Elementary Algebra (1971)
softwarepreservation.org
softwarepreservation.org
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.
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).
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).
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)
It's the approach used in APL and J, and I've certainly found it far preferable to the current math notation standard.
Looking at that table of contents, high school algebra classes were badass in the 70s!
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.