Radical Maths

This little puzzle popped up the other day and I thought it would make a good, simple example of how to deal with square roots.

First, a little terminology:  The word for root in latin is radix from which we get words like radius and radical. Mathematicians refer to the square root symbol √ as the radical symbol, and the stuff we’re taking the square root of they’ll refer to as what’s under the radical.

We start with this.

\sqrt{x + 15} + \sqrt{x} = 15

We can get rid of square root symbols by just squaring what’s under them, which will leave just the terms under the radical and nothing else.   Since this is an equation, we’ll need to do this to both sides:

(\sqrt{x+15} + \sqrt{x})^2 = 15^2 = 225

When we multiply this out, it looks like a real mess:

x + 15 + 2(\sqrt{x+15}\sqrt{x}) + x = 225

We can combine the two x’s to make this

2x + 15 + 2(\sqrt{x+15}\sqrt{x}) = 225

But, it isn’t all that much simpler.  Let’s try another approach.   Having two different radicals on the same side of the equation is what makes this messy, so, let’s move one of them to the other side of the equals sign.   We can move the \sqrt{x} term by subtracting it from both sides.

\sqrt{x + 15} = 15 - \sqrt{x}

Now let’s square both sides of the equation like we did before and see what we get:

x + 15 = 225 - (2)(15)\sqrt{x} + x = 225 - 30\sqrt{x} + x

I know, it doesn’t look all that much simpler, but, notice how there is a solitary x term on both sides?  We can get rid of that by subtracting x from both sides:

x + 15 -x = 225 - 30\sqrt{x} + x -x

leaving us with

15 = 225 - 30\sqrt{x}

It’s looking easier already!   Let’s subtract 15 from both sides, and then add 30\sqrt{x} to both sides.  We get

30\sqrt{x} = 225 - 15 = 210

Now we have simple terms for both left and right hand sides of the equation.   Divide both sides by 30 to get

\sqrt{x} = 210 / 30 = 7

Square both sides (yes, again) to get rid of the radical, and we end up with

x = 49

Let’s see if this is right.  Plug 49 in for x in the original equation:

\sqrt{49 + 15} + \sqrt{49} = 15

We can add the 49 and 15 under the first radical to get

\sqrt{64} + \sqrt{49} = 15

The square root of 64 is 8 (8 x 8 = 64) and we already have square root of 49 being 7, so

8 + 7 = 15

So, the answer checks out!

Totally RADICAL, eh?

What does the Quran really say about a muslim Woman’s Hijab? (TED Talk)

In recent times, the resurgence of the hijab along with various countries’ enforcement of it has led many to believe that Muslim women are required by their faith to wear the hijab. In this informative talk, novelist Samina Ali takes us on a journey back to Prophet Muhammad’s time to reveal what the term “hijab” really means — and it’s not the Muslim woman’s veil! So what does “hijab” actually mean, if not the veil, and how have fundamentalists conflated the term to deny women their rights? This surprising and unprecedented idea will not only challenge your assumptions about hijab but will change the way you see Muslim women.

Samina Ali is an award-winning author, activist and cultural commentator. Her debut novel, Madras on Rainy Days, won France’s prestigious Prix Premier Roman Etranger Award and was a finalist for the PEN/Hemingway Award in Fiction. Ali’s work is driven by her belief in personal narrative as a force for achieving women’s individual and political freedom and in harnessing the power of media for social transformation. She is the curator of the groundbreaking, critically acclaimed virtual exhibition, Muslima: Muslim Women’s Art & Voices.

Linear Thinking

I read this article in MotherJones a few months ago when it first came out. I thought the author’s analysis of how rural Trump voters think was particularly insightful given that many of the people the author came to know during his time among them largely agree with it. He summarized it this way:

You are patiently standing in the middle of a long line stretching toward the horizon, where the American Dream awaits. But as you wait, you see people cutting in line ahead of you. Many of these line-cutters are black—beneficiaries of affirmative action or welfare. Some are career-driven women pushing into jobs they never had before. Then you see immigrants, Mexicans, Somalis, the Syrian

Yard of Trump Supporter
Yard of Trump Supporter (photo: Stacy Krantitz)

refugees yet to come. As you wait in this unmoving line, you’re being asked to feel sorry for them all. You have a good heart. But who is deciding who you should feel compassion for? Then you see President Barack Hussein Obama waving the line-cutters forward. He’s on their side. In fact, isn’t he a line-cutter too? How did this fatherless black guy pay for Harvard? As you wait your turn, Obama is using the money in your pocket to help the line-cutters. He and his liberal backers have removed the shame from taking. The government has become an instrument for redistributing your money to the undeserving. It’s not your government anymore; it’s theirs.

What these poor souls fail to understand is that There Is No Line.

Yet, people have been sold the notion that there is a line, and it has become the basis for an otherwise unfounded sense of entitlement. “Good things come to those who wait.” How often do we hear that? Antithetically, we also hear “God helps those who help themselves.” Which is it?

Confused? Maybe that’s the point. Maybe that’s the purpose of constantly pounding conflicting messages into people’s heads: to confuse them. And if you make sure those same people never develop even a basic capacity for critical thinking, you can keep them befuddled, confused, dependent on authority figures to tell them what to do throughout their lives.

We liberals are told we don’t know how to talk to this part of the country, to these people. We’re told they feel disconnected, feel that their country has been taken over by those who they see as cutting in line, robbing them of their due. Well, to us liberals, they sound like petulant children, whining “It’s not fair!” when they feel they’ve been denied the cookie, the ice cream cone, equal turns (or time, right down to the last darn second) on the swing, or suffer any one of seemingly thousands of such injustices. The naive parent tries to reason with their child, tries to explain the how and why. Sooner or later, many of these parents (myself included) resign themselves to the reality that they are trying to reason with people who are unreasonable, and long-winded, thoughtful explanations are being ignored. The best answer turns out to be the simplest, and applies here, too: Life is not fair. Get used to that.

The overwhelming majority of people who think this way live in states that in fact receive more from the federal government than their residents pay out in federal taxes. If anything, those of us who live and work in “liberal” states, like New York or California, should be crying about how unfair that is. We don’t. Liberals operate on the principle that we’re all in this together, and that the purpose of any benevolent government is to be a tool for all of us to use to make life better for all of us. Does that mean each and every one of us will receive an equal portion? Ideally, sure; in reality, it’s not possible nor is it a reasonable expectation.

In an ideal world, life is fair, no one ever goes without, and everyone is always happy. I think the rural, benighted folk of the heartland need to grow up, need to understand that life indeed is not fair, the universe does not owe them anything, and that their life will be what they make of the opportunities chance bestows on them coupled with the character they exhibit when chance kicks them in the gut.

There is no line.

Horse lookin’ at you, kid

I have been on a horse precisely three times in my life. This photo was taken on the first such occasion. About five minutes afterward, the horse, named Winnie1, was running around the paddock at what seemed like about 100 mph with me hanging off to one side, held there only by my foot, caught in the opposite stirrup. I was four or five at the time.  My aunt  (whose horse it was) and uncle (the guy in the photo NOT on the horse) quickly ran over, got the horse under control and spent the rest of the day attempting, with varying degrees of success, to calm me down — or, at least get that silent, open-mouthed scream look off of my face before my parents showed up to take me home.

I did not get back on a horse again until I was maybe 20.  A friend from school owned a horse and asked if I’d like to come along with her to feed and brush it.  She also planned to ride her a bit around the paddock. I reckoned there’d be no harm (to me) in that, so, I went along. The horse was nice and calm (turns out it was faking, plotting when and how it would make its attempt on my life) and lulled me into a false sense of trust. My friend finished her short ride, and knowing my apprehension of horses, asked if I’d like to just sit in the saddle for a bit.  Sure.   Why not.

I was still alive and well after a whole two minutes on horseback, so she (my friend) asked if I’d like to have a short walk with her (the horse) around the paddock.  “Uh … ok.”  After all, I thought, I was little when that other horse nearly killed me.   I’m a full-grown adult now.  I’m big enough to control this animal.  And if not,  I’d at least be able to get in a couple of good hits before this one did me in.

Five minutes later, there I was hanging on for dear life as that fuckin’ horse went racing around the paddock at what seemed like around 100 mph.   “Pull on the reins and say ‘Whoa!'”, my friend yelled.  I pulled and pulled on the reins, to no effect. “Whoa!!”. Nothing. “Stop!! Fermo!! Halt!! стоите!!”  The horse was evidently deaf in addition to being homicidal.

Still moving at break-neck speed,  the horse turned and headed straight for the (closed) paddock gate.  Just when I thought my fear of injury or death couldn’t be any greater, my friend opened up a whole new world of pants-shitting terror for me when I heard her hollering, “Don’t let her jump!!!!”.

I decided the only chance I had of getting off alive was to jump off while the horse was in motion.  I got my feet out of the stirrups and vaulted2 as best I could off the horse, still holding on to the horn on the saddle.  My feet no sooner hit the ground than the horse stopped dead in its tracks, maybe a meter or two from the gate. It swung it’s head around to look at me with a murderous glint in its eye that said, quite unambiguously, “Next time, asshole.”

The third (and so far, last) time was on a visit to Turkey around 14 years ago. We were on the island of Heybeliada, in the Sea of Marmara near Istanbul. There are no cars there; only horses and horse-drawn carts. My companion had never been on a horse and was dying to try it. There was a guy there with several horses for hire and he was happy to lead the horse by its tether for those who wanted to ride but were untrained. I figured, what could go wrong? So, we hired a couple of them and he led us for two or three kilometers under a perfect blue sky along one of the lovely, quiet, car-less green avenues that follow the island’s shoreline.

Our guide saw that both horses were behaving well, and my friend seemed to take to riding like a duck to water. He saw that I had taken to riding like a duck to bowling.   Still, the horse was well-behaved as it continued clip-clopping along, so he decided to let go of the tethers.  Both of them.   My friend and her horse kept right on going, the horse happily obeying her gentle tugs this way or that on the reins, the two of them making shallow S-turns as they ambled along.  My horse plodded along lazily, disinterested, slightly behind hers.   I just held the reins nice and steady, making every effort to not remind the beast that it still had me on its back.   Several blessedly uneventful minutes passed and I started to think that maybe horses didn’t have it in for me after all.

Of course, he was just biding his time, waiting until the man wasn’t looking, at which point he turned right around and headed off toward … I don’t know where … a quiet spot to murder me, perhaps.  In any case, the man soon noticed I’d been kidnapped and quickly came back for us.  He gave me an apologetic smile as he took up the tether, and continued to lead the horse, with me still on it, for the remainder of our hired time.

I’m not kidding when I say they want to kill me.

To be fair, I have had equine encounters that were not life-threatening.  One could even go as far as to say that they were pleasant.   While visiting a long-time friend in Arizona some years ago, she introduced me to her horse, Mage.   We eyed each other cautiously at first, but it wasn’t long before she was letting me pat her on the forehead and taking a carrot from my hand3.  By the end of our visit, Mage and I were getting on famously.  The trick, I found, is in knowing where you and the horse stand with one another.  I stand on one side of the paddock fence, and the horse stands on the other.  No miscommunication, and no one gets hurt.

My most recent encounter was with a neighbor’s two Australian miniatures.   These look either like fat ponies or rather well-built donkeys. Regardless, they turned out to have the sweetest most gentle temperament of any animal I’ve been close to that’s larger than, say, a great dane.4  Still, I kept to my side, they kept to theirs, and we all got along just fine and all walked away with no injuries.   I hear they’d be great for keeping what passes for a lawn in our garden under control.   I’m even tempted to see if one day we might get one.  Trouble is, these adorable little creatures live for 40 or more years.   Given my present age the odds are that, for them, it would simply be a waiting game.

The Assumption of St. Ronald

McDonalds deserves to recover NO damages from the Tecoma 8, nor from anyone else for one simple reason: “assumption of risk“.

If you knowingly engage in an activity in which there is an obvious or clearly stated risk, you are not entitled to sue for damages in the event of injury sustained therein. Take for example, riding on a roller coaster. The operator of that ride must do “due diligence” to ensure the ride is safe and that no physical harm will come to those who ride it, such as may occur from malfunction of the ride or some component of it. However, they issue very clear warnings that people with certain medical conditions (e.g. heart problems, pregnancy) should not ride. They also point out that they are NOT responsible for items that may be lost during the ride (loose change, glasses, &c.)

If someone hops on the roller coaster, having seen the sign but decided to take the RISK of if it inducing a heart attack, they cannot expect to sue for damages should they, in fact, have an M.I. Same holds for the person who, say, looses spare change, or that expensive piece of jewlery that became dislodged and was ejected during the ride.

McDonalds boarded the Tecoma roller coaster knowing FULL WELL what they were getting into. They were told, in writing, in the media, on social networks, to their faces, by hundreds in person, and by thousands around the world: DO NOT BUILD IN TECOMA. You WILL be resisted.

But, they proceeded. They assumed the risk.

And now they want to recover damages due to construction delays and additional “unforseen” costs such as the security force they’ve employed? Who are they trying to kid, here? Oh, and they want attorney fees on top of that?

What HUBRIS!! What GALL!! What. A. Joke.

McDonalds KNEW there was a risk. They absolutely knew it. Their disregard, their arrogance, and their failure to live up to their own corporate propaganda — striving to earn the “trust” of communities — are all too evident. It is painfully clear that they knew what they were getting into. They assumed we weren’t serious. They assumed we’d grow weaker and give up. They assumed the risk.

They assumed too much.

McDonalds in Hot Water … Again

One of those so-called frivolous court cases people seem to recount often is the one in which McDonalds was sued by a woman who was scalded by coffee, purchased at McDonalds. The pop media made a joke of this, lampooning the woman as the archetypical “litigious American” who’ll sue anyone with deep pockets at the drop of a hat. In fact, it was an al-too-good example of just how proficient McDonalds PR department are at spin. The nearly 80-year-old suffered third degree burns (this is the kind where burn fully penetrates the skin) on her groin, buttocks, and thighs when the too-hot coffee spilled on her. She was hospitalised for over a week and required skin grafts and follow on treatment for the next two years. A jury awarded her over $2 million, but, in fact she received far less than this because McDonalds appealed the decision, tied her up in court for years, after which she finally settled for an undisclosed amount somewhere below $600,000. I’ll let you think about that for a moment whilst I set up for the next paragraph.

You see, McDonalds KNEW it’s coffee was too hot, but it’s management decided that they’d rather pay claims — over 700 of them, as it turns out — than lower the temperature of the coffee. They KNEW it, but because they have more money than decency, they chose to apply money, rather than decency, to the problem.

We have the same situation here in Tecoma, now. McDonalds KNEW it was not wanted up here. They claim the economics and zoning were favorable to building a store here. What they didn’t give any consideration to was that there is no DEMAND for one, here. I suppose they just reckon than if there’s no law against it, and it’s an area where disposable income per square kilometer is over some threshhold, they’ll put a store there, the community’s wishes be damned.

And now they’re suing us — specifically, eight of my neighbours and friends — essentially for putting up a huge resistance to their efforts to build here. They knew they weren’t wanted, and rather than do what a GOOD business does and simply look elsewhere to build, they bullied their way into our area, unwelcome, unwanted, and now unashamed of their rapacious tactics. They are claiming “injury” from not being able to build here. I say any such injury is SELF-INFLICTED. This is like the guy who jumps in front of an oncoming truck, gets run over, and then tries to sue the driver for the injuries he sustains.

There is no “law” that dictates limits on how hot coffee should or should not be. McDonalds could, in that case, claim that they did everything according to the law. And yet, through their negligence, as a result of their careless disregard for peoples’ well-being, they injured nearly 1000 people that we know of this way, at least one of them quite severely. They now find themselves in “hot water” with our community. WE didn’t put them there; they hopped in — no DOVE in — ignoring the very clear wishes of the community, and using legal thuggery to have its way in spite of our wishes.

They are feeling the “burn” and want to hold us liable for their injuries? Sorry, McDonalds. You got yourself into this soup, your own avarice drove you into a boiling hot caldron of your own making.

Acquainted with the Night

A poem by Robert Frost, with Dave and Angela in mind:

Acquainted with the Night

I have been one acquainted with the night.
I have walked out in rain — and back in rain.
I have outwalked the furthest city light.

I have looked down the saddest city lane.
I have passed by the watchman on his beat
And dropped my eyes, unwilling to explain.

I have stood still and stopped the sound of feet
When far away an interrupted cry
Came over houses from another street,

But not to call me back or say good-bye;
And further still at an unearthly height,
A luminary clock against the sky

Proclaimed the time was neither wrong nor right.
I have been one acquainted with the night.


So, originally, this blog was to be about my dabbling and forays into sustainable gardening and energy generation and storage.   However, I’ve decided to turn it into a more general compendium, largely due to my impending secession from a certain extremely popular social media site that shall remain nameless.  Once this election cycle is over, I’m done with that, however, I won’t be done commenting on politics or opera or sustainable living or whatever.