# Who used a trignometric formula to estimate pi to 16 decimal places?

• Drez1985
In summary, the formula used by Jamshid al-Kashi to determine pi to 16 decimal places involves the expression √(2(2r+r√3) and can be simplified to r√(2r+√3). This formula was found in the book "The Age of Genius - 1300-1800" by Michael J. Bradley and was used by al-Kashi by substituting the value of C_1 into the expression for C_2. The final simplified expression is C_2=r√(2r+√3).

#### Drez1985

How does √(2(2r+r√3) simpilfy to r√(2r+√3) ?

Read about the formula in a book about Jamshid al-Kashi, the great Iranian astronomer who used the above formula to determine pi to 16 decimal places.

All help is much appreciated.

Are you sure you wrote it correctly?
The only two cases when $$\sqrt{2(2r+r\sqrt 3)}=r\sqrt{2r+\sqrt 3}$$, are when r=0, or when $$r=\frac{\sqrt{35+16\sqrt 3}-\sqrt 3}{4}$$.

Did you mean $$\sqrt{r(2r^2+\sqrt 3 r)}=r\sqrt{2r+\sqrt 3}$$ ?

Yes, I'm sure; I wrote off the formula from a book called "The Age of Genius - 1300-1800" by Michael J. Bradley.

Jamshid al-Kashi used the following trignometric formula (as a way to estimate pi):

C_n=√2(2r+C_(n-1))

where C_1=r√3

By first substituting C_1 into the expression for C_2, supposedly, you will end up with:

C_2 = r√(2r+√3)

Was it any help?

P.s. Sorry for the messy notation, but when pasting from the excel equation writer onto this webpage, strange things happens...

## What is algebraic manipulation?

Algebraic manipulation is the process of rearranging algebraic equations and expressions to solve for unknown variables or simplify complex expressions.

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The basic rules of algebraic manipulation include the commutative, associative, and distributive properties. These properties govern how we can rearrange and combine terms and factors in an equation or expression.

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To simplify an algebraic expression, we can combine like terms, use the distributive property, and apply the rules of exponents. We can also use techniques such as factoring and the quadratic formula to simplify more complex expressions.

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