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Post by Eugene 2.0 on Aug 18, 2019 19:28:06 GMT
A not really normal math example that wanders around the Internet is not clear enough to solve it. I has, I'd say, a problem of priority of operations to start with. Here is the example:
8 ÷ 2(2+2) = ?
Usually, they say that the answer should be '16', because - according to the straight sequence - we need to complete a division firstly, and only after it we could multiply it with '(2+2)'. So, in this case we have something like this: A ÷ B x C.
An operation of multiplying is absent here, and it's not clear to see what '2(2+2)' means considering the priority status.
I wondered about two possible answers at the same time, but I was warned by the professionals that that had no sense. (x^2=9; x1 = 3, x2 = -3 has a sense though.)
What do you think about the example itself?
What do you think about the sign '÷'?
Is it correct to divide firstly and then to multiply?
And maybe we should change '÷' on '/'?
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Post by xxxxxxxxx on Aug 19, 2019 0:08:30 GMT
A not really normal math example that wanders around the Internet is not clear enough to solve it. I has, I'd say, a problem of priority of operations to start with. Here is the example: 8 ÷ 2(2+2) = ? Usually, they say that the answer should be '16', because - according to the straight sequence - we need to complete a division firstly, and only after it we could multiply it with '(2+2)'. So, in this case we have something like this: A ÷ B x C. An operation of multiplying is absent here, and it's not clear to see what '2(2+2)' means considering the priority status. I wondered about two possible answers at the same time, but I was warned by the professionals that that had no sense. (x^2=9; x1 = 3, x2 = -3 has a sense though.) What do you think about the example itself? What do you think about the sign '÷'? Is it correct to divide firstly and then to multiply? And maybe we should change '÷' on '/'? Rofl....that is the problem of mathematics...it depends upon an assumed starting point that is directed to another point than another. I assumed 1 because 2(2+2) is the number dividing 8 precisely because the parenthesis necessitate the relation to 2 specifically. If I am to divide first it is by the number 2(2+2) 8 ÷ 2(2+2) = x A. 8 ÷ (4+4) = 1 B. 8 ÷ 2(4) = 1 C. 8 ÷ (8) = 1 D. 4(2+2) = 16 E. 4(4 )= 16 F. (8+8)= 16 Why not 16? (8÷2)(2+2)=16 However if you are to take c as an answer it sets the foundation where each mathematical statement has multiple answers determined by the direction of the operations. Math is grounded in directional qualities in these respects. |
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Post by Eugene 2.0 on Aug 19, 2019 9:23:18 GMT
A not really normal math example that wanders around the Internet is not clear enough to solve it. I has, I'd say, a problem of priority of operations to start with. Here is the example: 8 ÷ 2(2+2) = ? Usually, they say that the answer should be '16', because - according to the straight sequence - we need to complete a division firstly, and only after it we could multiply it with '(2+2)'. So, in this case we have something like this: A ÷ B x C. An operation of multiplying is absent here, and it's not clear to see what '2(2+2)' means considering the priority status. I wondered about two possible answers at the same time, but I was warned by the professionals that that had no sense. (x^2=9; x1 = 3, x2 = -3 has a sense though.) What do you think about the example itself? What do you think about the sign '÷'? Is it correct to divide firstly and then to multiply? And maybe we should change '÷' on '/'? Rofl....that is the problem of mathematics...it depends upon an assumed starting point that is directed to another point than another. I assumed 1 because 2(2+2) is the number dividing 8 precisely because the parenthesis necessitate the relation to 2 specifically. If I am to divide first it is by the number 2(2+2) 8 ÷ 2(2+2) = x A. 8 ÷ (4+4) = 1 B. 8 ÷ 2(4) = 1 C. 8 ÷ (8) = 1 D. 4(2+2) = 16 E. 4(4 )= 16 F. (8+8)= 16 Why not 16? (8÷2)(2+2)=16 However if you are to take c as an answer it sets the foundation where each mathematical statement has multiple answers determined by the direction of the operations. Math is grounded in directional qualities in these respects.Thank you for the detailed answer. I would say we could receive '1' alternatively, if the formula is written vice versa: ? = (2+2)2 ÷ 8 Maybe I decided to read it from right to left. I see no true conditions to start from the left point. In this case, not only the direction of operations, but the way of how we're reading a formula might move us to different answers (surely, if there are more than one answer in the formula). If all the math grounds at numbers, and in turn any number is something like: 1+1+1+...+1 (A), Doesn't it mean that a direction is just an ad hoc instruction? For some reason, we need a tool to explicit our intentions (our thoughts?), and directions may be given to solve something or to draw a function?.. According to A, a direction may be shown as: a) for a certain '1' the left direction is where '+' is near "a nose" of '1'. ('1' has a nose at the top left corner of the digit); b) to move left: to put '+' near '1', and after to put the next '1' near '+' that has been added to '1' with 'a nose'); c) the right direction - is opposite to the left direction. As for me - there should be no "right" answer, because we could say "it is correct for the math" as long as we might answer 'according to whose math?'. Also, I think the example is nothing like a game puzzle. If I were a gamer I would try two answers, and if I had had just one chance to answer, there should be some tips (in the game) to help me to find the correct one. |
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Post by karl on Aug 19, 2019 12:51:36 GMT
8 ÷ 2(2+2)=1
(8 ÷ 2)(2+2)=16
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Post by xxxxxxxxx on Aug 19, 2019 14:15:31 GMT
8 ÷ 2(2+2)=1 (8 ÷ 2)(2+2)=16 That's how I see it. |
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Post by Eugene 2.0 on Aug 19, 2019 18:56:49 GMT
8 ÷ 2(2+2)=1 (8 ÷ 2)(2+2)=16 Hmm... I'm not sure, but don't we have to put a pair of brackets to the first example for some insurances? Instead of "8 ÷ 2(2+2) = 1", I guess, we need to write like this "8 ÷ (2(2+2)) = 1"? In this case, I hope, there's no way to confuse it by mistake further. And if this is true, the situation might be a little bit clearer: the formula is missing a pair of brackets. |
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Post by xxxxxxxxx on Aug 19, 2019 19:00:42 GMT
8 ÷ 2(2+2)=1 (8 ÷ 2)(2+2)=16 Hmm... I'm not sure, but don't we have to put a pair of brackets to the first example for some insurances? Instead of "8 ÷ 2(2+2) = 1", I guess, we need to write like this "8 ÷ (2(2+2)) = 1"? In this case, I hope, there's no way to confuse it by mistake further. And if this is true, the situation might be a little bit clearer: the formula is missing a pair of brackets. Works too. The brackets are fundamentally what determines the function. |
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Post by Eugene 2.0 on Aug 19, 2019 19:11:28 GMT
Hmm... I'm not sure, but don't we have to put a pair of brackets to the first example for some insurances? Instead of "8 ÷ 2(2+2) = 1", I guess, we need to write like this "8 ÷ (2(2+2)) = 1"? In this case, I hope, there's no way to confuse it by mistake further. And if this is true, the situation might be a little bit clearer: the formula is missing a pair of brackets. Works too. The brackets are fundamentally what determines the function.  Yeah, thanks. Reading Church "Introduction to Math Logic" it starts seeming clearly that brackets are really fundamental. At first sight one even feel some king of gross or something, because you're always asking yourself - where's the end of all these brackets?!! But later the system of points (e.g. (∃φ):.(∃x):φx:φy.φz. ⊃ .y = z) allows it to be stopped. |
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Post by xxxxxxxxx on Aug 19, 2019 22:09:42 GMT
Works too. The brackets are fundamentally what determines the function. Yeah, thanks. Reading Church "Introduction to Math Logic" it starts seeming clearly that brackets are really fundamental. At first sight one even feel some king of gross or something, because you're always asking yourself - where's the end of all these brackets?!! But later the system of points (e.g. (∃φ):.(∃x):φx:φy.φz. ⊃ .y = z) allows it to be stopped. Honestly I think the brackets are the most important part, they mimic the recursive properties of numbers (ie 2 is 1,1, same with 3 as 1,1,1) and set the beginning point for how the "string" is measured. You can take any property, math or otherwise, pick a different state of measuring it and get a new angle of awareness. I would go so far as to say a system can be made using only: () . → ○ For example: ○(∙1 ∙1) → ∙2 → ○(∙(0 → (∙∞ ∙-∞)) ∙(0 → (∙∞ ∙-∞))) → ○(∙1 ∙1) I will have to explain this further when I have time, it will seem obscure at first. Just think of 1 and 1 equals 2 but 2 equals and infinte number of arithmetic relations that cycles back to 1 and 1 but is not limited to it. |
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Post by Elizabeth on Aug 20, 2019 5:51:29 GMT
You must use PEMDAS.
Properly too.
And you'll get 16.
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Post by Elizabeth on Aug 20, 2019 6:36:29 GMT
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Post by Eugene 2.0 on Aug 20, 2019 8:39:12 GMT
Yeah, thanks. Reading Church "Introduction to Math Logic" it starts seeming clearly that brackets are really fundamental. At first sight one even feel some king of gross or something, because you're always asking yourself - where's the end of all these brackets?!! But later the system of points (e.g. (∃φ):.(∃x):φx:φy.φz. ⊃ .y = z) allows it to be stopped. Honestly I think the brackets are the most important part, they mimic the recursive properties of numbers (ie 2 is 1,1, same with 3 as 1,1,1) and set the beginning point for how the "string" is measured. You can take any property, math or otherwise, pick a different state of measuring it and get a new angle of awareness. I would go so far as to say a system can be made using only: () . → ○ For example: ○(∙1 ∙1) → ∙2 → ○(∙(0 → (∙∞ ∙-∞)) ∙(0 → (∙∞ ∙-∞))) → ○(∙1 ∙1) I will have to explain this further when I have time, it will seem obscure at first. Just think of 1 and 1 equals 2 but 2 equals and infinte number of arithmetic relations that cycles back to 1 and 1 but is not limited to it. Thank you for your works! I've been interested in a point as the end of a sentence (what's the difference between a sentence and a word-combination? what's the difference between a group of words and a word-combination? & is it possible for a word-combination to have sense? etc). What you've said about the brackets push me to a thought that brackets and points (in sentences) are closely a little; or, at least, something allows them to be seen as things that can be compared with each other). |
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Post by Eugene 2.0 on Aug 20, 2019 8:51:48 GMT
Let me explain more...also go left to right (or top to bottom) when doing PEMDAS correctly.    I'm very grateful to you, Elizabeth! Never heard of it... It's so shamy for me. PEMDAS works cool as a memorize tip. I should put it into my mind for good. Your conspectus look nice; also, your handwriting is really interesting too. Let me ask you - do you always write using this type of handwriting? The letters of your handwriting leaned to left a little. I knew some people with a similar handwriting and they all were persons with strong will, and also they were very friendly. |
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Post by karl on Aug 20, 2019 14:20:53 GMT
8 ÷ 2(2+2)=1 (8 ÷ 2)(2+2)=16 Hmm... I'm not sure, but don't we have to put a pair of brackets to the first example for some insurances? Instead of "8 ÷ 2(2+2) = 1", I guess, we need to write like this "8 ÷ (2(2+2)) = 1"? In this case, I hope, there's no way to confuse it by mistake further. And if this is true, the situation might be a little bit clearer: the formula is missing a pair of brackets.
That does clarify it, yes.
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Post by joustos on Aug 20, 2019 15:54:30 GMT
8-:-2(2+2) or 8/2(2+2)
In either case: the formula = (A) 8/[8] =1 by clearing the brackets first (B) (8/2)(2+2) = 4(2) + 4(2) = 16 by applying the distributive law The obvious contradiction results from the original improper formulation (as to what the writer means to say). Two possible intents: (brackets needed, as Eugene at al. have pointed out) 8/[2(2+2)]…=1 (8/2)[2(2+2)]…= 16 Conventual abbreviations, which we all are accepting, are usually not written down. E.g.: (2)(3) = 2 x 3. They have to be written down [incorporated] in a demonstrative treatise. Similarly, in a demonstrative treatise of the Pythagorean theorem, you would have to incorporate the fact that you are dealing with a right-angle triangle. All existing demonstrations fail in this respect, even if -- from Euclid on -- they are otherwise logically sound. (We assume the 90-degree angle, or, geometrically speaking, [CLUE:] the angle formed by two intersecting perpendicular lines.) Three lines on a plane whose lengths are 4, 5, and 6 need not be the sides of a triangle. Add this to your other problems, if you have time to spare. |
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