Teaching Approximations (was Re: Microcode, which is a no-go for

[Fred Cisin]

>> Few people (but most are right here) can recite PI to enough digits to 
>> reach the level of inaccuracy.  And those who believe that PI is 
>> exactly 22/7 are unaffected by FDIV.  (YES, some schools do still teach 
>> that!)

On Tue, 8 Jan 2019, Eric Korpela wrote:
> Really?  I find it hard to believe any schools taught that as anything 
> other than an approximation.

I first encountered it about 60 years ago, in fifth grade.  Our textbook 
said, "PI is about 3.1416 or 22/7."  Our teacher insisted that that 
sentence meant "PI is about 3.1416, or exactly 22/7."  I argued it.  I 
pointed out that 22/7 was about 3.1429, and "why would they say 'about 
3.1416' instead of 'about 3.1429' if it were actually 22/7?"  I got sent 
to the principal's office.  My father, who COULD recite a dozen digits of 
PI gave me a hard time about "staying out of trouble".

I thought that that was the last of 22/7, other than that my classmates 
had been subjected to "lies told to children".

In 1983 - 2013, I taught in the community college.  One of my courses was 
[BEGINNING] "Computer Math". The students were mostly high school 
graduates 18-25 years old, adults seeking career change or enhancement, 
and a rare few there for recreational education.

About every other semester, I would have a student who had been taught 
"exactly 22/7"!  One guy admitted that he had just never bothered to 
divide it out.  Once he did, he understood the concept of 
"approximation", did his homework, and found better ones, like 355/113.

A silly little exercise to get across the concept of approximation was to 
get them to divide 1 by 3, write down the result, then clear, and multiply 
that result times 3.  "What is WRONG with that calculator?" :-)  Once they 
grasped a comparison to "rounding", "approximation" wasn't so alien.


They all thought that they knew binary, but many were unaware of octal or 
hexadecimal!  Hardly any had any idea about NEGATIVE numbers! - "don't you 
just use a minus sign?", which I easily changed into the concept of a sign 
bit, followed by deriving one's complement and then two's complement by 
pointing out that it would be useful to have a bit pattern that we could 
add to a number to get zero. "The overflow?? WHAT HAPPENS when the 
odometer of your car gets to all 9's?  If you add one, you have a new car? 
Only way that I ever have 0 on MY odometer!  Trivial homework assignment: 
how many more miles do you need to drive to get zero on your odometer?"

Virtually NONE of them had any clue about how to represent non-integers, 
or even that non-integers COULD be represented in binary!
THAT is further exacerbated by the WINDOZE Calculator that switches to 
INTEGERS-ONLY in "Programmer" mode!

I put my usual grid of 8 boxes on the board, then a dot and another grid.
"It is NOT a 'decimal point'!  You can call it a 'period', but it's really 
a 'binary point' or more generally, a 'radix point'"
We did a whole bunch of simple fractions, such as 3.5, 1.375, etc.
Then I talked about ridiculously large or small numbers, and reminded them 
of "scientific notation", and we launched into floating point 
representation.  One of my homework assignments was the IEE single 
precision floating point bit pattern for PI.


And, why bother to memorize a dozen digits of PI, when you could just 
memorize a few hundred characters of a macro that will produce it?
Or wait for a infinite series to converge.

--
Grumpy Ol' Fred     		cisin at xenosoft.com