(Note - if you'd rather watch a video than read this, (and I don't blame you) see The Discovery That Transformed Pi)

Sir Isaac Newton is undoubtedly one of the most brilliant mathematicians in all of history.

In 1666, Isaac Newton was 23 years old, and had invented calculus the year before (though it had not yet been published).  He was quarantined due to an outbreak of bubonic plague, and was toying around with binomial expansions - when you take (1+x) and multiply it by itself.  You get the first 1 + 2x + x², then 1+3x+3x²+x³, etc.  The coefficients can be shown in tabular form as the binomial pyramid, which many high school math students will recognize:

This was well known in Newton's time, and there was even a formula to compute the coefficients of an nth power binomial, where n was a positive integer (and ONLY a positive integer).  Newton, having nothing better to do, since ARFCOM, Netflix, and the internet had not been invented yet, decided to muck around with it a bit.

Because I don't want to simply type a transcript of the video, and because you don't want to read it, I'll condense a lot from here on - it's still quite long, however, since as noted in the title, his changes were widespread and monumental.

Newton first managed to break the formula for binomial expansion by plugging the number -1 into it for n.  He determined that this then gave a formula for 1/(1+x), where the result was 1 - x + x² -x³ +x⁴..., and proved that it did indeed work  If you plug in -x instead of x into the formula and switch it around, you may recognize a basic formula for the infinite geometric series where A(1+x+x2+x3+...) = A/(1-x).  You may have learned a form of this it in a class on economics, since it can be used to represent and calculate loan payments for any given interest rate (and length of payment schedule).  A quick bit of fiddling and he expanded it to cover all negative values of n.

Now for the REALLY useful part.  He broke the formula again, by putting a fractional value of n into the formula.  As it turns out, he was able to prove that this, too, works, and he now had a way to calculate any square root, simply by a little manipulation to turn (for instance) the square root of 3 into the square root of (4 times (1-1/4)), or 2 times the square root of (1 - 1/4), such that you were always dealing with values less than 1.  This provided a very quick and efficient way to calculate square roots, using only multiplication and addition/subtraction of fractions.

And for the mind-blowing part - he looked at the graph of a circle, and noted that a quarter circle is described by the equation Y=(1-x²)^1/2.  He now had an equation for Y that involved an infinite series of positive integer powers of X, which could be quite simply integrated (I believe I mentioned that he had invented calculus the year before) to find the area under that curve, which was known to be Pi/4.  Thus, he could calculate Pi to any arbitrary precision by simply adding higher powers of X into the binomial expansion formula into the integration.



Since in the formula, he was integrating for x from 0 to 1, the math was simple (1 to pretty much any power is in fact just 1), but the numbers didn't shrink all that fast, which meant that he needed a pretty large number of terms to get Pi to be very precise.  So he had another stroke of genius.  He decided to only look at the portion of the circle from x=0 to x=1/2.  This meant that every time the power of x increased by one in his formula, the actual contribution to Pi dropped by a factor of 2 - and those powers increased quickly - by the fifth term of his formula, the value of that term had dropped by a factor of 512, so the precision of the calculation increased VERY quickly.  That area could be calculated as Pi/12 plus 3^1/2/8, so a quick bit of algebra gave a formula for Pi.

In the early 1600's, a mathematician by the name of Ludolph van Ceulen had calculated Pi using the then-best method by using polygons, as originally done by Archimedes.  Ludooph had dedicated 25 years to calculating Pi, ending up with a polygon with over 4 quintillion sides (2⁶², to be precise), and calculated Pi to 35 decimal places.  Newton's new method needed only 50 terms and a few days of hand calculation to replicate that feat.

Mike
Edit:  1666...

Not bad for a guy over 200 years old at the time.

Attached File

Born in 1642.

The mathematician Gottfried Leibenitz invented “The calculus of variation” overnight.  An entire branch of calculus in a few hours.....

Primarily for plotting the “best” path for an object to take between 2 point in 3 dimensional space.  Not always a straight line given different conditions and media.  Pretty amazing actually, and actually invented differential and integral calculus independent of, and virtually simultaneous to Newton’s discovery of same