Pascal’s Triangle, an interesting number patterns in mathematics.

ProtonsTalk ^ | Pravallika

[shobk]

Pascal’s Triangle has many applications in mathematics and statistics, including its ability to help you calculate combinations.

[dayglored]

> For them to come up with 16 the question would have to read what is 8/2 X (2+2).

Ah, but that is what the problem DOES read. The implied multiplication operation between the first '2' and the parenthesized (2+2) is not somehow granted higher priority than the division that precedes it. It's just a multiplication. It does not bind the '2' any tighter for the fact that it's implied.

The simple fact is that the problem is intentionally stated poorly, in order to encourage ambiguity and multiple interpretations.

The other simple fact is that the rules of order of operation are by agreement for the sake of notational clarity. One could construct a different set of rules, and as long as it was consistent, it would be valid.

But mathematicians have agreed upon one set of rules, and it's best to abide by them if you want to be understood accurately.

[dayglored]

> Although, on further reflection, I have sometimes seen PEMDAS interpreted as P-E-M-D-A-S, as opposed to P-E-[MD]-[AS], which would give a result of 1.

It's unfortunate that "My Dear Aunt Sally" got stuck in everyone's head, because "Darn My Shoes Ache" is just a valid.

What I learned on my way to a BS in Physics was [MD] and [AS], but I also learned that if you want to be sure, add some parentheses to force the order you intend, even if they were superfluous, because that way you were sure your intention is clear.

[Reeses]

I believe those characteristics are true because Math is a description of the nature of Being. If it exists, then Math describes it. I'm not a mathematician, but was trained in Physics, which is Math applied to the physical universe that comprises Objective Reality. Or as some have said, Math is the language in which Physics is written.

Math can be used as a modelling tool for reality to great accuracy but never forget models are simplifications. Models are always at least partly wrong, and sometimes very wrong.

Like God, mathematics is beyond space and time. It has eternal life with no beginning or end. It's an alien world from outside our universe that we can visit but never communicate with.

The unusual conditions that led to the Big Bang must have extremely long odds for ever happening. However when there's nothing available to measure time, everything that can happen, happens all at once. Maybe God created the universe because he needed a clock.

[jamaksin]

Think “PLICKO” ... and Normal Distribution ... and ...

[Enterprise]

We were also taught to factor the numbers, with 8 being the top number and 2(2+2) being the bottom number. When we take 8/2(2+2) it becomes thus:

_____8______
___2(2+2)___

Divide 8 by the multiplier 2 and the result is:

_____4______
____(2+2)___

Then:

_____4______
_____4______

The result being 1.

Using factoring you cannot possibly come up with an answer of 16.

I am neither a teacher nor a mathematician. I am in total agreement with you that weasels can take a basic equation like this and try to confuse someone. Going back to what I posted, for the answer to come out to 16 the equation cannot be written as 8/2(2+2). I would sit down with the person if it was possible, and write it out, as above, to show where the misconception is, and suggest they take an algebra class.

[SgtHooper]

Exactly correct! It is a numerator-denominator issue first!

[dayglored]

> Math can be used as a modelling tool for reality to great accuracy but never forget models are simplifications.

Quite true. What we think of as Math, whether simple things like 2+2=4 or bigger things like transcendentals and the cardinality of infinite sets (aleph-null, etc.), is really just a way (a model if you wish) that our human brains can use to grasp and work with the underlying concepts.

E.g. The number '2' connotes a quantity, but what is the essential nature of "two-ness"? It is at once both self-evident, and yet mysterious.

Christianity has a similar topic in the Holy Trinity. Christianity is monotheistic, yet there is a Trinity, giving two millennia of religious scholars and spiritual seekers the mystery: "How can Three be One? How can One be Three?" Quantity does not always map easily to essence.

So I do agree that Math is a modelling tool. I would differ with your assessment of it being at least partly "wrong", and opine instead that we should never confuse the model with the underlying conceptual essence. Math, and mathematical notation, are just ways for humans to get things done, and perhaps, to begin to understand the nature of God, if one is so inclined.

> Maybe God created the universe because he needed a clock.

Now THAT is a marvelous philosophical idea, worthy of cogitation. Excellent!

[Hardastarboard]

I’m waiting for Joe Biden to do a YouTube video explaining it.

[dayglored]

I think the "weaselly" aspect of the problem as stated is that the slash '/' causes the eye to imagine that it's written in fractional notation. Using the divide symbol '÷' would be less troublesome in this regard.

Now, if the slash were large, and the '8' was written higher, then the fractional nature would have been more evident:

8 /
 / 2(2+2)

But then it wouldn't be confusing, and it wouldn't go viral.

I file this viral "conundrum" under: "Play stupid games, win stupid prizes." I.e. IMO it's just a poorly and confusingly stated, and thus stupid, problem.

Makes about as much sense as:
Q: "What is the difference between a duck?"
A: "Because one feet is both the same."

:-)

[Enterprise]

Well, it’s been quite a few years, but in my education if you showed me 8/2(2+2) I would understand it as algebra, and as SgtHooper explains, the 8 being the numerator and the 2(2+2) being the denominator. If it were a math problem, it would be confusing to write it like this: 8÷2 X 2+2 and expect the answer to be 1.

[Mr. K]

Anything within the parentheses gets done first so 2+2 = 4.

The 2 connected to the parentheses is an operation on the contents within the parentheses. So 2×4 = 8.

Last, the division is the numerator 8 divided by the denominator 8.

So the answer is 1

[Mr. K]

I was taught that 2 connected is an operation on the parentheses.

so 2(2+2) = (2(2+2))

Is that wrong?

[Mr. K]

That seems to be three of us in this thread that have a BS in physics.

[dayglored]

> I was taught that 2 connected is an operation on the parentheses. so 2(2+2) = (2(2+2)) Is that wrong?

It's a matter of context. Written as above, by itself, that's correct.

What causes a difference in how you evaluate the overall expression is the "slash" that precedes the multiplication.

If you interpret the slash as a fraction line (making the 8 the numerator and everything else the denominator), then as you state, the multiplication binds to the parenthesized (2+2), because it's all in the denominator, and the answer is 1.

But if you interpret the slash as a division operation, no fraction involved, then the evaluation proceeds left-to-right, the division happens BEFORE the multiplication, and the answer is 16.

The latter interpretation is the accepted standard.

The problem as it went viral was intentionally written poorly, so as to encourage ambiguity and multiple interpretations.

[dayglored]

> That seems to be three of us in this thread that have a BS in physics.

LOL, plenty to go around.

My degree was from Drexel U in 1974. I actually entered as a EE major, but the first-year Physics courses were seductive and I switched majors, and got my degree in Physics rather than EE. But in the last two years of study, I realized that I really was much more EE than Physics, and so when I graduated I got a job doing EE and computers rather than go for the MS and PhD in Physics.

I don't regret the extra Physics courses I took, even though practically speaking they were tits on a boar hog in terms of employment. They were an invaluable blessing for the math skills, the discipline, the problem solving skills, etc. that they gave me.

[21twelve]

“Math can be used as a modelling tool for reality to great accuracy but never forget models are simplifications.”

Back in college the first day the prof had the wave equation written on the board - it took up the entire board. He said something like “Now this is obviously too long to work with. But, if we assume a homogeneous earth we can get rid of this (erases a bunch of lines at the end)....

He kept going with numerous assumptions and got it down to two main parts of the equation IIRC. I’m guessing that with modern computers they can do the math fast enough to account for more variables now. (If they know what those variables are!)

For the answer we had to get it to whatever decimal place and show that, but then for the final answer (say a seismic velocity) we rounded it to something. 8,723.24 ft/sec -—8,700 ft/sec. “Remember that wave equation? No way you are figuring out the correct number to the 100th!”

Of course those are conventions. In the above example, I’m not sure how widespread the conventions were. I’m guessing if it was a math professor giving me the test (rather than a geophysicist) my answer of 8,700 would have been wrong.