
Elementary Algebra (1971) - tosh
http://www.softwarepreservation.org/projects/apl/Papers/ElementaryAlgebra
======
hackermailman
He also wrote a book on elementary functions as a sequel to this to act as a
precalc text
[http://code.jsoftware.com/wiki/Doc/Elementary_Functions_An_A...](http://code.jsoftware.com/wiki/Doc/Elementary_Functions_An_Algorithmic_Treatment)

This is an interesting book, uses multiplication tables flipped around to show
the pattern of zeros that make up the cartesian coord system, introduces
determinants and matrix operations, monadic functions used as function
arguments, tracing functions/analysis, proof by induction.. wish this was my
highschool algebra text.

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dptd
Anyone read this? How is it? I was never good at math and I have it on my
"todo" list. Go back to some material from my high school and undergrad and
fill the gaps. ;)

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hackermailman
If you want to fill gaps, any typical undergrad calculus text will do such as
Stewart's _Early Transcendentals_ book. You do enough of those 8,000+
exercises and your highschool gaps will fill themselves. My favorite beginner
math books are Thomas VanDrunen's _Discrete Mathematics and Functional
Programming_ because it's entirely done in SML, and Apostol's Calculus because
you end up doing so many exercises you absolutely will never make a silly
algebra mistake in a proof or forget a trig identity ever again. Often Apostol
will just defer to endless calculating in the chapter exercises if he doesn't
have anything he wants to add to the material, this is really good practice if
your basic math education is shit like mine was.

~~~
dptd
Awesome! Thanks a lot!

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mathattack
I’m curious what the historical significance of this is. I’ve seen a lot of
cool math resources here lately.

~~~
btilly
The author is the inventor of the programming language APL, and the formulas
are working APL programs.

~~~
mathattack
Ahh - nice!

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cjebbs
"when two or more functions occur in succession with no parentheses between
them, the rightmost function is executed first. " \-- this is wrong....at
least according to traditional order of operations....

~~~
shoo
that sounds like a standard convention of mathematical notation to me

if you're composing some function g : X --> Y with a function f : Y --> Z then

let x be some element of X.

then f g (x) = f(g(x)) = f(y), say, where y = g(x).

~~~
irishsultan
Yes, but a * b + c is not normally interpreted as a * (b + c), in this text it
is.

1+2x3+4x5 evaluates to 1+2x23 which in turn evaluates to 1+46 which is 47. In
the normal way it would be 1+6+20 or 27.

(edit: messed up on the order of operations and added example from the book).

~~~
NPMaxwell
That looked really weird to me too. Iverson is finessing the whole issue of
order of operations by assuming there is no order of operations. To see what
the issue is, see
[http://www.slate.com/articles/health_and_science/science/201...](http://www.slate.com/articles/health_and_science/science/2013/03/facebook_math_problem_why_pemdas_doesn_t_always_give_a_clear_answer.html),
where the author says that the CORRECT order of operations is PEMDAS and then
uses "PEDMAS" in the same article.
[https://en.wikipedia.org/wiki/Order_of_operations](https://en.wikipedia.org/wiki/Order_of_operations)
lists a variety of mnemonics that are inconsistent with standard current
software. I'm amused that if you put 1-3^2 into the Google calculator, it will
helpfully rewrite your equation as "1-(3^2)". I gather that the PEDMAS order
of operations was an invention related to the development of an early software
language and would have been very recent in 1971 when Iverson was writing.

~~~
irishsultan
It's not that PEDMAS was a new thing in 1971, if you look at the article about
the timeline of Parsing
([https://news.ycombinator.com/item?id=16856694](https://news.ycombinator.com/item?id=16856694))
then you'll see that the earliest programming languages didn't have operator
precedence, and that this annoyed users of those languages.

It's probably either because parsing operators with precedence wasn't really a
solved problem in 1964 when Iverson first started working on APL, or because
Iverson disagrees with the normal precedence rules. If you look at Notation of
a tool of thought (also linked on HN in the recent past ), then there is this
passage (when comparing APL with normal math notation):

> In the interpretation of composite expressions APL agrees in the use of
> parentheses, but differs in eschewing hierarchy so as to treat all functions
> (user-defined as well as primitive) alike, and in adopting a single rule for
> the application of both monadic and dyadic functions: the right argument of
> a function is the value of the entire expression to its right. An important
> consequence of this rule is that any portion of an expression which is free
> of parentheses may be read analytically from left to right (since the
> leading function at any stage is the "outer" or overall function to be
> applied to the result on its right), and constructively from right to left
> (since the rule is easily seen to be equivalent to the rule that execution
> is carried out from right to left)

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electricslpnsld
> The present text treats the usual topics expected in a second course in high
> school algebra.

Looking at that table of contents, high school algebra classes were badass in
the 70s!

~~~
fao_
Almost all schools and universities have different criteria for what is
considered 'high-school' / 'undergrad' / 'post-grad'. I've seen calculus
treated as a post-graduate subject, and as a high-school subject -- the same
concepts, no less.

~~~
btilly
Calculus as a post-graduate subject?

That would either be real analysis, in which case "the same concepts" is a
highly misleading description at best, or else it was not a math department.

That said, browsing a syllabus for a real analysis course, you could be
pardoned for thinking it was the same material as a Calculus course. You would
be wrong, but you will see a lot of the same keywords.

The difference is that things which are claimed in the Calculus course, such
as the mean value theorem, actually get rigorous proofs from the 13 standard
axioms for the real numbers. The first 9 being the usual rules of arithmetic
for fields. The next 3 make it into an ordered field. And most of the
attention goes to the 13'th axiom, _If a non-empty subset has an upper bound,
then it has a least upper bound._

