1y ago

Glitch in the matrix

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For anyone like me who has math as their worst subject: PEMDAS.
PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations. PEMDAS stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.
So we gotta do it in the proper order. And remember, if the number is written like 2(3) then its multiplication, as if it was written 2 x 3 or 2 * 3.
So we read 8/2(2+2) and need to do the following;
Read the Parentheses of (2 + 2) and follow the order of operations within them, which gets us 4.
Then we do 2(4) which is the same as 2 x 4 which is 8
8 / 8 is 1.
The answer is 1. The old calculator is correct, the phone app which has ads backed into it for a thing that all computers were invented to do is inaccurate.
- Well that's just wrong... Multiplication and division have equal priorities so they are done from left to right. So: 8 / 2 * (2 + 2)=8 / 2 * 4=4 * 4=16
  
  This but unironically
  
  Implicit multiplication takes priority over explicit multiplication or division. 2(2+2) is not the same thing as 2*(2+2).
  
  Correct! 2(2+2) is a single term - subject to The Distributive Law - and 2x(2+2) is 2 terms. Those who added a multiply sign there have effectively flipped the (2+2) from being in the denominator to being in the numerator, hence the wrong answer.
  But it's not called "implicit multiplication" - it's Terms and/or The Distributive Law which applies (and they're 2 separate rules, so you cannot lump them together as a single rule).
  
  Not quite, pemdas can go either from the left or right (as long as you are consistent) and division is the same priority as multiplication because dividing by something is equal to multiplying by the inverse of that thing... same as subtraction being just addition but you flip the sign.
  8×1/2=8/2 1-1=1+(-1)
  The result is 16 if you rewrite the problem with this in mind: 8÷2(2+2)=8×(1/2)×(2+2)
  
  I've never had anyone tell me operations with the same priority can be done either way, it's always been left to right.
  
  I've always heard it that way too but I think it is for consistency with students, imo Logically, if you are looking at division = multiplying by inverse and subtraction = adding the negative, you should be able to do it both ways. Addition and multiplication are both associative, so we can do 1+2+3 = (1+2)+3 = 1+(2+3) and get the same answer.
  
  But subtraction and division are not associative. Any time you work on paper, 2 - 2 - 2 would equal -2. That is, (2-2)-2=0-2=-2. If you evaluate right to left, you get 2-2-2=2-(2-2)=2-0=2
  
  Correct, subtraction and division are not associative. However, what is subtraction if not adding the opposite of a number? Or division if not multiplying the inverse? And addition and multiplication are associative.
  2-2-2 can be written as 2 + (-2) + (-2) which would equal -2 no matter if you solve left to right, or right to left.
  In your example with the formula from right to left, distributing the negative sign reveals that the base equation was changed, so it makes sense that you saw a different answer.
  2 - (2 - 2) = 2 + ((-2) + 2) = 2
  
  2-(2-2)
  But you broke the rule of left-associativity there. You can go right to left provided you keep each number with the sign to it's left (and you didn't do that when you separated the first 2 in brackets from it's minus sign).
  
  I’ve never had anyone tell me operations with the same priority can be done either way, it’s always been left to right
  It's left to right within each operator. You can do multiplication first and division next, or the other way around, as long as you do each operator left to right. Having said that, you also can do the whole group of equal precedence operators left to right - because you're still preserving left to right for each of the two operators - so you can do multiplication and division left to right at the same time, because they have equal precedence.
  Having said that, it's an actual rule for division, but optional for the rest. The actual rule is you have to preserve left-associativity - i.e. a number is associated with the sign to the left of it - and going left to right is an easy way to do that.
  
  8÷2(2+2)=8×(1/2)×(2+2)
  No, that's wrong. 2(2+2) is a single term, and thus entirely in the denominator. When you separated the coefficient you flipped the (2+2) into the numerator, hence the wrong answer. You must never add multiplication signs where there are none.
  
  8 / 2 * (2 + 2)
  That's not the same as 8 / 2 (2 + 2). In the original question, 2(2+2) is a single term in the denominator, when you added the multiply you separated it and thus flipped the (2+2) to be in the numerator, hence the wrong answer.
- Uh.. no the 1 is wrong? Division and multiplication have the same precedence, so the correct order is to evaluate from left to right, resulting in 16.
  
  The real correct order is to use brackets to remove ambiguity.
  
  Exactly, these types of problems are designed to make people confused and discuss PEMDAS and drive social media engagement.
  
  There's no multiplication in this question - multiplication refers literally to multiplication signs - only division and brackets, and addition within the brackets. So you have to use The Distributive Law to solve the brackets, then do the division, giving you 1.
- The problem with this is that the division symbol is not an accurate representation of the intended meaning. Division is usually written in fractions which has an implied set of parenthesis, and is the same priority as multiplication. This is because dividing by a number is the same as multiplying by the inverse, same as subtracting is adding the negative of a number.
  8/2(2+2) could be rewritten as 8×1/2×(2+2) or (8×(2+2))/2 which both resolve into 16.
  
  You left out the way it can be rewritten which most mathematicians would actually use, which is 8/(2(2+2)), which resolves to 1.
  
  Division is usually written in fractions
  Division and fractions aren't the same thing.
  fractions which has an implied set of parenthesis
  Fractions are explicitly Terms. Terms are separated by operators (such as division) and joined by grouping symbols (such as a fraction bar), so 1÷2 is 2 terms, but ½ is 1 term.
  8/2(2+2) could be rewritten as 8×1/2×(2+2)
  No, it can't. 2(2+2) is 1 term, in the denominator. When you added the multiply you broke it into 2 terms, and sent the (2+2) into the numerator, thus leading to a different answer. 8/2(2+2)=1.
- not to be That Guy, but the phone is actually correct... multiplication and division have the same precedence, so 8 / 2 * 4 should give the same result as 8 * 4 / 2, ie 16
  
  but the phone is actually correct
  No, it's actually wrong.
  8 / 2 * 4
  It's 8/(2x4). You can't remove brackets unless there is only 1 term left inside.
- P E M D A S
  vs
  P E M/D A/S
  The latter is correct, Multiplication/Division, and Addition/Subtraction each evaluate left to right (when not made unambiguous by Parentheses). I.e., 6÷2×3 = 9, not 1. That said, writing the expression in a way that leaves ambiguity is bad practice. Always use parentheses to group operations when ambiguity might arise.
- You are correct. This is the right sequence of operations done here.
- Ignore the idiots telling you you're wrong. Everyone with a degree in math, science or engineering makes a distinction between implicit and explicit multiplication and gives implicit multiplication priority.
- The problem is that the way PEMDAS is usually taught multiplication and division are supposed to have equal precedence. The acronym makes it look like multiplication comes before division, but you're supposed to read MD and as one step. (The same goes for addition and subtraction so AS is also supposed to be one step.) It this example the division is left of the multiplication so because they have equal precedence (according to PEMDAS) the division applies first.
  IMO it's bad acronym design. It would be easier if multiplication did come before division because that is how everyone intuitively reads the acronym.
  Maybe it should be PE(M/D)(A/S). But that version is tricky to pronounce. Or maybe there shouldn't be an acronym at all.
  
  but you’re supposed to read MD and as one step
  You can do them in any order at all - M then D, D then M (hence the acronym BEDMAS), or all in one - what does matter is not treating Distribution as though it's Multiplication (which refers literally to multiplication signs), when in actual fact it's the first step in solving Brackets.
- PEMDAS evaluated from left to right. If you followed that you’d get 16. 1 is ignoring left to right.
  
  1 isn't ignoring anything. 16 can be arrived at by ignoring any one of multiple order of operations rules.
- You're a lifesaver, thank you so much. I actually didn't know about PEMDAS, I was never taught it before...
- Turns out I’m wrong, but I haven’t been told how or why. I’m willing to learn if people actually tell me
  Well, I don't know what you said originally, so I don't know what it is you were told was wrong - 1 or 16? 😂 The correct answer is 1.
  Anyhow, I have an order of operations thread which covers literally everything there is to know about it (including covering all the common mistakes and false claims made by some). It includes textbook references, historical Maths documents, worked examples, proofs, memes, the works! I'm a high school Maths teacher/tutor - I've taught this topic many times.

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