(a+b)2=a2+2ab+b2
(a+b)3=a3+3a2b+3ab2+b3
(a+b)4=a4+4a3b+6a2b2+4ab3+b4
Wait a minute. For (A+B) 12, it is obviously hoped that the coefficient of a7b5 in its expansion can be obtained without a dozen complicated calculations of (A+B) square. Long before Newton was born, people had put forward and solved the binomial expansion problem. China mathematician Yang Hui discovered the secret of binomial as early as13rd century, but his works were not known to Europeans until modern times. Witt also demonstrated the binomial problem in XI's proposition in the preface of Introduction to Analytics. But this great discovery is usually named after Blaise Pascal. Pascal noticed that the binomial coefficient can be easily obtained from the arrangement we now call "Pascal triangle":
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 2 1 35 35 2 1 7 1
etc
In this triangle, each new number is equal to the sum of the left and right numbers. Therefore, according to Pascal's triangle, the value of the next line is
1 8 28 56 70 56 28 8 1
For example, the table value 56 is equal to the sum of two numbers 2 1+35.
The relationship between Pascal's triangle and (A+B) 8 expansion is very direct, because we provide the necessary coefficients for the last behavior of the triangle, namely
(a+b)8=a8+8a7b+28a6b2+56a5b3
+70a4b4+56a3b5+28a2b6+8ab7+b8
As long as the value of the triangle is extended downward by several lines, the coefficient of a7b5 in the expansion of (A+B) 12 is 792. So the practicability of Pascal's triangle is very obvious.
After studying binomial expansion, young Newton invented a formula, which can directly derive binomial coefficients without extending the triangle to the required straight line. Moreover, his inherent belief in the persistence of patterns makes him think that he can correctly deduce something like (a+b) 2 or (a+b) 3.
This form of binomial.
One more thing needs to be said here about the problem of fractional exponent and negative exponent. We know that in primary school
These relationships.
Newton expounded the binomial expansion listed below in a letter written to Gottfried Wilhelm Leibniz, a great man of his time, in 1676 (this letter was forwarded by Henry Oldenberg of the Royal Society). Newton wrote:
The question of "whether the index is an integer or (for example) a fraction, positive or negative". A, B, C, etc. Represents the previous letter in the expansion in the formula.
For those readers who have seen the modern binomial expansion, Newton's formula may seem too complicated and unfamiliar. But as long as you study it carefully, any questions of readers can be solved. Let's take a look at it first.
appear
Perhaps, this form looks familiar.
We might as well apply Newton's formula to solve some concrete examples. For example, when expanding (1+x) 3,
This is exactly the columnless coefficient of Pascal's triangle. In addition, since our original index is a positive integer of 3, the extension ends in the fourth term.
However, when the exponent is negative, Newton had a completely different situation before. For example, expand (1+x)-3. According to Newton's formula, we get
Or simplified to
The right side of the equation never ends. Applying the definition of negative exponent, this equation becomes
Or its equivalent equation.
Newton cross-multiplied the above formula and eliminated similar terms, confirming that
( 1+3x+3 x2+x3)( 1+3x+6 x2- 10x 3+ 15x 4-……)= 1
Newton squared the infinite series on the right side of the equation just to test this seemingly strange formula, and the results are as follows:
therefore
This confirms.
The result is the same as Newton's original deduction.
Newton wrote: "It is very simple to use this theorem to calculate the square root." For example, suppose we ask
Now, substitute the square root on the right side of the equation into the first six terms in the binomial expansion marked with the symbol (). Of course, the x in the original formula should be changed to 29 here, so I
The first six constant terms are given. If we take more terms in binomial expansion, we will get a more accurate approximation. Moreover, we can also use the same method to find cubic roots, quartic roots and so on.
Continuous calculus.
Don't be surprised. What is really surprising is that Newton's binomial theorem tells us exactly which fractions we should use, and these fractions are obtained completely mechanically without any special opinions or cleverness. This is obviously an effective and ingenious way to find the source of any power.
Binomial theorem is one of the two necessary premises of the great theorem that we will discuss soon. Another premise is Newton's countercurrent number, which is what we call integral today. But the detailed explanation of countercurrent number belongs to calculus, which is beyond the scope of this book. However, we can use Newton's words to illustrate his important theorem and give one or two examples to illustrate it.
Newton put forward the problem of countercurrent number in his book Analysis with Infinite Equations written in 1669, but this book was not published until 17 1 1 year. This is the first time Newton put forward the problem of countercurrent number. He handed the paper around to several colleagues in the math department. For example, we know that isaac barrow once read this paper. In a letter to an acquaintance on July 20, 1669, he wrote: "... a friend of mine ... is very talented on these issues. He once brought me several papers. " The first rule that Barrow or any other reader of Analysis encounters is as follows.
Let the base of any curve AD be AB and its vertical longitudinal side be BD, and let AB = X,
Bd = y, let a, b, c, etc. Is a known quantity, m and n are integers. Then:
The graphic area in the x point. According to Newton's law, the area of this graph is
According to Newton's formula, the area is 12x2. For this result, the triangle area formula can be easily used.
Newton further explained the second law of analysis, "If the value of y consists of the sum of several terms, then its area is equal to the sum of each term." For example, he wrote, Qu
Then Newton used two tools: binomial theorem and flow number method to find the area under a curve. By using these two tools, he can easily solve many complicated mathematical and physical problems, and what we will see is how Newton used these two tools to give a new life to an old problem: calculating the approximation of π. In the postscript of the fourth chapter, we trace back some history of this famous number, and confirm the contributions of some scholars, such as Archimedes, Vedas and Lu Dolf von Thuren, in calculating a more accurate π approximation. 1670 or so, this problem has attracted the attention of isaac newton. He studied this ancient problem with his wonderful new method and made brilliant achievements.