At least according to an entry in the notebook of then almost 15 year old Richard Feynman, the most remarkable formula in the world is: |
At least according to an entry in the notebook of then almost 15 year old Richard Feynman, "the most remarkable formula in the world" is: |
where e is the base of the natural logarithm, i is the imaginary unit (an imaginary number with the property i2 = -1), and π is Archimedes' Constant (the ratio of the circumference of a circle to its diameter). The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. |
where e is the base of the natural logarithm, i is the imaginary unit (an imaginary number with the property i2 = -1), and π is Archimedes' Constant Pi (the ratio of the circumference of a circle to its diameter). The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. |
* The number π is a constant related to our world being Euclidean (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences). |
* The number π is a constant related to our world being Euclidean, on small scales at least (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences). |
where e is the base of the natural logarithm, i is the imaginary unit (an imaginary number with the property i2 = -1), and π is Archimedes' Constant Pi (the ratio of the circumference of a circle to its diameter). The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748.
Feynman found this formula remarkable because it links some very fundamental mathematical constants:
The formula is a consequence of Euler's formula from complex analysis, which states that
for any real number x. If we set x = π, then
and since cos(π) = -1 and sin(π) = 0, we get
and