Pi

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Bishal21

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Which of the following is closest to the actual value of pi in mathematics?
A)22/7
B)3.1413
C)3.1415
 
Bishal21 said:
Which of the following is closest to the actual value of pi in mathematics?
A)22/7
B)3.1413
C)3.1415
Click to expand...
Get a calculator and compare them! That will give you as many digits of pi, and of 22/7, as you need.

Or are you required to do this by hand?
 
Bishal21 said:
Which of the following is closest to the actual value of pi in mathematics?
A)22/7
B)3.1413
C)3.1415
Click to expand...
[imath]\pi \approx 3.1415926535897932384626433832795028841971693993751058209749445923...[/imath]
 
One way to compare numbers is by looking at their differences. We use subtraction for that.

pi, correct to five decimal places:
3.14159

Approximate differences between pi and the given three numbers:

22/7 - 3.14159

3.14159 - 3.1413

3.14159 - 3.1415

The closest numbers are those whose difference is smallest.

?
 
Bishal21 said:
The answer is given as 22/7 which I'm not able to understand.
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That answer is not correct. If you look at the differences, you'll see that 22/7 is the farthest away from pi. So, it's the worst of the three.

By the way, 355/113 is a very good approximation for pi, better than any of the others. Using pi correct to eight places:

355/113 - pi = 0.00000027

?
 
Otis said:
That answer is not correct. If you look at the differences, you'll see that 22/7 is the farthest away from pi. So, it's the worst of the three.

By the way, 355/113 is a very good approximation for pi, better than any of the others. Using pi correct to eight places:

355/113 - pi = 0.00000027

?
Click to expand...
They say other 2 numbers are terminating decimals while 22/7 gives a non terminating decimal and hence more accurate.
 
Otis said:
That answer is not correct. If you look at the differences, you'll see that 22/7 is the farthest away from pi. So, it's the worst of the three.

By the way, 355/113 is a very good approximation for pi, better than any of the others. Using pi correct to eight places:

355/113 - pi = 0.00000027

?
Click to expand...
Wow, the things you learn on this forum!
 
Bishal21 said:
They say other 2 numbers are terminating decimals
Click to expand...
That is true.

Bishal21 said:
They say ... 22/7 gives a non terminating decimal
Click to expand...
That is true.

Bishal21 said:
They say ... hence [22/7] more accurate.
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That is false. It's the size of the number that matters, not its decimal form.

Counterexample: Which is closest to 0.3?

0.31
0.32
1/3

1/3 is a repeating decimal, yet 0.3333... is not closest to 0.3000...

Using approximation to four decimal places:

0.3333 - 0.3000 = 0.0333
0.3100 - 0.3000 = 0.0233

233 ten-thousandths is less than 333 ten-thousandths, so the terminating decimal 0.31 is closer to 0.30 than the repeating decimal 1/3.

?
 
Bishal21 said:
They say other 2 numbers are terminating decimals while 22/7 gives a non terminating decimal and hence more accurate.
Click to expand...
Who are "they"? Why do you trust them?

It is true that 22/7 is more accurate than the commonly used approximation of 3.14; but adding a couple decimal places changes that.
 
Bishal21 said:
They say other 2 numbers are terminating decimals while 22/7 gives a non terminating decimal and hence more accurate.
Click to expand...
Whoever ”they” may be, they know nothing about mathematics.

They probably heard that 22/7 is a better approximation than 3.14 and heard some pseudo-mathematical “explanation.” Idiots.

The explanation is that

[math]\dfrac{22}{7} - \pi \approx 0.00126\\ \pi - 3.14 \approx 0.00159[/math]
WHICH IS WHAT OTIS HAS BEEN SAYING.

As Dr. Peterson points out, adding a single decimal unit changes that.

[math]\pi - 3.141 = 0.00059.[/math]
 
Last edited:
In defense of 22/7

You can do mental math with 22/7 (0.04% error) much faster - even with assumption of pi ~ 3 (~5% error) - good enough to appear "smart".

Actually the % error for assuming pi ~ 22/7 (~0.0402%) is lower than that of pi ~ 3.14 (~0.0507%).

By the way, my statements above does not quite address the problem posed in OP.
 
@Subhotosh Khan Oh Khan of khans, you are of course correct that 22/7 is an approximation that is good enough for most practical purposes. But that is not at all the questions raised by the OP. The initial question was which of three approximations was more exact. He had that wrong, and Otis explained why. He then suggested that the best approximation of pi must be a rational number that cannot be expressed exactly in a finite number of decimals. That is truly absurd because there is no definition of "best approximation" to an irrational number.
 
JeffM said:
@Subhotosh Khan Oh Khan of khans, you are of course correct that 22/7 is an approximation that is good enough for most practical purposes. But that is not at all the questions raised by the OP. The initial question was which of three approximations was more exact. He had that wrong, and Otis explained why. He then suggested that the best approximation of pi must be a rational number that cannot be expressed exactly in a finite number of decimals. That is truly absurd because there is no definition of "best approximation" to an irrational number.
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I presumr you mean "best rational approximation" to an irrational number.

The "best approximation" to any number is that number itself.
 
HallsofIvy said:
I presumr you mean "best rational approximation" to an irrational number.

The "best approximation" to any number is that number itself.
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Because pi cannot be expressed in the life of the universe, it has no physical correlate and so exists only in the realm of ideas. You cannot use “pi itself” in any computation.
 
I don't know what you mean by "cannot be expressed in the life of the universe". I would say that pi can be clearly "expressed" by "the ratio of the circumference of any circle divided by the radius of that circle." If you mean that it is an infinitely continuing decimal, that has nothing to with the existence of the number itself or whatever you mean by "physical correlate". The number \(\displaystyle \sqrt{2}\) has an infinitely continuing decimal and is the length of the diagonal of a square with side length 1. If you agree that "1" has a "physical correlate" then you must agree that "\(\displaystyle \sqrt{2}\)" has a "physical correlate" and that is no different from \(\displaystyle \pi\).
 
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