Transcendental number, a mathematical concept, refers to numbers that are not algebraic numbers.The existence of transcendental number is determined byFrancemathematicianLiouville (Joseph Liouville, 1809~1882) was first proved in 1844.On the existence of transcendental number, Liu Weir wrote the followinginfinite decimal : a=0.110001000000000000001000... (a=1/10 ^ (1!)+1/10 ^ (2!)+1/10 ^ (3!)+...)Algebraic number, but a transcendental number.Later, in memory of him, people first proved transcendental numbers, so the number a is calledLiouville number。[1]
Transcendental number is a number that cannot be taken as the root of a polynomial equation with rational coefficients[2], that is, a number that is not an algebraic number.Euler said: "They are beyond the scope of algebraic methods." (1748).
French mathematician in 1844Liouville (J. liouville, 1809~1882) first proved the existence of transcendental numbers.ErmettAndLindermanIt has been proved that e and π are transcendental numbers.
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Transcendental number is a number that cannot be taken as the root of a polynomial equation with rational coefficients. Its definition is just the same asAlgebraic numbercontrary.Two famous examples: pi=three point one four15926535…、Natural logarithmBottom e=2.718281828.It can be proved that there are infinite transcendental numbers.In real numbers, except for algebraic numbers, all others are transcendental numbers, but transcendental numbers are not necessarily real numbers, such as the famous Euler formula
In
It is an imaginary transcendental number.Real numbers can be classified as follows: real numbers are divided into real algebraic numbers and real transcendental numbers.The set of all transcendental numbers is aUncountable set。This implies that the transcendental number is an infinite set.However, there are very few transcendental numbers found today, because it is very difficult to prove that a number is transcendental.
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After the proof of Liouville number, many mathematicians devoted themselves to the study of transcendental numbers.1873, French mathematicianHermit(Charles Hermite, 1822~1901)Base of natural logarithmThe transcendence of e makes people have a clearer understanding of transcendental number.In 1882, German mathematician Linderman proved thatPiIt is also a transcendental number (completely negated“Round to Square”Possibility of drawing).
In the process of studying transcendental numbers, David Hilbert once proposed a conjecture: if a is an algebraic number that is not equal to 0 and 1, and b is an irrational algebraic number, then a ^ b is a transcendental number (the seventh question in Hilbert's problem).
This conjecture has been proved, so we can conclude that e and π are transcendental numbers.
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The number in real number other than algebraic number, that is, it does not satisfy any integral coefficient polynomial equation
(n is a positive integer,
The number ≠ 0).It is not difficult to prove the existence of transcendental numbers in theory, and we know that there are a large number of transcendental numbers.But it is extremely difficult to construct a transcendental number or demonstrate that a number is a transcendental number.At present, only a few numbers (such as π, e) have been proved to be transcendental, and the research on the transcendence of other numbers of interest is a matter of great concern to mathematicians.[3]
Several examples
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π
π. In China, it is also called ring rate, circularity, pi, etc.
The first one to get π≈ 3.14 is GreekArchimedes(about 240 BC), the Greeks were the first to give the exact value of π decimal in the last four placesPtolemy (about 150 BC), Zu Chongzhi (about 480 BC), a Dutch German mathematician, used inscribed and circumscribed regular polygons to calculate the π value in 1610, and calculated π to 35 decimal places through a 2-sided polygon, which took his whole life. In 1630GreenbergleyUsing Snell's improved method to calculate π to 39 decimal places is the most important attempt to calculate π using the classical method.
All the above are classical methods to calculate π value.
Dash first calculated the exact 200 digits of π.
It is worth mentioning that Dash was born in Hamburg in 1824 and died after only 37 years. He was a lightning calculator and the most remarkable manual calculator. He completed the multiplication of two 8-digit numbers in 54 seconds, two 20 digit numbers in 6 minutes, and two 40 digit numbers in 40 minutes;He worked out the square root of a 100 digit number in 52 minutes.Dash's extraordinary computing ability has helped him create sevenLogarithmic tableAnd the factor table of numbers from 7000000 to 10000000 has been fully used.
In 1706, William James of England first used the symbol π to represent the circumference and diameter of a circleratioBut it was only after Euler adopted this method in 1737 that π was widely used in this case.
In 1873, William Shanks, an Englishman, used Maxin's formula to calculate π to 70 digits.
In 1961, American Resch and D. Sanx used an electronic computer to get 100000 digits of π value.
e
In the middle school mathematics book, it is proposed that the logarithm with e as the base is calledNatural logarithm。So what's the practical significance of e?
In 1844, the French mathematician Liu Weier first speculated that e was a transcendental number, and it was not until 1873 that the French mathematician Hermit proved that e was a transcendental number.
In 1727, Euler first used e as a mathematical symbol. Later, after a period of time, people decided to use e as a mathematical symbolBase of natural logarithmTo commemorate him.Interestingly, e is the first lowercase letter of Euler's name. Is it intentional or accidental?It is not available now!
The application of e in natural science is no less than π value.As in atomic physics and geologyDecay lawOr the age of the earth.
In useZiolkovsky formulacalculationrocketE is also used for speed, and for calculating the optimal interest of savings and biological reproduction.
Like π, e also cares about unexpected places. For example, "divide a number into several equal parts. To maximize the product of each equal part, how to divide?" To solve this problem, we need to deal with e.The answer is to make the equal parts as close to the e value as possible.For example, 10 is divided into 10 ÷ e ≈ 3.7 shares, but 3.7 shares are not good, so it is divided into 4 shares, each of which is 10 ÷ 4=2.5. At this time, the product of 2.5 ^ 4=39.0625 is the largest. If it is divided into 3 or 5 shares, the product is less than 39.E is so miraculous.
In 1792, 15-year-old Gauss discovered the prime number theorem:prime numberIs approximately equal to the reciprocal of the natural logarithm of N;The greater the N, the more accurate this rule will be. "It was not until 1896 that this theorem was established by French mathematicianshadamard It was proved by Bu San, a Belgian mathematician of almost the same period.There are still many advantages based on e.If prepared based on eLogarithmic tablebest;CalculusThe formula also has the simplest form.This is because only e ^ xderivativesIt is itself, that is, d/dx (e ^ x)=e ^ x.
