How to Convert Pie and Sqrt into Fractions?

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Discussion Overview

The discussion revolves around converting the values of trigonometric functions involving π (pi) and square roots into fractional representations. Participants explore various methods and tools, including calculator functions, to achieve this conversion, while addressing the nature of irrational numbers and approximations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that π is an irrational number and cannot be expressed as a fraction, only approximated.
  • Others suggest that while π cannot be converted to a fraction, trigonometric values like sin(π/4) and sin(π/3) can be approximated as decimals and then converted to fractions.
  • A participant mentions that sin(π/4) is approximately equal to √2/2, indicating a method for converting decimal approximations back to fractional forms.
  • There are discussions about using calculators, specifically the TI-83 Plus, to find fractional equivalents, with some participants noting its limitations in converting decimals to exact fractions.
  • One participant proposes using the unit circle to identify common sine and cosine values, which often involve square roots in their fractional forms.
  • Another participant describes a method of squaring the decimal approximation to find recognizable fractions, such as relating 0.866025404 back to √3/2.
  • Some participants express frustration over the lack of clarity in the original question and suggest that the inquiry is about converting decimals rather than the irrational numbers themselves.

Areas of Agreement / Disagreement

Participants generally agree that π is irrational and cannot be expressed as a fraction. However, there are multiple competing views on how to handle the conversion of trigonometric function outputs into fractional forms, and the discussion remains unresolved regarding the best method to achieve this.

Contextual Notes

Limitations include the dependence on calculator capabilities, the assumption that participants are familiar with the unit circle, and the unresolved nature of converting decimal approximations to exact fractions.

Who May Find This Useful

This discussion may be useful for students and educators in mathematics, particularly those interested in trigonometry, irrational numbers, and calculator functionalities.

JasonKit
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Turning pie & sqrt into...

I am desperately need help how to turn pie and sqrt to fraction?

Here is my question, punch in sin(pi/4) and the answer would be .7071067812
I know how to use fraction mode I'm no noob. I tried that feature and it didn't works.
Any brilliant idea would be appreciated!:smile:
 
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JasonKit said:
I am desperately need help how to turn pie and sqrt to fraction?

Here is my question, punch in sin(pi/4) and the answer would be .7071067812
I know how to use fraction mode I'm no noob. I tried that feature and it didn't works.
Any brilliant idea would be appreciated!:smile:

Cherry pie or apple? :smile:

Just kidding, pi is an irrational number, i.e. there is no fraction equivalent, only an approximation to a certain decimal place.

Square root is a function not a number.
 
In fact,

it can be turn to fraction, friend of mine, show it to me and I forgot how to do it, o let me see how long is been 2 YEARS AGO. If you have any genius idea please help me out not intimidated me.

If you know this that .7071067812 is equal to sqrt 2/2.

I just need help how decimal turn it to fraction
Let me give you another example:

sin(pi/3)= .8660254038

what is the fraction for this?
 
Last edited:
JasonKit said:
it can be turn to fraction, friend of mine, show it to me and I forgot how to do it, o let me see how long is been 2 YEARS AGO. If you have any genius idea please help me out not intimidated me.

Your friend was wrong. Again, Pi is an irrational number.

http://en.wikipedia.org/wiki/Π
 
stewartcs said:
Your friend was wrong. Again, Pi is an irrational number.

http://en.wikipedia.org/wiki/Π

are you even trying the problem that I provide above?
 
JasonKit said:
it can be turn to fraction, friend of mine, show it to me and I forgot how to do it, o let me see how long is been 2 YEARS AGO. If you have any genius idea please help me out not intimidated me.

If you know this that .7071067812 is equal to sqrt 2/2.

I just need help how decimal turn it to fraction
Let me give you another example:

sin(pi/3)= .8660254038

what is the fraction for this?

So your question is not what is the fraction equivalent of Pi, but rather what is the fraction equivalent of sin(pi/3). Two different things.

All you need is the Unit Circle. See attached...
 

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it helps a little bit, I do know this unit circle and I even memorize it.- However I still need to know how to do it on the calculator which ti-83 plus.
 
The ANS-->FRAC function on a TI-83 Plus won't give you the fractional equivalent of that. It's beyond its capabilities. The Voyager 200 (and possibly the TI-89) will though.
 
  • #10
Thanks for the look out, I'll find a way. I did it once, i just forgot which feature it does that.
 
  • #11
If sin(pi/3) is approximately .8660254038, then it is obviously (and again, approximately) equal to the fraction 8660254038/10000000000. Now you can try to reduce this fraction, for example by looking for the unique prime factors of 8,660,254,038 and 10,000,000,000 and cancelling the common ones; or, which is the same thing, by dividing 8,660,254,038 and 10,000,000,000 by its g.c.d. (in the ti-83, this is math-num-9:gcd). Since gcd (8660254038 ,10000000000) = 2, and 8660254038 / 2 = 4330127019, and 10000000000 / 2 = 5000000000, the number .8660254038 equals the fraction 4330127019/5000000000.

But notice three things:
1) .8660254038 equals the fraction 4330127019/5000000000, but none of them equals sin(pi/3). Both of them only approximate sin(pi/3), up to a certain numbers of decimals. With less decimals, the approximation would have been 8/10, or 86/100, or 866/1000, or ...
2) Everything they have been telling you is true; these numbers are irrationals, and you can only approximate them with a fraction. On the other hand, if what you wanted was to convert a given decimal like .8660254038 to a fraction, that can be done as above, but has nothing to do with sin(pi/3), except as an approximation.
3) Of course this is just a guess of what you want, because you are not explaining yourself very well, and by repeating "it can be done" the question doesn't get any clearer.
 
  • #12
This sort of thing is why calculators should be banned in math classes...

Jason, your calculator can't convert the decimal representation to a fraction, because sin(pi/3) is not a rational number.

\sin \frac{\pi}{3} = \frac{\sqrt 3}{2}
 
  • #13
Thanks for all your help, I lost contact with my friend, he is the one who show me the trick on ti-83 plus. I saw it and he taught me that. Its been awhile.
 
  • #14
There might be an "exact mode" or something like that. It won't convert a decimal into "sqrt(3)/2", but it should give exact formulas for common sine and cosine values. Try asking on a TI-83 forum?
 
  • #15
If the angle is one of the most commonly used ones (i.e. the ones listed on the unit circle), you can do this.

Enter sin(pi/3)...the calculator will tell you it is 0.866025404.
Since you know that the common sine/cosine values are things that have a square root in the numerator, you might want to square the answer that the calculator gave you and see if it's something you recognize. For instance, take
(0.866025404)^2 = 0.75
which you recognize as 3/4. Since you squared it, now let's square root it to find
sqrt(3/4) = sqrt(3)/2
which I assume is the symbolic answer you're looking for.

Another similar trick you could use if you want to know the symbolic form of arcsin(0.866025404) = 1.04719755
Since I know the answer will likely have a factor of pi in the numerator, take
1.04719755/pi = 0.333333333
which you recognize as 1/3. So, 1.04719755 = pi/3.

Or you could just remember the unit circle!
 
  • #16
Another similar trick you could use if you want to know the symbolic form of arcsin(0.866025404) = 1.04719755
Since I know the answer will likely have a factor of pi in the numerator, take
1.04719755/pi = 0.333333333
which you recognize as 1/3. So, 1.04719755 = pi/3.

This is actually pretty slick
 

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