientist tried to do precise work without a table of logarithms, generally a fairly thick book because the more entries, the better. It makes for boring reading, but engineering was utterly dependent on logarithms, and for precision the book was necessary.

The table is (structurally) one dimensional: you look up one number and get one answer. However, formatted in the book, you’d see something looking pretty doggone two dimensional. Here’s an example, one page out of an eighteen page table:

This was done to save space, paper and bindings. This is a four decimal table, covering 1000-9999, this page only goes to 1500 so we need another 17 pages.

Nonetheless, we can work an example (as long as I’m careful to pick one where both numbers and their product will show up here.

What is 11 x 13?

Start by realizing that 11 is between 10 and a 100. Its characteristic will be 1, because it’s more than 10¹ but less than 10². Likewise with 13. Remember those characteristics.

To look them up seek out 1100 in the table above. It’s in the row marked 110 and the column marked 0 (110 (as a string of symbols) with 0 tacked onto the end is 1100.) That’s 04139, meaning our mantissa is 0.04139 (space is saved by omitting the leading zero, which is always a zero, and the decimal place, which is always there). Then seek out 1300, the same way, and it gives 11394.

OK, add the two mantissas, 04139 and 11394 to get the mantissa of the answer, 15533. Now you can hunt for it in the body of the table, and you’ll see that it’s in row 143, column zero. Almost. The number shown is 15534, but that’s much closer than the numbers either side of it (15503->1429 and 15564->1431) So your answer is 1430.

But you’re not quite done. This number did not “roll over”, the sum of the mantissas stayed less than 100000. So simply add the two characteristics, 1+1 = 2. Your answer is greater than 10², but less than 10³. So with an answer of a “bare” 1430 you need to stick the decimal point after the 3, to get 143.0, a number between a hundred and a thousand.

And indeed 11 x 13 = 143.

There are more nuances to the table, including ways to squeeze one more digit of precision out of it by interpolating (and the table gives you little “cheat tables” off to the right margin to help with that). But you should get the general idea and I don’t want this to be a class on how to use the table.

A table is good for lots of precision, one part in ten thousand or even a hundred thousand with a big enough table (and even better if you interpolate).

But it takes time, you have to thumb through the book three times for every multiplication, and you have to very carefully guard against reading from the wrong column or row (and that is very easy to do!).

[By the way, one of my tables…is actually an Army technical manual from the early 1960s. It’s full of logarithms of sines and cosines…because artillerymen need to multiply sines and cosines of angles. In 1960 the army was still focused on breaking things and killing people, rather than on “woke” crap…so if they put a technical manual’s cover around tables of logarithms and tables of logarithms of trig functions…you can bet this stuff was genuinely useful for things Deplorables should care about.]

Mechanical Help for Addition

If you don’t want to be as accurate, you can use a mechanical aid…its precision will be limited by how good your eyeball is, and how many marks are on the mechanical aid.

So how would such a thing work?

Let’s start by imagining a mechanical aid for addition and subtraction.

You can sort of fake this with two rulers…for this centimeters is better, but just make sure both rulers use the same unit of measure.

What’s 5.5 + 3.7? Set your first ruler down on the desk. Find 5.5 centimeters on this ruler (and now you see why metric rulers are better for this). Now set the second ruler on top of the first ruler, so that zero on the second ruler is directly on top of the 5.5 on the first ruler. Now on the second ruler, find 3.7. Now very carefully figure out what number on the first ruler is under that 3.7, it should be 9.2 centimeters.

Below is a schematic diagram…pretend they’re transparent rulers and I’ve flipped one over (so the numbers are backwards). The black line is how you read off the answer.

Similarly, to subtract 4.6 from 6.2, put the 4.6 on one ruler, over the 6.2 on the other, then go to the zero point on the first ruler; it’s over 1.6 on the second ruler. You’ve now subtracted lengths to do subtraction.

What you’ve done is added two lengths together to get a third length, but it’s on a ruler so you’ve automatically measured that length, and it represents your answer.

A slide rule simply has two outer “rulers” and a middle “ruler” that slides back and forth between them, so you can readily position 0.0 on the slidey part over 5.2 on the stationary part, then go look to see what is next to the 3.7 on the slidey part.

Slide rule manufacturers even have standard names for those two scales, they call them A and B (and in fact they’re often not present on a real slide rule).

Mechanical Help for Multiplying and Dividing

OK, but I wanted to multiply and divide, not add and subtract.

Yeah, but I really hope you didn’t sleep through the part about how you can add logarithms to multiply numbers.

So use your A-B scale slide rule to add the logarithms–or rather the mantissas. That way all you have to do is find a trained chimp to look up the logarithms in the book. (You might need a slightly smarter chimp to translate the result back because you will likely want to interpolate. Since chimps might fight over the books, you will want two copies of the table.)

But…we’re missing out on the true power of the thing.

Let’s multiply 800 x 60. Log(800) is 2.90309. Log(60) is 1.77815. So line up your rulers at 9.0309 and 7.7815 centimeters. Actually, you’re going to want meter sticks, to get more precision…and since this will run off past the end of the ruler you’ll want a double length ruler, if one of them has twice the range of the other it can’t “overflow” but if your answer is in the second half of the big stick, you need to add an extra one to your characteristics, you get 4.68124.

However…ANY time you go to 7.7185 on the ruler…it’s because you’re really interested in 6.0000. and ANY time you go to 9.0309 on the ruler, it’s because you find 8.0000 fascinating at that moment. That’s true whether the 6 (or 8) is one of the numbers you’re multiplying…or the answer (product).

So take a bit of whiteout, put a dab on that spot of the ruler, and write a dang 6 (or 8) there! Let the ruler do the lookup for you! You want an 8, find it on the damn ruler instead of discovering you want 9.03!

Go through and re-work the scale so it’s not marked in logarithms, but rather the numbers they are the logarithms of. You end up with a scale that runs from 1 (the logarithm of which is zero) through 10 (the logarithm of which is 1). But you know how to make that work for numbers outside that range, right? Characteristic and mantissa.

Now you can go directly to the slide rule, find your numbers (not their logarithms) marked on the ruler where their logarithms would be ordinarily, slip the sticks as described, and your answer is the product, no need to go do a reverse-lookup in the tables. You follow the same procedure you did for adding…but you get the product not the sum. Pretty slick, huh?

Now of course writing the scale this way gives you this funky-looking scale where the numbers close to 1 are spaced far apart, 3 is almost halfway between 1 and 10, and 7, 8 and 9 are bunched up together at the right end. This is called a logarithmic scale, because the numbers are spaced according to their logarithms, rather than their actual values (which makes the ordinary arithmetic scale).

These paired logarithmic scales, called C and D by the manufacturers (who apparently all followed the same standard), appear on just about every slide rule ever made.

Logarithmic scales in general (and not just on slide rules) have all sorts of other interesting properties, to the point where graph paper can be set up this way instead of as an even “grid,” and people used to seek that out, calling it “log paper.” [If Johannes Kepler had had some and a bit of training on what a straight line on such paper meant, he could have saved himself YEARS of work.] Well, maybe some other time.

Well…just one. If you double the length of your 1-10 logarithmic scale, you can simply duplicate the first scale into the second. The second part actually runs from 10 through 100, not ten through 20. 20 is actually at 2 on the second half of the scale. The distance from 1 to 10 is the same as the distance from 10 to 100, which is the same as the distance from 100 to 1000. Or you can check, the distance from 1-2 is the same as the distance from 2-4, which is the same as the distance from 4-8. These actually represent different bases; the difference in the bases is manifested in the relative length on the rule. A slide rule for actual use by engineers will have the C/D scales run from 1 to 10.

By putting other scales on slide rules, you can do square and cubes, and square and cube roots on them (and you don’t even need to slide the stick). A scale, the same physical length as the main scale but which runs from 1-100 will help you with squares and square roots, just for instance.

The downside is, no slide rule you can hold in your hands can reliably give you more than three digits of precision. (If you’re lucky and your numbers are closer to 1 and 2 than to 8 and 9, you can often squeeze another digit out of the thing.) And it takes skill to truly use one…there is a video out there of someone raising a fractional number to a power that is itself the quotient of two non-whole numbers, and he did it pretty fast, in fewer steps than I would have guessed. I am no slide rule virtuoso by any means…though I could use one half-assedly if I had to.

On the upside the user had to know what you were doing. You had to keep in mind what numbers you were actually multiplying and would likely catch one that was off by a factor of ten…or a hundred…because it was on you to track it. Unlike the guy asleep behind the modern adding machine/cash register/calculator who might not have any idea he accidentally fatfingered an extra zero as he tells you your burger and fries will cost you $67.90. Yes, that’s an advantage, not a disadvantage, in spite of the fact it meant that a lot of people had to learn more math in order to use the things. And by “learn” here, I mean truly understand it, not just be able to parrot it until memory-holing it after the exam.

Love them or hate them, slide rules are no joke…we won two world wars with engineering made possible by them–and understanding them can deepen your appreciation of mathematics.

OK…that’s enough digressing (I hope). Back to the “main” story next time.

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 !!!