syzar

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.

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.

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.

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.

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.

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.

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.

The distribution of resources is grossly unequal.