What is the correct value of $\pi$

Question

I have seen that:

1. $\pi = 22/7$

2. $\pi = 3.14\ldots$

3. $\pi = 17 - \sqrt{192}$.

4. $22/7 \gt \pi$

5. $22/7 \lt\pi$

My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or myself is totally wrong? Please discuss.

I think, if any one is wrong, then whole mathematics may be wrong, especially things which we dealt with $\pi$. These questions there from long back in my mind. I think, by your help, I can end by your solutions or reasoning.

Thanks in advance.

Answer 1

1) False. Sadly, some teachers tell students that $\pi$ is $\frac{22}{7}$.

2) Impossible to decide, depends on what is intended by the dots. The number $3.14$ (no dots) is a slightly worse approximation to $\pi$ than $\frac{22}{7}$. But the decimal expansion of $\pi$ does begin with $3.14$, and then continues.

3) Doesn't even need checking, $\pi$ is transcendental while the given number is algebraic. But one might note that in fact it is a worse approximation to $\pi$ than either $\frac{22}{7}$ or $3.14$.

4) True, $\pi$ correct to $5$ decimal places is $3.14159$ while $\frac{22}{7}\approx 3.142857$.

5) False, since in fact 4) is true.

Answer 2

Note that 1, 4, and 5 cannot all be right or all wrong; in fact, exactly one of them must be true, and the other two must be false, simply by trichotomy: given any two real numbers $a$ and $b$, either $a=b$, or $a\lt b$, or $a\gt b$, and exactly one holds.

And of course, 1 and 3 cannot both be true at the same time. So you know that there are at least two false statements, and at least one correct statement. So the assertion that "if any one is wrong then all of mathematics may be wrong" does not follow; at least two must be wrong if mathematics is to not be all wrong.

$\pi$ is a transcendental number. That means that it cannot cannot equal a number obtained from integers and rationals through the use of addition, subtraction, multiplication, and root extraction.

Two does not give you an exact value, so it is neither true nor false as written, though if you believe that the exact value of $\pi$ begins as "$3.14$" before continuing, then you may want to mark is sort of true.

Answer 3

Each of the examples you give are approximations of $\pi$. They are not actually equal to $\pi$ (I'm ignoring the $\dots$ in C). Also, none of A, B, or C are equal. These approximations are rarely used in pure mathematics and find more applications when an approximation of $\pi$ is necessary (e.g. if you are an engineer). So the fact that these statements are false does not affect mathematics significantly. (your concern that the falsity of these statements will undermine mathematics)

In fact, $\pi$ is a transcendental number, so it is both irrational (it cannot be written as a fraction) and cannot be written as a square root of a number.

Also, $22/7 > \pi$, so it cannot be less than $\pi$ or equal to it.

Answer 4

http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80

$$ 0<\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \, dx = \frac{22}{7} - \pi. $$

Anyone who knows first-year calculus can quickly evaluate this integral, and that proves that $\pi$ is not $22/7$.