It’s strange how one can read something many times and miss the complete stupidity of it, in math at any rate.
This is from the CCSSM Grade 8:
The Number System 8.NS (Grade 8)
Know that there are numbers that are not rational, and approximate
them by rational numbers.
2. Use rational approximations of irrational numbers to compare the size
of irrational numbers, locate them approximately on a number line
diagram, and estimate the value of expressions (e.g., π2). For example,
by truncating the decimal expansion of √2, show that √2 is between 1 and
2, then between 1.4 and 1.5, and explain how to continue on to get better
I need an approximation to √2. Just get me the decimal expansion, please. Oh, and I need it to 73 decimal places.
Do I have to explain to the authors of this garbage that the only way I am going to get anywhere with √2 is by a process of successive approximation, NOT THE OTHER WAY ROUND ! !
And just try doing this for pi.
“I know that there are irrational numbers”. “How do you know that?”. “Because my teacher told me”.
And where will I encounter π2 ? Or “estimate the value of pi-e”.
And when we get to High School we find:
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational;
that the sum of a rational number and an irrational number is irrational;
and that the product of a nonzero rational number and an irrational
number is irrational.
I find real difficulties explaining the last point.
I am not proposing that we go as far as Cauchy Sequences or Dedekind cuts, but if they cannot do a better job than this the topic is best stopped at “√2 is irrational and here’s why”. How many students can prove that √2+√3 is irrational?