It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution",
By the time one reaches calculus they have been taught that complex solutions are valid solutions. They just aren’t real solutions. Therein lies one of the problems teachers of mathematics have. Conveying the concept of the answer depending on what the current algebraic object one is working on. We have to hand wave do some brain washing because the nuances involved are far too complicated for the students to understand at this level.
It is clear you are not a mathematician. When we write something like:
x^2 + x + 1 = 0
And say solve it we are definitely not asserting that the two expressions are the same. Indeed they are not the same polynomials and if your view were correct we wouldn’t spend time teaching how to solve the equation. There are values for which the two polynomials evaluate to the same number. Those are the solutions.
EDIT: In mathematical logic class one talks about predicates and you learn to think of equations as assertions that two expressions are the same. However, as people typically use and think about math they don’t think in these terms. Indeed, the graphical interpretation of an equation in one variable lends itself to the idea that solving an equation, in essence, is finding values of x that make two functions have the same value.
It is also equally clear that you haven’t taught basic mathematics to innumerate students. When students are taught to solve basic linear equations we include in our instruction that they can encounter situations like:
x+1 = x
And that they can see there is no solution because they reduce the equation to solving 0=1 and that equation has no solution.
You are in an absurd position when you think
0x = 1
is an equation but that
0=1
is not. I doubt that when you simplify:
x^2-2x - (x^2 -2x)
You write 0x^2 + 0x. What I wrote about solving equations has an important word in it. Namely “essence”. In essence…. I was not providing a mathematically rigorous definition. Indeed, the rigorous definition is far beyond the scope of students of basic mathematics. So we have to teach them the essence of things.
0x is the same thing as 0 so it appears my example is a good one in that you yourself don’t fully understand the concepts involved. This isn’t pejorative.
Suppose I said solve
x = x+1
You then subtract x from both sides and end up with
0 = 1
Then you conclude that the original equation has no solution. I’m guessing that you wouldn’t realize that the reason we conclude that the original equation has no solution is because the two equations
x = x + 1
and
0 = 1
have the same solution set since adding the opposite of x to both sides is a solution set preserving operation. It transforms a given equation into a new equation with the same solutions and clearly 0=1 has no solution. That is, 0=1 is a perfectly valid equation.
The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. To find an example of an equation with no solution just find two expressions that are never equal to each other.
Not everyone can learn calculus or even basic algebra. A person with a severe intellectual disability isn’t going to be able to learn these subjects. Between severely intellectually disabled and Einstein is a boundary between those who can and those who can’t. I don’t know where the boundary is but it exists.
Students don’t fully grasp concepts at the early stages and they never will. Very few students who master the quadratic formula understand that this formula allows you to factor second degree polynomials. Many students will correctly solve x^2+x+1 = 0 while simultaneously believing that this polynomial doesn’t factor. Here’s a simple problem that tests understanding,
A second degree polynomial with leading coefficient 3 has zeros of -1 and 2. Find all the terms of the polynomial.
Most students can’t do this. Even most calculus students can’t do it.
We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It’s only with a certain level of mathematical maturity that one is able to understand the full import of even basic concepts.
I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. They’ll play around with x’s in various complicated looking expressions. Even something as basic and fundamental as the meaning of an equation eludes people at this level even though they have been dealing with equations for years.
Well, such is my experience teaching math at a community college for over 20 years.
I think it’s clear what is meant by the title. In case it’s not to you then “aftermath” refers to: aftermath of the start of the war and the accompanying sanctions and ongoing global realignment.
I have no medical expertise but I do know that overwhelmingly those who do have expertise took the “vaccine” and the booster shots. It does not matter to me what word you use to denote the COVID vaccine. I do hope anti-vaxers become rare. They are a menace and are incredibly selfish and self-centered.
I agree with your overall point but at this time there are, thankfully, few believers of Scientology. Hopefully at some point in the future encountering an anti-vaxer will be as rare as encountering a Scientologist.
Let’s say we are in base 3. Typically the symbols used to construct representations for numbers using the standard positional notation would be the numerals {0, 1, 2}. We could use any 3 symbols but these three work well since most people are comfortable with standard base 10 notation. When one says 0 is 0 in base 3 (or any other base) what the mean is precisely this:
To represent the number 0 in base 3 we just write: 0.
To represent the number 1 in base 3 we just write: 1.
To represent the number 3 in base 3 we must write: 10.
In general, for any positive number n the symbols used to construct representations for number will be {0, 1, 2,. . ., 9,. . ., n-1}. In base n the number 0 is represented as: 0. In base n the number 1 is represented as 1. The number n is represented as: 10.
Thus it makes sense to always use the symbol 0 and 1 as part of the set of numerals in whatever base you are in and if one does this then 0 is 0 and 1 is 1.
Don’t have to take off your shoes with pre-check and I’ve found that the probability of being groped during a secondary screening is far lower with pre-check.
So Gazprom is a monopoly within Russia. Germany is not solely the reason for the EU sanctions. Other countries were involved in the sanctions decision. The Baltics and Poland, etc. have been far more vocal about wanting sanctions.
The fact is Russia hasn’t been stably non expansionist for centuries and it is they who have been aggressors in Moldova, Ukraine, Belarus, and Georgia. At some point it’s a reasonable response by those threatened by Russia (the Baltics) to respond by demanding action by their allies. Russia alone is at fault and it alone deserves ire.
The moral imperative clearly lies with those desiring sanctions. Europe is going through the painful process to divest itself now of Russian resources. This will pay dividends in the future.
Russia’s economy, from what I’ve read, is contracting much more with sanctions than without them. I think it’s clear and obvious that the EU did not impose sanctions on Russia for the purpose of increasing prices of food and fuel for poor people in the U.S. The EU and the U.S. have announced the reason for sanctions. There are secondary effects of the sanctions and I believe that food price increases are more a result of global supply chain issues and Russia preventing Ukraine from fully exporting its products.
Which other Russian companies supply Russian gas to Europe in significant quantities? I thought Gazprom was pretty much the only company doing this but I have only a cursory knowledge of this matter.
The point of sanctions is a combination of two things. One is to punish a country and the other is to make a moral stance. If in the course of punishing the country a particular company makes more money that does not necessarily negate the efficacy of the sanctions or necessarily negate the morality of the sanctions.
It is meaningful to point out. As the poverty rate declines worldwide more and more people will start consuming like Americans do. There aren’t enough resources in the world to sustain that long term unless we start mining asteroids and find a cheap way to send our garbage to the sun.
Russia’s economy is contracting so in what sense are sanctions not working? They aren’t 100% effective but they do appear to be having an effect and long term the effects, if sanctions remain, will force Russia into a pseudo vassal client of China.
The point OP made was that speech patterns can influence thoughts/ideas. The point wasn’t that their kids don’t have any biases or don’t have a new set of pejoratives they use. It was an argument being made that perhaps political correctness does have a place or is useful. That’s what I deduced from OP’s comment.
But what does this have to do with the point made in OP’s comment? They gave an instance of changed attitudes based off of language usage. They weren’t saying their kids don’t marginalize anyone.
I don’t understand your point. The person you responded to isn’t generalizing to all people based off of school age kids. They noticed something about attitudes and speech in their kids versus when they were growing up and this observation comports with well a known fact. Namely, that regulation of speech can frame peoples’ thinking.
None of us are experts in everything and we need to try to make sense of things as we go about life. We all use examples/experiences in our lives to make general conclusions about the efficacy of policies.
I’m interested in the reasons for downvoting this comment. What is written is thought out and done in good faith. I don’t like regulation of word usage but it is true that propagandists can be effective at changing thinking patterns by changing speech patterns.
By the time one reaches calculus they have been taught that complex solutions are valid solutions. They just aren’t real solutions. Therein lies one of the problems teachers of mathematics have. Conveying the concept of the answer depending on what the current algebraic object one is working on. We have to hand wave do some brain washing because the nuances involved are far too complicated for the students to understand at this level.