GingerAle
Oct 14, 2018

#17**+3 **

^{... UNfortunately....we get a lot of folks posting stuf which is UNclear! Parentheses and brackets are NECESSARY to get correct answers to postings (and not make an engineering mistake that kills people !)}

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I agree that the postings from many students are unclear. It’s obvious these students lack the prerequisites for the posted questions; including an understanding of basic mathematical hierarchical conventions. Without these prerequisites, they will not understand where to place parenthetical operators, or they believe them to be unnecessary even when included in the original question.

These deficits point to an apparent flaw in lower-level education of students for these hierarchical conventions. Based on my observations, the debate on these conventions are usually by second-year college/university students and reaches a crescendo every other year. It may be because the crescendo carries over to first year students, that when they become second-year students it’s mostly a moot point and last year’s news.

As for engineers, especially those who have the credentials and experience to create or certify anything that might cause death to a population, or the destruction of expensive science experiments, it’s reasonable to assume they are well versed on such hierarchies. In addition, all mathematics and algorithms are peer reviewed and subjected to tests to catch transient math and logic errors. This process works well, as indicated by the rare failure. Most failures, when they occur, are usually traceable to a sequence of errors where no single event by itself would have caused the failure.

In modern computing, redundant uses of parenthetical operators are a trivial matter, but in the early days of computerized control, mnemonic and variable storage space was always at a premium and *maxed-out*, so superfluous parenthetical operators were not an option.

The article below gives insight to the computer related testing and contingency plans in use for the Apollo 11 moon landing.

https://arstechnica.com/science/2019/07/no-a-checklist-error-did-not-almost-derail-the-first-moon-landing/

I think it is fascinating we could land men on the moon, collect samples, and bring everything back intact, in an era when the population standard thought of computers and space travel as mostly science fiction. It’s a tribute to the *hierarchy* of human intelligence—never forgetting the occasional genetically enhanced chimp.

GA

GingerAleAug 4, 2019

#12**+2 **

This debate seems perpetual. The issue returns at least once every school year.

https://web2.0calc.com/questions/help-with-pemdas

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*Another reason I prefer the rule "only use the item immediatley adjacent to the operator as the operand" is that we can use a similar rule for exponents. In the case of exponents, the rule would be "only the item immediately preceding the caret is the base."*

*4^2^3 *

*Should 4^2^3 be 4^(2^3) or should it be (4^2)^3 ? *

*If we just stick with the left-to-right rule, we would get*

*4^2^3 = 16^3 = 4096*

**Note that the web2.0calc calculator resolves this from the left to right or ascending order, where the resultant product becomes the BASE of the next exponent. **This operation is easy to see in the display above the calculator. This is the only exception to the hierarchical order of operations. The reason for this is probably because the standard product of exponents to a base is well more common in physics and engineering than conventional stacked powers.

*If we use the rule "only the item immediately preceding the caret is the base," we would get*

*4^2^3 = 4^8 = 65536*

*Here, WolframAlpha says the answer is 65536:*

**Wolfram follows the (official) stacked power convention, where stacked powers are (exponentially) multiplied from the right to left (from the top down), and the resultant product becomes the EXPONENT to the base number. Generally, stacked power and power-towers are used in advanced, theoretical mathematics. **

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*According to the answer there, a/bc means a/(bc) which goes against my idea.*

*Hmm, I don't like that! But I guess it is the case. Though I still think 1/4(3) = 3/4*

**For a/bc, the convention will equal a/(bc) **

**Implicit multiplication of variables takes precedents over division – a noted exception, dating back to the late 1960s, to the normal convention of mathematical hierarchy. Herr Massow’s calculator is the only one I know of that that does this, and it’s probably because it allows the use of variables. (Comments from LancelotLink –2015)**

Pasting a=20;b=2;c=5; a/(b)(c) in the calculator will result in (2).

Pasting 20/2*5 in the calculator will result in (50)

Note that pasting 20/(2)(5) will result in (2), but pasting this 20/(2)*(5) will result in (50). The parenthesis without an operator between them triggers the variable precedents.

Note also, that the convention for *precedents of variables multiplication applies *only to the first two variables.

Posts related to hierarchy-order of operations:

https://web2.0calc.com/questions/help-with-pemdas

https://web2.0calc.com/questions/4-2-x-2-2-x-56#r8

GA

GingerAleAug 3, 2019