afair, multiplication was always before division, also as addition was before subtraction
It's BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
But multiplication and division are whichever comes first right to left in the expression, and likewise with subtraction.
Although implicit multiplication is often treated as binding tighter than explicit. 1/2x is usually interpreted as 1/(2x), not (1/2)x.
a fair point, but aren't division and subtraction are non-communicative, hence both operands need to be evaluated first?
It’s commutative, not communicative, btw
whoops, my bad
1 - 3 + 1 is interpreted as (1 - 3) + 1 = -1
Yes, they're non commutative, and you need to evaluate anything in parens first, but that's basically a red herring here.
ok, i guess you're right
It's BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
But, since your rule has the D&M as well as the A&S in brackets does that mean your rule means you have to do D&M as well as the A&S in the formula before you do the exponents that are not in brackets?
But seriously. Only grade school arithmetic textbooks have formulas written in this ambiguous manner. Real mathematicians write their formulas clearly so that there isn't any ambiguity.
That's not really true.
You'll regularly see textbooks where 3x/2y is written to mean 3x/(2y) rather than (3x/2)*y because they don't want to format
3x
----
2y
properly because that's a terrible waste of space in many contexts.
You'll regularly see textbooks
That's what I said.
You generally don't see algebra in grade school textbooks, though.
12 is a grade. I took algebra in the 7th grade.
Grade school is a US synonym for primary or elementary school; it doesn't seem to be used as a term in England or Australia. Apparently, they're often K-6 or K-8; my elementary school was K-4; some places have a middle school or junior high between grade school and high school.
I don't know why you're getting lost on the pedantry of defining "grade school", when I was clearly discussing the fact that you only see this kind of sloppy formula construction in arithmetic textbooks where students are learning the basics of how to perform the calculations. Once you get into applied mathematics and specialized fields that use actual mathematics, like engineering, chemistry and physics, you stop seeing this style of formula construction because the ambiguity of the terms leads directly to errors of interpretation.
Sure, the definition of grade school doesn't really matter too much. Because college texts are written in ways that violate pemdas.
For example, if f(x,y)=x2+yx, then (∂f/∂x)y=2x+y, and (∂f/∂y)x=x. We can extend this idea to higher derivatives: ∂^2f/∂y^2 or ∂^2f/∂y∂x. The latter symbol indicates that we first differentiate f with respect to x, treating y
as a constant, then differentiate the result with respect to y, treating x as a constant. The actual order of differentiation is immaterial: ∂2f/∂x∂y=∂2f/∂y∂x.
Notice: ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x).
What an interesting error to point out in support of pemdas.
Clearly the formatting of a paragraph of text in a textbook full of clearly and unambiguously written formulas discussing the very order of operations itself compared to the formatting of an actual formula diagram is going to be less clear. But here you've chosen to point to a discussion of why the order is irrelevant in the case under question.
Your example is the conclusion of a review of mathematics.
First we shall review some mathematics.
...
The actual order of differentiation is immaterial:
The fact that the example formula is written sloppy is irrelevant, because at no point is this going to be an actual formula meant to be solved, it's merely an illustration of why, in this case, the order of a particular operation is "immaterial".
Even if ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x), it doesn't matter because "∂2f/∂x∂y=∂2f/∂y∂x". So long as you're consistently applying pemdas, you're going to get the same answer whether you derive x first or y.
However, when it's time to discuss the actual formulas and equations being taught in the example text, clearly and unambiguously written formulas are illustrated as though copied from Ann illustration on a whiteboard instead of inserted into paragraphs that might have simply been transcribed from a lecture. Which, somewhat coincidentally, is exactly what your citation is.
Under PEMDAS, ∂2f/∂x∂y = (∂2f/∂x) * ∂y = ∂2∂y/∂x
Multiplication VS division doesn't matter just like order of addition and subtraction doesn't matter.. You can do either and get same results.
Edit : the order matters as proven below, hence is important
If you do only multiplication first, then 2×3÷3×2 = 6÷6 = 1.
If you do mixed division and multiplication left to right, then 2×3÷3×2 = 6÷3×2 = 2×2 = 4.
Edit: changed whitespace for clarity
4 would be correct since you go left to right.
2nd one is correct, divisions first.
I was taught that division is just inverse multiplication, and to be treated as such when it came to the order of operations (i.e. they are treated as the same type of operation). Ditto with addition and subtraction.
afair, multiplication was always before division, also as addition was before subtraction
It's BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
But multiplication and division are whichever comes first right to left in the expression, and likewise with subtraction.
Although implicit multiplication is often treated as binding tighter than explicit. 1/2x is usually interpreted as 1/(2x), not (1/2)x.
a fair point, but aren't division and subtraction are non-communicative, hence both operands need to be evaluated first?
It’s commutative, not communicative, btw
whoops, my bad
1 - 3 + 1 is interpreted as (1 - 3) + 1 = -1
Yes, they're non commutative, and you need to evaluate anything in parens first, but that's basically a red herring here.
ok, i guess you're right
But, since your rule has the D&M as well as the A&S in brackets does that mean your rule means you have to do D&M as well as the A&S in the formula before you do the exponents that are not in brackets?
But seriously. Only grade school arithmetic textbooks have formulas written in this ambiguous manner. Real mathematicians write their formulas clearly so that there isn't any ambiguity.
That's not really true.
You'll regularly see textbooks where 3x/2y is written to mean 3x/(2y) rather than (3x/2)*y because they don't want to format
properly because that's a terrible waste of space in many contexts.
That's what I said.
You generally don't see algebra in grade school textbooks, though.
12 is a grade. I took algebra in the 7th grade.
Grade school is a US synonym for primary or elementary school; it doesn't seem to be used as a term in England or Australia. Apparently, they're often K-6 or K-8; my elementary school was K-4; some places have a middle school or junior high between grade school and high school.
I don't know why you're getting lost on the pedantry of defining "grade school", when I was clearly discussing the fact that you only see this kind of sloppy formula construction in arithmetic textbooks where students are learning the basics of how to perform the calculations. Once you get into applied mathematics and specialized fields that use actual mathematics, like engineering, chemistry and physics, you stop seeing this style of formula construction because the ambiguity of the terms leads directly to errors of interpretation.
Sure, the definition of grade school doesn't really matter too much. Because college texts are written in ways that violate pemdas.
Look, for example, at https://www.feynmanlectures.caltech.edu/I_45.html
Notice: ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x).
What an interesting error to point out in support of pemdas.
Clearly the formatting of a paragraph of text in a textbook full of clearly and unambiguously written formulas discussing the very order of operations itself compared to the formatting of an actual formula diagram is going to be less clear. But here you've chosen to point to a discussion of why the order is irrelevant in the case under question.
Your example is the conclusion of a review of mathematics.
...
The fact that the example formula is written sloppy is irrelevant, because at no point is this going to be an actual formula meant to be solved, it's merely an illustration of why, in this case, the order of a particular operation is "immaterial".
Even if ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x), it doesn't matter because "∂2f/∂x∂y=∂2f/∂y∂x". So long as you're consistently applying pemdas, you're going to get the same answer whether you derive x first or y.
However, when it's time to discuss the actual formulas and equations being taught in the example text, clearly and unambiguously written formulas are illustrated as though copied from Ann illustration on a whiteboard instead of inserted into paragraphs that might have simply been transcribed from a lecture. Which, somewhat coincidentally, is exactly what your citation is.
Under PEMDAS, ∂2f/∂x∂y = (∂2f/∂x) * ∂y = ∂2∂y/∂x
Multiplication VS division doesn't matter just like order of addition and subtraction doesn't matter.. You can do either and get same results.Edit : the order matters as proven below, hence is important
If you do only multiplication first, then 2×3÷3×2 = 6÷6 = 1.
If you do mixed division and multiplication left to right, then 2×3÷3×2 = 6÷3×2 = 2×2 = 4.
Edit: changed whitespace for clarity
4 would be correct since you go left to right.
2nd one is correct, divisions first.
I was taught that division is just inverse multiplication, and to be treated as such when it came to the order of operations (i.e. they are treated as the same type of operation). Ditto with addition and subtraction.