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Elementary Algebra (1971) (softwarepreservation.org)
97 points by tosh 11 months ago | hide | past | web | favorite | 20 comments

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...

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.

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. ;)

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.

Awesome! Thanks a lot!

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

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

Ahh - nice!

I recommend Kahn Academy if you want to go back and relearn math.

"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....

Iverson explained the reasoning behind this in an appendix to a book he wrote in the 60's: http://www.jsoftware.com/papers/EvalOrder.htm

It's the approach used in APL and J, and I've certainly found it far preferable to the current math notation standard.

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).

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).

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..., 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 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.

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) 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)

This is APL.

> 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!

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.

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.

Which part is unusual for 2nd-year high-school algebra?

Looks about right for early high school math.

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