Unveiling Newton’s Binomial Power Series Discovery: Thinking about questions and searching for patterns inspired Newton to discover the link between infinite sums and curves.
Isaac Newton was not known for his generosity and his hatred of his adversaries was well-known. However, in a letter addressed to his adversary Gottfried Leibniz that is now known as the Epistola Posterior, Newton appears to be charming and almost like a friend.
In it, he relates the story of his early time which was when he was beginning to master math. He recounts the moment he made an important discovery by comparing areas under curves and infinite sums through a process of guessing and rechecking.
The logic he explains in the letter is so cute and easily understood that it makes me think of the patterns-guessing games that children enjoy playing.
The story began when a young Newton was reading John Wallis’ Arithmetica Infinitorum, a seminal mathematical work from the 17th century. Wallis included a new and inductive method for determining the value of pi and Newton determined to create something like it.
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He began by tackling the issue of determining the size of an “circular segment” that has a width adjustable to x. This is the region that lies beneath the unit circle defined by y=1-x2, which is above the area that runs horizontally that runs from the x-value to. In this case, x can be any number ranging from 1 to 0 while 1 represents the diameter that the circle has.
The area of an unicircle is pi as Newton well understood, therefore, when x = 1 is the area that lies under the curve is equal to a portion of the unit circle. which is p4. However, for different values of x it was not known.
If Newton could discover a method to calculate the area under the curve for any possibility of the value an x , it might provide him with an unimaginable method to approximate pi.
This was his initial idea. However, he came across something much better that could replace complex curves with infinite amounts of building blocks that are simpler and made from powers of x.
Newton’s initial step was to consider reasoning through analogy. Instead of looking directly at the circle’s area segment, he studied the dimensions of similar segments which were bound by these curves:
y0=(1-x2)02,y1=(1-x2)12,
y2=(1-x2)22,
y3=(1-x2)32,
y4=(1-x2)42,
y5=(1-x2)52,
y6=(1-x2)62.
Newton realized that the areas of the curves of the list of powers with whole numbers (like 22=1 and 02=0) could be simple to calculate because the algebraic equations are simplified. For instance,
y0=(1-x2)02=(1-x2)0=1.
Similarly,
There is no similar simplification available for the equation of the circle -it is y1=1-x2=(1-x2)12as well as for the other curves using half powers. In the beginning nobody knew how to determine the area of each of these curves.
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Fortunately, the regions under the curves that had whole-number powers were easy to calculate. Consider the equation y4=1-2×2+x4. A popular rule of thumb at the time for these functions permitted Newton (and any other person) to determine the area fast by using the following formula: for any whole-number strength that is greater than n, the area of the curve y=xn in the range from 0 to x can be calculated by the formula xn+1n+1. (Wallis knew this rule using his inductive approach, and Pierre de Fermat proved it definitively.) With this rule in mind, Newton knew that the area of the curve of y4 would be x-2×33+x55.
This same rule enabled him to calculate the area of the other curves that have whole-number power in the above list. Let’s use the formula An to calculate the area beneath the curve yn=(1-x2)n2 where n=0.1,2. ,… . By applying the rule, you will get
A0=x
A1=?
A2=x-13×3
A3=?
A4=x-23×3+15×5
A5=?
A6=x-33×3+35×5-17×7
and and so and so on. Newton’s clever idea was to fill the gaps and try to figure out A1 (the series that covers the undiscovered portion in the segment with a circular shape) by looking at the patterns he saw within the different series. One thing was evident that each An started with the letter x . This suggested changing the formulas as follows:
A0=x
A1=x-?
A2=x-13×3
A3=x-?
A4=x-23×3+15×5
A5=x-?
A6=x-33×3+35×5-17×7.
To replace the following batch of questions mark, Newton looked at the 3×3 terms. With a bit of license we can determine that even A0 contained at least one cubic term because we could rewrite it as A0=x03x3. According to what Newton stated to Leibniz that he noticed “that the terms 03×3,13×3,23×3,33×3, and so on. and were in arithmetic sequence” (he was talking about the numbers 0 1 2, 3 numbers of the numbers).
Suspecting that this arithmetic progression might extend into the gaps as well, Newton guessed that the entire sequence of numerators, known and unknown, ought to be numbers separated by 12(0,12,1,32,2,52,3…) “and hence that the first two terms of the series” he was interested in — the still unknown A1, A3 and A5 — “ought to be x-13(12×3),x-13(32×3),x-13(52×3), etc.”
So, at this point it is suggested by Newton the idea that A1 should be the first to begin
A1=x-13(12×3)+….
It was a good beginning however, he still needed to improve. When he began to search for other designs, Newton noticed that the denominators of the equations often had odd numbers that were in ascending order. For example, take a look at A6, which contains 1 3, 5, and 7 as denominators. The same pattern was used for A4 and A2. It’s not difficult. This pattern appears to have persisted across the denominators for all equations.
It was left to look for an underlying pattern among the numerators. Newton looked over A2, A4, and A6 and saw something. In A2=x-13×3, he noticed an 1 multipliing the x, and a second 1 in the form 13×3 (he did not notice its negative signification at the moment).
In A4=x23x3+15×5, he observed numerators of 1,2 and 1. In A6=x-33×3+35×5-17×7 He saw numerators 1 3, 1. These numbers are familiar to anyone who has looked up Pascal’s triangle, which is a triangular arrangement that, in its simplest form it is made by adding the numbers on top starting with 1 at the bottom.
Instead of calling attention to Pascal, Newton referred to these numerators as “powers of the number 11.” For instance eleven 2 = 121 that corresponds to the 2nd row of the triangle as well as 3 = 1131. 3. = 1331 that is the third row.
These numbers are now known as binomial coefficients. They arise when you expand the powers of a binomial like (a+b), as in (a+b)3=1a3+3a2b+3ab2+1b3. With this formula in hand, Newton now had an simple method of writing A2, A4, A6 and the rest of the equally-numbered numbers, including the A‘s.
Then, to extend the results to half-powers as well as odd-numbered subscripts (and eventually reach the sequence he desired A1), Newton needed to extend the triangle of Pascal to a completely new system which was half-way between each row.
To calculate the extrapolation Newton formulated an overall formula for the binomial coefficients of every row in Pascal’s triangular — row Mand then boldly plugged into the number m = 12. Amazingly, it worked. It gave him the number of number of numerators in the series he was looking for to create a unit circle. A1.
In Newton’s own terms, is his summation for Leibniz about the pattern he observed inductively until this point in the debate:
I started to think about how the denominators 1,3 5 7 7, etc. were in arithmetical sequence and their numerical coefficients numerators alone were to be investigated. In the alternately presented areas, they were the figures of power of 11 … which is first ‘1’, then “1, 1” and thirdly 1, 2, 1′ and fourthly ‘1,3 3 1’; and fifthly ‘1, 6 4 and so on.
And so I began to investigate what the other figures in this series might be constructed from the initial two figures. I discovered that by adding m to on the other figure, remainder could be derived by the constant multiplying the terms of this series.
m-01xm-12xm-23xm-34xm-45, etc.
… In this way, I applied this rule to interposing series within series and as for the circle there was a second word, 13(12×3) I set m=12 and the two results were
12×12-12, or -18
+116,
116×12-34, or -5128,
until all the way to. From there, I realized that the circle that I wanted to use was
x-12×33-18×55-116×77-5128×99 etc.
In the end, by adding x=1 to x, Newton could obtain an infinite sum for the number p4. It was an important discovery however, it was later discovered that there are more efficient methods to approximate pi the use of an infinite sum such as Newton himself quickly discovered after his first foray into the realm of infinite sums known as power series. In the end, he was able to calculate the initial 15 digits of pi.
Recalling the problem of the circular segment Newton recognized that the equation for the entire circle (not just the area beneath it) could be represented using an inverse power series. All he had to do was remove the denominators and decrease the powers of x to one in the power sequence shown above. This led him to conclude that
To determine if this result was correct, Newton multiplied it by itself: “It became 1-x2, remaining terms vanishing due to the continuing series until infinity.”
Removing ourselves in the details we can see many lessons in this book about problem solving. If you find a problem too difficult, modify the way you approach it. If you find it too narrow, broaden it. Newton was able to do both and achieved results that were more significant and effective than the results he initially wanted.
Newton didn’t fixate incessantly on the quarter of an arc. He was looking at a larger, more general form, any circular section of width x. Instead of limiting himself with x = 1, he let the x value to vary between 0 and 1.
This revealed the binomial nature of the coefficients of his series. He also discovered the unusual number of numbers appearing in Pascal’s Triangle and their generalizations that let Newton discover certain patterns Wallis and others missed. These patterns provided Newton the insight he required to formulate his theory on power series more broadly and widely.
In the course of his work afterward, Newton’s power series offered him the Swiss Army knife for calculus. With these tools, he could perform integrals, discover the roots for algebraic equations and determine the value of cosines, sines and logarithms. He said, “By their help, analysis is able to reach, I would almost say, all the problems.”
The moral of the story: changing an issue is not cheating. It’s creative. It could be the secret to something more.
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