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January 13th, 2019, 04:40 PM   #91
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One wouldn't have thought that Australia had much depth in eroticism. Their reputation is rather more neanderthal than that in such matters.
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January 13th, 2019, 06:13 PM   #92
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Originally Posted by AplanisTophet View Post
So now the question is, how does this ruin all of mathematics again?
Yes and no. Mathematically, of course not. The real numbers are defined (real number axioms) and constructed (Dedekind cuts, etc.) without using decimals. Then we prove that every real number has one or two decimal representations as you described. The real numbers are logically prior to the useful-but-flawed decimal notation. Or radix notation in general. There are other notations such as continued fractions that don't have these kinds of problems.

On the other hand when it comes to the way we teach math, we create a lot of confusion in the minds of students. We tell them in high school that a real number is an "infinite decimal expression," whatever that's supposed to mean at that level. We tell them there's this number $\pi$, which is a special one of these "infinite decimals" that has something to do with circles. Remember, these are all students who have never seen a non-numeric symbol used to represent a constant before. In algebra they learned to find "x" and most of them eventually accept that. But now this $\pi$ is Greek letter, another thing they've never seen before. But it's not a variable, it's some particular number that's special because it's "infinite," an impression a lot of people come away with. No wonder people are so confused and most of them hate math.

By the way I believe that the poor teaching of the real numbers in high school is one of the main sources of confusion over Cantor's diagonal argument. People in general have a very shaky grasp of the real numbers and infinite decimals in the first place. The whole enterprise seems bogus to them. And it's not their fault. When I'm in charge, the math educators are in big trouble.
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Last edited by Maschke; January 13th, 2019 at 06:27 PM.
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January 14th, 2019, 12:09 AM   #93
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Originally Posted by Maschke View Post
Yes and no. Mathematically, of course not. The real numbers are defined (real number axioms) and constructed (Dedekind cuts, etc.) without using decimals. Then we prove that every real number has one or two decimal representations as you described. The real numbers are logically prior to the useful-but-flawed decimal notation. Or radix notation in general. There are other notations such as continued fractions that don't have these kinds of problems.

On the other hand when it comes to the way we teach math, we create a lot of confusion in the minds of students. We tell them in high school that a real number is an "infinite decimal expression," whatever that's supposed to mean at that level. We tell them there's this number $\pi$, which is a special one of these "infinite decimals" that has something to do with circles. Remember, these are all students who have never seen a non-numeric symbol used to represent a constant before. In algebra they learned to find "x" and most of them eventually accept that. But now this $\pi$ is Greek letter, another thing they've never seen before. But it's not a variable, it's some particular number that's special because it's "infinite," an impression a lot of people come away with. No wonder people are so confused and most of them hate math.

By the way I believe that the poor teaching of the real numbers in high school is one of the main sources of confusion over Cantor's diagonal argument. People in general have a very shaky grasp of the real numbers and infinite decimals in the first place. The whole enterprise seems bogus to them. And it's not their fault. When I'm in charge, the math educators are in big trouble.
The entire education of mathematics nowadays is rubbish. I don't claim that my high school education was of a very high level, but as somebody interested in education of mathematics, I have seen the level of education drop every year. It's horrible.

The knowledge of the students entering my university is extremely bad. The students are still very bright and smart, but have been taught very badly.
I don't blame the students, I don't blame the teachers (their job is very difficult as it is), but I blame the curriculum and the so-called educators. They have succeeded in taking everything mathematical out of math. Math has been reduced to a series of plug and chug exercises and "rules to memorize". No critical thinking involved. No proofs or demonstrations involved (except in geometry, where the level of the "proofs" just is.... horrible).
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January 14th, 2019, 08:20 AM   #94
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Now we are on the subject, the following is a must read for anybody interested in education https://www.maa.org/external_archive...lin_03_08.html

The points he brings up are extremely important. Sadly, he doesn't really give many good and viable solutions.
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