Isn't it equal to 1?
π 2 2 / 7
Which of the following is true regarding the number above?
Note: π is the ratio of the circumference to the diameter of a circle.
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57 solutions
Moderator note:
Yes, this is essentially the proof that pi is greater than 22/7 . To increase the bounds of π , simply increase the powers of 4 to any multiples of 4 and express the integrand as a proper fraction via long division and apply d x d ( tan − 1 ( x ) ) = 1 + x 2 1 .
Without resorting to integrals, can you prove that 7 2 2 > π ? Hint: Exhaustion.
Great! , but how did you think of that?
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I didn't think of that maths researchers thought of this, the integration came into IIT-JEE paper as a multi -correct problem with the options :
1)22/7-pi 2)22/7 3)pi 4)0
Interestingly many people did 1),4).
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Interesting questions leave their footprints in our minds.
Hmm, good proof, but it seems needlessly complicated.
Another way to look at it is 22/(7*pi)
7 times pi is 21.987... But once you see the first decimal, it doesn't matter. You know it's not going to be greater than 22. So 22 divided by anything less than 22 but greater than zero is greater than one, so the answer is "greater than one."
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Yeah, but it says "You are not given the value of pi". So, how could you know that "7 times pi is 21.987"? Of course I know the value of pi, but I think we are not alowed to use it.
Thnx...for easiest solution
Challenge Master. How is this less circular than just using the value of pi and doing the division? Either way you assume a value of pi, or in this case when evaluating the integral arctan(1) = pi/4 which you still need to use some given value for pi.
This is just a very inefficient and round about way of doing a simple division problem.
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Funny how people tend to take such a simple problem and turn it into such a difficult solution
He/she means that we don't need to know the value of pi at all to prove that it's lesser than 22/7. The rest of the "solutions" already applied the fact that pi = 3.14159..., so their proofs are circular.
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If you dont know the value of pi how can you claim it is less or greater than 22/7? 22/7 / birds >1 Prove it.
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@Jeff Wallace – That's the point, I don't need to know the value of pi but I know that it's less than 22/7.
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@Pi Han Goh – The pi symbol could represent anything in one reading of the problem statement; it would just be a free variable, so indeterminate is a respectable choice.
But I see that it is quite an interesting question if you assume that pi represents the ratio of circumference to diameter of a circle.
My question is how did we know the derivative of the arctangent. I may not recall correctly (as you can see by my age I am reaching far back into memory) but believe I was taught this identity without proof. Does it depend on some argument whereby we could evaluate pi directly?
@Pi Han Goh – This solution relies on a given value of pi though. How else would they prove that 4*arctan(1) = pi = C/d? Its the same pi? did they compare out to the infinite decimal place?
They just magically know the values match up? They dont show it in their answer..
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@Jeff Wallace – Let me repeat again, C/d is not the only formula to determine the value of pi. Ronak's integral shows that it is strictly positive, and it evaluates to 22/7 - pi. And because we know that the integral is positive, then 22/7 - pi > 0, or 22/7 > pi, or (22/7)/pi > 1. Did you even read his solution? Or you just ignore it because it doesn't coincide with your reasoning?
Almost every human who can create a Brilliant.org account can efficiently do division. In order to post a solution, one needs to put something new and go beyond the usual. Moreover, many people had already posted the simple division solution
22/7 is greater than pi so 22/7 divided by pi equals >1
Well, when you make the statement that you are not given the value of pi, then pi can be any number (including a negative number) and thus there isn't enough information to make a conclusion. Anything else and you are making assumptions.
Excuse me -- you have pi >22/7 in your comment. Not Pi<22/7.
(22/7)(1/π) >1
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22/7 & pi are of equal value....
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Except they aren't, exactly. Neither stop at 3.14. They keep going, and they're different from the third decimal place (π = 3.14159..., 22/7 = 3.14285...)
Great, but in the problem is stated that the value of pi is not given, therefore it is defined as a variable, which can be substituted for any number
This is an extremely specific solution that you essentially have to be aware of an know ahead of time in order to solve or understand it. That makes it completely useless in terms of practical math or problem solving. Given the fraction 22/7 and some arbitrary number (the term pi is used in the question but it clearly states we are not given the value of pi. It does not so clearly state if that means it is a random and unknown number or if it is just such a common term that we should know it is 3.14.... ) How would you be able to probe without calculus and with out knowing the value of pi that this is correct? Or at least how would you get to that integral which proves it? How do you get the integral you used without pulling it out of thin air or looking it up on wikipedia?
I didn't see the option- 'greater than 1'. It was down the scroll so I had to tick none of the above. Lol! How lazy am I! I didn't even scroll it down.
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Ummmm, its the first choice... You had to scroll passed it to get to your answer of "none of these choices".
I would like to know because I want to learn, where did u find that function and why did u use it to prove it, I did it by dividing but that's not the right way. Thank you
Nice explanation , but I am still trying to understand it
Or, you know, 7/22 = 3.1428 pi = 3.1415. Obviously the first is higher.
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Yeah, that's how I figuered it. I must be the retard for doing the easy math.
I understand the answer, but could the question be edited? When it asks "Which of the following is true?" I believe ideally there should be nothing following other than the choices. Here, the following statement is "Pi is the ratio of the circumference to the diameter" (a true statement). This question is made clearer simply by moving this line above the equation.
Funny, when phi intended as 3,14 ..... It means pi represents the ratio of the circumference of the circle (2phi * r) to diameter (2r). It also means that the real value of phi 3.14 .... it comes from the decimal expression of 22/7, so it should phi = 22/7 = 3.14 .... and It's not phi < 22/7 or otherwise.
Why is willing to spend time proving by using an integral approach or even arc tan. You can't apply phi which represents the ratio of circumference of a circle to its diameter by leaving it's basic philosophy (https://en.wikipedia.org/wiki/Pi) and refracted by the application of phi in various other mathematical operations that contain little bias.
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@Alimun Bidzatissudur No.Pi doesn't come from the decimal expansion of 7 2 2 .In fact, 7 2 2 is used as an approximation for Pi.And they're not equal.
π = 3 . 1 4 1 5 9 2 6 5 3 . . . . .
7 2 2 = 3 . 1 4 2 8 5 7 1 4 2 8 5 7 . . . . .
And lots of scientists and mathematicians spend time on apparently "useless" problems such as the Sphere Packing Problem,the Four Color Theorem and so on.Loads of research and individual and collaborative effort have also gone into computing the value of Pi and studying it's properties.So you cannot claim that there is nobody willing to spend time on such problems.
22/7 is 3.14... Which is pi. So, pi divided by pi is 1. But I don't understand why it's not right
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22/7 = 3.14, yes, but it's not exactly 3.14 If you extend it, it's 3.1428.... Whereas pi is 3.14159. 3.1428/3.14159 will result in a figure greater than 1
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I love simplicity in math. Congratulations! Awesome solution!
You're not wrong. But there's more than 2 decimal place for the values of pi and 22/7. Yes the answer is 1.000..... But it's not exactly 1
awesome reply
Isn't 22/7 a recurring number whereas pi is irrational but we Know it's less than 22/7 since said number has every seven digits beyond the decimal point in the same order, over and over, while pi has been worked out.. You don't need algebra considering you can work out each separately, to seven decimal places.
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I don't think you know how math works... 3.1498989898... and 3.1412121212... are both recurring numbers but one is bigger than pi and one is smaller. What logic did you use to determine that irrational numbers are always smaller than repeating ones?
For the sake of simplicity, simply type 22/7 ÷ π in your Scientific Calculator, the answer you get is greater than 1
22/7 is 3.14..... Which is pi. How come pi divided by pi isn't 1
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You drew the '...', that should have clued you in. And did you just say pi is exactly 3.14? Seriously?
So first off, pi is not 3.14, secondly, 22/7 is not 3.14. Instead pi is 3.1415... and 22/7 is 3.1428...
@Abhishek Sharma i do not agree with your point. pi is an approximation of 22/7 . so no matter how how far you go in identifying decimals places of 22/7 it will still be called pi. we use approx value because its a irrational no , they are both just the same no with names being different until we came with a proof where they both changed values.
and the integral you have provided is also an approximation. integrals generate approx value in itself.this is not a proper reasoning to justify .
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Nope aditya you are strongly incorrect in your argument Pi is an exact value no matter what which can be represented suitably by integral, and if you think 22/7 is equal to pi even for a blink of a second then I have to tell you, you have to learn maths all again.
Also integrals are not an approximation but finite riemann sums are but as we take the limit we get the EXACT value of the integral.
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Let me remind you I am not stating about the nature of pi. Although it is exact but for practical reason we use approx value.and if you are so certain about answer I strongly urge you to provide exact value of pi so we can judge the proper reasoning.and if not don't give your question paper it's not a good proof of answer.
22/7 is a nearest rational representation of pi. it is not equal to pi. pi is a ratio of a circumference of circle to its diameter.
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Not true, see the link given, you could increase the accuracy of the rational representation of pi. For example 1 1 3 3 5 5 is a better approximation compared to 7 2 2 . This gigantic one is even better: 2 3 5 , 9 5 3 , 5 1 7 , 8 0 0 7 4 1 , 2 6 9 , 8 3 8 , 1 0 9 . The link given in the Challenge Master Note also demonstrate how to increase the accuracy of the bound.
22/7 is an approximation of pi.
pi is a ratio between circumference and diameter of a circle.
Either way I agree with you that the integral isn't that great of a proof. If anything its less valuable than just dividing since with the simple division you waste less time.
By long division, you can find that to 4 decimal places, 7 2 2 = 3.1429...
We know that pi is 3.1416... to four decimal places
(if you don't believe me, you can evaluate the maclaurin series of 6*arctan(x) for x= 3 1 to 8 terms for this accuracy)
So since 3.1429>3.1416, 7 2 2 >pi
So 7 2 2 /pi>1
Moderator note:
Nicely done! For the sake of variety, can you think of a solution that doesn't resort to calculus? Hint: construction.
Great job Paul, when different vies are used to explain a mathematical problem the easiest explanation is usually chosen as the correct solution. You and I Agree. Great job!
Thatz great really
If we see the geometrical proof then the string that we assume to fall in the 7 th part out of the 22 parts on the circumference of the circle is nearly less than it so the solution is greater than 1.
Moderator note:
Can you further elaborate on it?
I see a lot of people using pure algebra to solve this problem. What about where pi originally comes from, a circle? Here is a link to how Archimedes shows that 22/7 is greater than pi http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html I honestly would like to see more interesting geometric takes on this problem, I could do one but do not have much time xD
because 3 x 7 = 21. in the problem is 22 therefore it is greater than 1
Moderator note:
Wrong. What is so significant about 3 × 7 = 2 1 ? Why not 4 × 6 = 2 4 > 2 2 ?
i think he means, challenge master, that pi rounds to 3 (1 s.f.) so he used that like that.
22/7 = 3 & 1/7, which is 3.1429 with 4 decimal places of accuracy
pi = 3.1416 with 4 decimals of accuracy
3.1429... > 3.1416
Therefore, (22/7)/pi > 1
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
Why is this circular reasoning? I don't get what's wrong, please help me. @Brilliant Mathematics
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Hi Vinayak, the author of the solution has already implicitly know the value of π , which defeats the spirit of this question. Which is to prove the inequality without knowing the numerical value of π to begin with.
22/7 is 3.1428.... Pi is 3.1415...
3.1428>3.1415 so it's greater than 1.
The question says 22/7*3.14=? ..... i.e. 22/21.99... and hence it is greater than 1,
It's easy to calculate that 22/7 = 3 with remainder 1. This 1 is also over 7, so becomes 1/7. Meaning that 22/7 = 3 and 1/7.
So, to solve it, we just need to know whether 1/7 is larger or smaller than 0.1415 [as pi = 3.1415...]
To make it a seemingly easier calculation, we'll just call it 100/7. Which is 14 and 2/7. By the same calculation (200/7), this is larger than 28.
So, by reverting these to their proper factors, 1/7 is higher than 0.1428.
Therefore, 3 and 1/7 is greater than pi. So 22/7 divided by pi is greater than 1.
Challenge Master: It seems to me that your "Begging the Question" reply is in itself begging the question.
Besides: It doesn't say "Prove your answer." If it did, I would not have been able to do so. The answer is still "Greater than one." And THAT is what the question was.
No need to hit the problem with a sledgehammer (say pulling out integrals) when a flyswatter is all you need. Here's a simple solution (something I'd get my undergraduates to do if this were an exercise): bound pi. Since 3 < pi < 3142/1000, consider the left side and the right side. If both are greater than 1, then (22/7)/pi > 1 (*).
(a) LHS case: (22/7)/3 = 22/21 > 1.
(b) RHS case: (22/7)/(3142/1000) = 22000/21994 > 1.
With (a) and (b), by (*), (22/7)/pi > 1 and have proven the proposition. This is a fun but simple technique to get students to try bounding arguments. Note that this argument worked due to all the values involved being constants.
Side note: If people are anal about the observation that pi < 3142/1000, just do the following (though this should be trivial to observe using contradiction). Assume pi >= 3142/1000. This holds if and only if pi 1000 >= 3142, and if and only if pi 1000 - 3142 >= 0. The fractional parts of both are <1, so we can proceed by taking their integral parts. Take the difference of their integer parts and you get 3141 - 3142=-1<0, which is a contradiction! It's straightforward to do the LHS of the inequality as well if necessary using a similar argument by instead considering their fractional parts.
Hope this helps!
I love your methodology but you're contradiction proof still assumes the value of pi to be 3.1415926..., in order to avoid using the exact value of pi, many answers try to use arct(1)*4 or anything else that would give us the value of pi. I like your proof, in fact you can make this solid by adding some geometry into it. Try to find two bounds you can prove using geometric shapes that are close to circles, use the same logic and you have a solid argument on your hand!
Lol circular argument or not they got the correct answer with wetwear... As did I
22/7=3.1428.... Pi=3.1416...
I think the answer is obvious...
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
Yah, why bother with integrals when the answer is obvious...?
@Challenge master. I went out and turned a large bar to be very concentric. I then measured the diameter and circumference to use when evaluating pi =C/d the math just came out that way...
Easiest solution: pi is not exactly 22/7(or 3.142857) but it is 3.1415926535 which is less than 22/7 thus 22/7 divided by pi is greater than 1
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
When dividing by 7, the digits repeat in this order: .142857 where 1/7=0.142857, 2/7= 0.285714, 3/7= 0.428571 and so forth (#/7 equals remainder and that is the remainder lowest number of .142857). As a result, we know that pi=3.1415926535. 22/7 has a remainder of 1 so it is 3 and 1/7 which is equal to 3.142857. Comparing the thousandths places, 2>1 so (22/7)/pi is greater than 1.
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
22/7/pi=1.0004 which is obviously greater than 1
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
Just convert 22/7 into a decimal. Knowing Pi as a decimal you can see the fraction using the decimals is greater then one. No need to overdose on math logic
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
22/7>pi so therefore more than one if in the numerator which it is
Moderator note:
But how did you know that 7 2 2 > π in the first place?
We know that π =3.141592... and 22/7=3.142857...
π 2 2 / 7 = 3 . 1 4 1 5 9 2 . . 3 . 1 4 2 8 5 7 . . .
Since denominator is less than the numerator.So, the value of fraction must be greater than 1
Therefore the answer is the value is greater than 1
Moderator note:
Aditya Anilkumar is right. Your argument is basically a circular argument. It's like saying 7 2 2 is greater than π because it says so.
You cannot really argue like that. How do you know that pi=3.14159.....?
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Maybe Vishal went out and measured the circumference and diameter of a big round object and found the decimal himself? Just like dividing 22/7...
The only answer I haven't seen shut down was the integral proof but how do you know 4*arctan(1) = pi? Maybe an identity?
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Even if he did, he failed to mention it in his solution.
Yours is wrong too.
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@Pi Han Goh – Every answer is wrong unless you use pi = C/d and go measure some precise round stock, then do the math yourself. Every answer I have seen shut down relies on some given value of pi, but also every answer that is accepted also relies on some given value of pi. In the Leibniz formula for pi did you evaluate all of the terms to get pi/4?
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@Jeff Wallace – That's not the only way to measure pi, see the second top answer. The top answer does not require to calculate any decimal places of pi at all.
" but also every answer that is accepted also relies on some given value of pi." How so?
"In the Leibniz formula for pi did you evaluate all of the terms to get pi/4?", nope, you just need to show that pi/4 = arctan(1).
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@Pi Han Goh – Please show me how pi/4 = arctan(1) without using any identities or constants.
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@Jeff Wallace – I don't see why you're being so adamant about this. Did you even bother reading the link I sent you? Or you simply don't understand? Or you're just trying to find simpler ways to understand why 22/7 > pi?
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@Pi Han Goh – I did see the link. Using that series to obtain pi with 10 decimal places you have to evaluate "about 5 billion terms". You could spend your whole life dealing with that. The problem is dismissing the much simpler use of pi as a mathematical constant, which you could take the value, or easily calculate out to enough decimal places to prove 22/7 > pi in favor of taking someone else's word that arctan(1) = pi/4. Both answers are correct, and have sound logic. Why is one not accepted?
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@Jeff Wallace – You're missing the point of what I'm trying to make again. I'm trying to show you a connection that pi/4 is equals to arctan(1). With Ronak's solution, you will eventually have to go through the working to show that there will be integral ∫ 1 + x 2 1 d x . I did not in any point mention that we should calculate pi to any decimal places. Let me repeat this again: His solution basically shows that because the integral is strictly positive and it exactly equals to 7 2 2 − π , then we can make a conclusion that 7 2 2 > π or π 2 2 / 7 > 1 . There is no place in his calculation to determine the value of pi. He just shows that it's larger than pi, that's all to it. We are not concern with the value of pi, I don't care whether you can evaluate it to 10 decimal places or more, we just need to show that it is less than 22/7.
How would you "calculate out to enough decimal places to prove 22/7 > pi"? Like I said, there's more way to prove the inequality is true without finding the value of pi. You logic is not accepted because you already accepted that pi = 3.1416.... (which is wrong). It's like saying that you proved that 3 8 6 6 9 2 7 + 4 1 1 4 1 3 7 = 4 4 1 8 4 9 7 simply because it's not true, hence the circular argument. It's equivalent of saying that it's raining because it's raining. You have already made up your mind that pi is less than 22/7 because pi is less than 22/7. Where is your reasoning? Is it because pi = 3.1416?... If so, how would you obtain that? If you said C/d, then prove it! Don't say "hey! it's 3.1416 because of some concrete block gibberish". Even if that is what you did, show your proof!
'pi' is approximately but not exactly equal to the fraction 22/7. It is just an approximation which is never equal and always slightly less than 22/7.
NOTE: 'pi' is irrational while 22/7 is rational (since it can be written in the form of P/Q, where P and Q are integers and Q is not equal to zero) though both seem to be equal.
Moderator note:
Do you have proof that it is "always slightly less than 22/7"?
not exactly a convincing solution, this one
When we look at the thousandths place of each part of this problem that proof is there in decimal form.
Let's rewrite this as 7 π 2 2 . Now the question is simple. How does 7 π compare to 2 2 ? Well, 7 π ≈ 2 1 . 9 9 1 4 . . . so it is less than 2 2 so our fraction is less than 1
Since π=3.14159265359.... Then 22/(7π) =22/21.9911485751.... ≥1
π 7 2 2 = 3 . 1 4 1 5 9 2 6 . . . 3 . 1 4 2 8 5 7 1 . . .
Thus, 7 2 2 > π
Therefore, π 7 2 2 > 1)
No need of the integral. Take calculator and know 22/7 > 3.1415.
Or if you are mediocre like me, just type it in a calculator. I'm only grade ten so I don't know anything about other people's complicated proofs.
22/7 = 3.142857 recurring.
π =3.1416...
Therefore (22/7) / π > 1
22/7/pi ; law of fraction 22/7 x 1/3.14= 22/7(3.14) = 22/21.98=1.0091; its graeter than 1
22 / 7 is slightly greater than π , therefore, their ratio is greater than 1.
If the value 'pi' isn't known than the result isn't known........
Also the integral outcome of the tan.
hey, can't it be thought without INTEGRATION ??
Know that pi=3.14 ...... (to infinity) Observe that 7*3.14 = 21.98 Now since pi is an infinite number, we can observe that 21.9.... will extend to infinity. However, although it approaches 22, it will never reach there.
ex. (calculating up to first five digits of pi you have)
7*3 =21
7*3.1=21.7
7*3.14 = 21.98
7*3.141 = 21.987
7*3.1415 = 21.9905
As one can see, as you calculate more numbers of pi into the equation, the result gets bigger but in smaller increments. Thus, although it approaches 22, it will never reach there. Ergo, the answer is greater than one.
I just did 22 divided by 7 to get 3.1428... and I know pi is 3.1415... That would mean 3.1428 divided by 3.1415, and a bigger number divided by a smaller number is always greater than 1.
22/7= 3.1428... pi=3.1415..
3.1428../3.1415..= Approximately 1.00041
1.0041... > 1
22/7 / TI = 3.1428571/3.14159265 = 1.0004025 greater than 1
22/7/pi equals (22/7) (pi/1). (22 pi) should be greater than 7 (22* approximately 3.1459). Because of this, the result of that multiplication divided by 7 should be greater than 1. This is how I got to the conclusion. :')
I mean there are two ways to hold your nose- one is how everyone would normally do and the other is taking your arm around the neck and holding from the other side.
Simply, you move the 7 down with the π so you get 22 over 7π. Approximately, we know that π is 3.14, and that multiplied by 7 is 21.98. So now we have 22 over 21.98, which is slightly larger than one.
But your answer depends on you knowing that \pi is 3.1.415.... - which we were told in the question that you don't know - i.e. the answer you need to use is one where you never use 3.141579..... (or any approximation of it)...
22/7 divided by pi becomes 22/7 multiplied by 1/pi becomes 22/(7)(pi) by using calculator, the answer is more than 1
22:7=3.14285714.>>>>>>>>>>>>PI IS=3.14159
pi = 3.141592653 | 22/7= 3.142857
It's just fraction has a an exact number of what it suppose to have and pi is only take the exact number in just a few decimal places
3 x 7 = 21. in the problem is 22 therefore it is greater than 1
where did you get 3 from ? π isn't 3 - it might approximate to it (if you ignore all the decimal places) - but the question didn't ask if the equation was approximately greater than 1.
Since 22/7 = 3.1428,while value of PIE is 3.1415,there fore value of given question is more than 1 .Ans
Just use calculator to find 22/7 = 3.1428... which is more than pi (3.14159...).
The solution below is true, but there is a simpler method
Everyone knows that 21/7 = 3. However, many people don't know that 1/7 is .142857 recurring. Therefore, 22/7 is 3.142857, which does NOT equal pi. Pi is 3.14159265 3.142857 is greater than 3.14159, therefore, the solution is greater than one.
For the simplest possible solution, remember that pi (up to 5 decimal places) is 3.14159.
Then do simple division of 22/7. It yields you 3.14285
Now, let's do the last step. Divide the quotient of 22/7 and the value of pi up to 5 decimal places.
3.14285/3.14159
The value of pi is clearly lower than 3.14285. Looking at this example: 4/3 = 1.33
A larger number on top will yield a number > 1. (very simple)
Thus, the final quotient is > 1
22/7/π is nothing but 22/7*3.14 if we simplified the answer will be1.0009099181 which is greater than 1
Hi Hari in response to statement that you made you will find the sum to be 1.00040249943.
Take care my friend. You did good young grass hopper!
I honestly didn't believe that you can solve this problem in so many ways.
Can u give the solution of this integral plzzz
Man I simply used a little funny trick. The question was 22/7divided by pie.. Ok 👌 I have the value for 22/7 as 3.14285 also, we write pie in decimal as 3.14159 ..so 3.14285/3.14159=1.0004010 which is greater than 1 so answer is greater than 1
Simple. You can rewrite ((22/7)/pi) as (22/(7*pi)).
7 times pi is 21.987... (Keeps going) But once you see the first decimal, it doesn't matter. You know it's not going to be greater than 22. So 22 divided by anything less than 22 but greater than zero is greater than one, so the answer is "greater than one."
Just divided 22 by 7 and found it to be slightly bigger than the approximate value of pi. That's how the answer is greater than 1, even if it is by a slight margin.
My thought process was that since 22/7 is an approximation it had to be slightly bigger than pi. So by logic the bigger number over the smaller one would equal to greater than one. Idk if that's sound logic :/
3141/1000 is an approximation also - but it is smaller the π - you can't assume that all approximations are greater than the values they are approximating
It's quite simpler than it looks. You have 22/7 all over pi. If you multiply 22 by pi you get 66.11, if you divide that by 7 you get 9.87 which is greater than 1
Your method is calculating 22/7 times pi, not 22/7 divided by pi
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well if its 22/7 dividing pi then 22/7/pi = 22*pi /7 = 66.11 /7 = 9.87 which is greater than 1
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I'm sorry but I don't think that's right - 22/7 divided by pi = 22/(7 pi) not (22 pi)/7. If you want to check type it into a calculator and see that (22/7) divided by pi = 1.0004
just punch it in the calculator
That's what I did. It's a very easy question. It's interesting though.
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Agreed. Still don't understand why people have to resort to using integrals to solve this question.
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Seems to me the problem is not the question but what is Pi?
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@Andy Brewerton – A mathematical constant which denotes the ratio of a circle's circumference to its diameter.
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@Martin Kok – As a decimal
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@Andy Brewerton – 3.1415.... It is a non terminating number.
22/7 is 3.14286 Meanwhile, Pi is 3.14159
Therefore, the quotient is greater than 1.
For God's sake, why can't you guys keep it simple! 22/7 = 3.14285 and π = 3.1416 And so for this fraction (22/7 ÷ π ), the numerator is greater than the denominator, hence the result is always greater than 1. For the sake of simplicity!
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
What I used too. Works fine, save the cal for more complex problems.
I am not familiar with calculus; so I just took expressed the values of 22/7 and Pi in decimals to 5 decimal places each.
22/7 = 3.14285 Pi = 3.14159
Then I just compared the two values and found 22/7 greater than Pi.
Therefore, since the numerator (22/7) is greater than the denominator (Pi), the fraction is greater than 1.
Moderator note:
This is a circular argument. You had already made up your mind that π < 7 2 2 in the first place.
So, of course I know pi to be 3.14159... but in the problem it states that we are not to know the value of pi making me understand it as a variable. There are no other conditions for pi (as a variable) so I could understand the "None of these choices answer."
Observe that 1 + x 2 x 4 ( 1 − x ) 4 ≥ 0
Hence ∫ 0 1 1 + x 2 x 4 ( 1 − x ) 4 > 0
The integral results into 7 2 2 − π
Hence we have 7 2 2 − π > 0
Finally we have 7 2 2 > π