Hello. I understand (sort of) that pi is an irrational number, which goes on to an infinite number of decimal places without repeating. What I would like to know, however, is; what is its formula, i.e what would I have to key into my trillion-digit calculator to get the answer 3.141592653589...? I know that 22/7 is an approximation, but is the actual formula something like that, that a reasonably intelligent non-mathematician could understand? Thank you very much. Sam Kelly, age 57, Oldham Lancashire

The circumference (perimeter) of a circle is given by the formula
C = 2 (pi) (r), where r is the radius. Since radius is half of the diameter, d = 2 (r), so C = (pi) d.
Dividing both sides by d to get pi by itself, pi is the circumference of a circle divided by the diameter.

Pi is irrational, meaning that it goes on forever, without repeating numbers and cannot be represented as a fraction.

In everyday life, we often use 22/7 as an approximation of pi. Other people use 3.14.

I have memorized what my calculator gave me plus one more decimal, 3.1415926536.

Others have memorized it to thousands of digits and people have spent their lifetimes calculating pi to more and more decimals. Books have been written about pi.

Some schools celebrate Pi Day, since Pi begins with 3.14, it is celebrated on March 14th.

Good luck,

Paul


https://en.wikipedia.org/wiki/Pi

All integers are rational numbers. The easiest way to see that is to write an integer as a fraction with 1 as a denominator; 4=4/1=8/2; 5=5/1.
Any number that can be written as a ratio of two integers numbers is by definition a rational number ( it is a ratio---rational).
Any decimal number with a periodic decimal part can be cast as a fraction, so it is rational.
An irrational number is a number that cannot be written as a fraction.
Examples: PI, sqRT(2), Sqrt(3), SQRT(5), e. There is an infinite number of them.

22:7

An integer (whole number) or a fraction, as opposed to an irrational number which would have many decimal places, like the constant Pi.

There are a lot of irrational numbers. In fact there're an infinite number of them. One is called "pi". Another is called "e". Another is the square root of two. Do you have a particular one in mind?

pi is a transcendental number, which means (among other things) that it has an infinite number of decimal places. You can see the first one million digits here. By the way, it's 3.14159265...

pi is an irrational number, with an infinite number of decimal digits. (It has an infinite number of digits in any base.) As such, it cannot be represented exactly in any finite calculator.

Not all decimal representations have equivalent fractions. Take pi. Its approximate value as given by the Casio FX9750II is 3.141592654. This is just an approximation because the string of decimal digits in pi has no end. It continues . Such a number that cannot be represented by a ratio of two integers is not rational, not a fraction. It is called irrational.Other such numbers are ?2, ?3,?5, e the base of natural logarithms.
As a matter of principle, if a number is called irrational, it is because it is not rational (not a fraction), and it makes no sense to try to cast it into fraction form.

However, if you truncate an irrational number so as to keep only a finite number of decimal digits, then that truncated number can be represented by a fraction, but qualitatively speaking, it is not the original irrational number.
Thus if you decide to use 3.141500000000000000....0 as an approximate value of pi then you write
3.1415=31415/10000 and the calculator will convert it to the mixed fraction 3u283u2000. To effect this conversion, you press the [FD] key to the right of [a b/c]. If you want an improper fraction 6283u2000 you press [SHIFT][[FD].

As to the value of sin 45 you quote, you should be aware that it is valid for an angle unit in radians.
It is an approximation because the radian unit is based on the value of pi, which, as I said above, is an irrational number.
If you use the degree angle unit, sin45 deg is actually equal to 1/?2 or ?2/2 and this is clearly irrational, and it cannot be represented as a fraction. In conclusion it makes no sense to try to cast it into such form.
On the other hand, sin 30 deg or cos 60 deg are both exactly equal to 1/2. The calculator will convert them to a fraction.

On a more general note, the calculator will always be able to convert a fraction into an approximate decimal number, but the converse may not be possible. It depends on the number of display space available on the screen.

The sine of 6pi is indeed zero. The calculator works with twelve decimal digits while pi is an irrational number with an infinite number of decimal digits. Thus the calculator cannot actually calculate the sine of 6pi, instead it calculates the sine of 18.8495559215. The sine of that number is close to zero but not quite, about 0.000000000002 which is what the calculator shows. This is a natural consequence of using finite machines to deal with infinite numbers.

In degrees, 3*360 is 1080 exactly and thus the calculator can produce an answer of exactly zero.

You'll notice that the calculator also gives a nonzero result for the sine of 4pi.

The "proper" fix for this is something called "argument reduction." If the argument lies outside the range of 0 to 2pi (or alternatively, the range -pi to pi), add or subtract multiples of 2pi until the argument is inside the range. So in this case, subtract 6pi from 6pi to get zero, then take the sine of that.

Bear in mind that argument reduction won't always work, since it too is limited to twelve-digit arithmetic.

Hello,

Sorry to disappoint you but you cannot do it due to the fact that the representation of irrational (SQRT(2), ..) and transcendental (e, pi, etc.) numbers on a calculator or even a computer is approximative : a number like sqrt(2) has an infinite number of decimal digits while the number of internal digits a calculator uses to store results is limited (10, 12, 15 or even 24).
Once an irrational or a transcendental number has been evaluated numerically it generally ceases to be exact.

If you want to manipulate algebraic expressions, you need a calculator that has a CAS (computer algebra system) operating system and a symbolic algebra option.

Some exemples of calculators with a CAS are The TI nspire CAS, the HP 48 series, HP 49, HP 50, Casio ClassPad3xx, the TI 89titatium, the TI Voyager 200PLT, the TI 92 and Ti 92Plus. The ones with a fullblown keyboard might not be accepted on certain standardized test.

You may also try to get some emulators on the Internet for the HPs or the TI 8xx and play with them on the computer.

Hope that helps.