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[#]MathsMonday
This week we come back to the #Mathematics order of operations for the coup de grace for those still clinging to the idea that the answer to 8/2(1+3) can be anything other than 1 #MathsIsNeverAmbiguous
Sometimes - very rarely, but sometimes - people cite a #Maths textbook to me that they THINK supports their point and not mine. Spoiler alert: it never ends well for them. π I usually have a look through said #Math textbook for anything interesting...
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2/7
This time I found some VERY interesting things! I'll get to that shortly, but first, earlier in the book...
This person was citing it to me, because they were - they thought - proving my claim that "no Maths textbook ever says 'implicit multiplication'" wrong. Spoiler alert: my claim remains unchallenged (because of course it does - "implicit multiplication" is a made-up rule by those who have forgotten the actual rules of Terms and The Distributive Law, which has been my point all along)β¦
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3/7
The textbook is Modern Algebra : Structure and Method, Book 1 by Dolciani, Mary P; Berman, Simon L; Freilich, Julius (1965) https://archive.org/details/modernalgebraboo00dolc They CLAIMED that page 36 is about "Implicit multiplication"... DESPITE the fact it clearly says "Products", and on page 37 it says "Terms", NOT "implicit multiplication" (because products aren't "multiplication", they're THE RESULT of a multiplication. If a=2 and b=3 then axb=2x3, multiplication, but ab=6, a Term, the product of 2x3)β¦
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4/7
The first interesting thing is on page 37, where it discusses Terms, and specifically points out that 3(x-y) is ONE term (which of course is the first reason why 2(1+3) is entirely in the denominator).
Now, back at the bottom of page 36 they have several examples of products, all of which go straight from a(b) to ab - doesn't even bother showing the intermediate step of (axb)! e.g. 5(2)=10, 5(3)=15, etc. This helps emphasize that a(b) is a product, not "multiplication" (no multiply sign)...
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5/7
And in fact Khan Academy do the exact same thing. In this video https://www.youtube.com/watch?v=GiSpzFKI5_w&t=28s, after explaining that Parentheses have top priority, he goes straight from (5)(4) to 20, leaving out the intermediate step of (5x4), using the same colour-coding as Parentheses, showing that Distribution happens at the first step, and that (5)(4) represents the Product 20, not "multiplication" 5x4. The only time he uses the multiply/divide colour is for the lone division in the expression...
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6/7
Now, coming back to Modern Algebra and the coup de grace for the "it's not (only) 1" crowd...
On page 282 there are some exercises. The student has been asked to express the questions as fractions. Look at questions 15 and 16 - 1Γ·x(x-1) and 1Γ·y(y+4)!
So, firstly that outright kills any claims of "you wouldn't see it written like that in a Maths textbook", because here it is written EXACTLY like that in a Maths textbook!
Secondly, this textbook has answers in the back, so let's look...
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7/7
And there we have it - x(x-1) and y(y+4) have both been treated as single terms (exactly as the textbook had already said they should be), and are entirely in the denominator, as we have discussed many times already. π These are both bracketed terms - i.e. factorised terms - thus solved at the FIRST step in order of operations, NOT the "Multiplication" step, which refers LITERALLY to multiplication symbols, nothing else.
I think I can finally put this thread to rest nowπ Feel free to share
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1/7
[#]MathsMonday
Should've known better than to think my last instalment would end the #Mathematics order of operations arguments. The #Math deniers always making up new excuses to avoid being wrong (sigh). So, this week I'm debunking the claim that the #Maths order of operations rules are dependent on notation (despite having already covered that the rules are universal - https://dotnet.social/@SmartmanApps/112019943359120289). Spoiler alert: they're not...
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2/7
When people say that (they believe that) the order of operations rules are dependent on notation, they usually cite Reverse Polish notation (RPN) as an example, so I'm going to address that example specifically (but proves the point in general). For those who don't know what RPN is, this is how it works...
It works with 2 numbers and an operation at a time, or one number and your last result with the operation. So let's say we wanted to do 2+3x4...
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3/7
We start with what we want to do first (for this example), which, as per the order of operations rules, would be the multiply, so we write that as 3 4 x. Then the product, 12, will thus be our last result, and now do what we want to do with that and add the 2, thus, 2 3 4 x +. In other words, the machine calculating this reads until the first operator, then uses the previous 2 numbers for it, then reads to the next operator, and applies the next number with it to our result...
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4/7
So we go from 2 3 4 x + to 2 12 + to 14, which of course is the correct answer for 2+3x4. If instead I wanted to do (2+3)x4, then I would have to change the expression to 4 2 3 + x, so that the next step becomes 4 5 x, and then 20 - again the correct result. Note that RPN doesn't use brackets. To make it do 2+3 first, I had to make that the innermost operation. In other words, when I do that, it IMPLICITLY IS BRACKETED! The same is true of every operation...
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5/7
When we did 2+3x4, and I did it in RPN as 2 3 4 x +, what RPN was REALLY doing behind the scenes was 2+(3x4), which of course we don't need those brackets with our normal order of operations rules. So how did I get the correct answer to 2+3x4 with RPN when it doesn't have brackets? By DOING THE MULTIPLICATION FIRST, as per the ORDER OF OPERATIONS rules (sigh). In other words, if I had done 2 3 4 + x, I would've got the WRONG answer...
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6/7
So the ONLY way we get right answers out of RPN is to OBEY THE ORDER OF OPERATIONS RULES when crafting the expression. i.e. it's on the user to make sure they're obeying the order of operations rules. If you want to do (2+3)x4, then the part in brackets has to be the innermost part. If I want 2+3x4, then I have to have the multiplication innermost. It's on me to put them in the right order for what I want it to do, because in the background it's putting brackets around each operation...
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7/7
Which is an opportune time to remind everyone that the worded order of operations rules start with the declaration "In the absence of grouping symbols...", and what does RPN do? Put brackets around each operation! It performs 2 3 4 x + as (2+(3x4)), and it's on me to do that multiplication first if that is what I want! In other words, RPN 100% obeys the order of operations rules. The order of operations rules are universally true (Brackets, Multiply, Add), REGARDLESS of the notation used.
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1/7
[#]MathsMonday #Maths #Math
I want to discuss some #Mathematics terminology, as I've seen some people tripping up on it, particularly around what an "Expression" is, which I'll get to shortly...
First up, Pronumeral, but also commonly called "Variable", though strictly speaking that's not true, since sometimes Pronumerals are constants, like in a linear equation, y=mx+b, where "m" (gradient) and "b" (y-intercept) are both constants. In this case "pro" means "substituted for"...
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2/7
Next is Terms/Products, though in different contexts may also be referred to as Monomials or Operands. Terms are made-up of one or more pronumerals and/or a number. A Product is THE RESULT of a multiplication. If a=2 and b=3, then axb=ab, 2x3=6, axb=2x3, ab=6, though sometimes textbooks will show you the intermediate step(s) of working out, ab=(2)(3)=(2x3)=6, the important thing here is that a Term/Product is a SINGLE NUMBER, so there needs to be brackets in any working-out steps...
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3/7
The most concise definition I've seen of "Term" is separated by operators (plus, minus, multiply, divide) and joined by grouping symbols (brackets, vinculum, exponents). I mention this because sometimes there's a context-specific definition, like in a "Collecting like terms" chapter, which only mentions plus and minus, and some people incorrectly read between the lines that they AREN'T separated by multiply and divide...
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4/7
But in the context of collecting like terms we have ALREADY DONE ALL MULTIPLIES AND DIVIDES, hence they're not mentioned in those chapters. Some people then incorrectly read between the lines here that Terms AREN'T separated by multiply and divide, but we have already seen a definition which doesn't even mention signs at all(!), as per several textbooks. Terms are separated by ANY operators. We can see then that 3(x-y) is a SINGLE TERM, as the only operator is INSIDE BRACKETS...
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5/7
Lastly, Expressions. Expressions are made up of operators and operands (Terms/Products/Monomials). So, 1Γ·2 is an expression, 1 divided by 2, whereas Β½ is a single term, grouped by a vinculum. In a single line we can represent the latter by putting it in brackets, (1Γ·2). It's important to understand that a Pronumeral CANNOT CONTAIN AN OPERATOR, UNLESS it's in Brackets, otherwise that would make it an Expression, NOT a Term. So 1Γ·2(b+c)=1Γ·(2b+2c), whereas (1Γ·2)(b+c)=(Β½b+Β½c)...
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6/7
And one of the big issues there is some people - even some textbooks(!) - calling a Term "an Expression", but then ALSO calling an Expression "an Expression"! This is relevant to the pre-20th Century definition of Division, which was everything (the EXPRESSION) to the left divided by everything (the EXPRESSION) to the right, but then, sometime before the late 19th Century, it was changed to the TERM to the left divided by the TERM to the right....
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7/7
So 3+1/2+6 in the OLD way meant (3+1)/(2+6)=4/8=Β½, but NOW means 3+(1/2)+6=9Β½! And the result of people not knowing the difference between an expression and a term is you end up with Youtubes and blogs who think ab=axb, NOT (axb), and that only the first NUMBER/PRONUMERAL is in the denominator of 1/ab, so THEY think it's 1/axb=b/a, NOT 1/(axb), even though just looking at Maths textbooks can quickly debunk that... but they never looked in any Maths textbooks! I call them the #disinformati π
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@SmartmanApps RPN calculators are completely unbothered by the 'controversies' around order of operations, as the interpretation is offloaded to the brain attached to the fingers.
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@dm319
Yeah that's right, but in doing that some people have come to the incorrect conclusion that it doesn't obey the order of operations rules, when in fact it's them doing it when they type it in. π I had one person claim that since RPN doesn't have brackets (it does, in the background) then you can't have Distribution... so I promptly showed him how to do Distribution in RPN π 8/2(1+3)=8 2 1 3 + x /
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