What abstract  beauty  the numbers have！Comparisons are often made with music and poetry. In fact，math is an art to a larger extent and seems an art to exist in the world. Mathematical development is directed and assessed by aesthetic principles.So,math is possessed of wonderful harmony. It is universally acceptable truth that math shows dazzling abstract beauty before people. An abstract beauty abstraction and image, simple and complex, approximation and rigor, induction and deduction, to say briefly, a symmetric contradiction unity. The abstract beauty is intrinsic and presumptive.We can not create it, can only feel it and discover it.

The math beauty would be ineffable to some mathematical workers, to say nothing of the ineffability of general people.

Why could the mathematicians have discovered so much mathematical beauty?

The mathematicians with being music or poet in soul are young and vigorous, inbued with vitality, be quick-witted. They often glitter with dynamism of  intelligence. Their strong nerve center network are more refined and developed ,which some substructure are harmoniously connected to a priori mathematical idea. The synpses of the relation between nerve cells are bursting with vim and vigur and in time release agile neurotransmitters or/and neuromodulators related to bio-electric current to treat the afferent information.The energy supply to the thought activities of their brain are ample and high efficiency. On the other hand, they impassioned to probe the secrets of math. They keep on fighting in spite of repeated setbacks and tenaciously resolve the mathematic brain-teasers. Even though they may be alone, they would be overwhelmed by extreme joy of math beauty till they have no regret to be haggard.

According to the facts mentioned above, we should say the faculty and willpower of mathematicians surpass general people so that they are able to presume and discover the mathematical beauty.

We can not have a good swim in the sea of math, but we should be able to stroll along the sea beach and pick up a few shells by accident.

It is different from people to love the math beauty as songs.The examples describe below are merely to belong to myself recreational activities which give a hurried and cursory glance at the number beauty. For one number cited, ten thousand numbers may have been left out. There must be some errors in it. I hope knowledgeable persons would give me direction or guidance.

1.      Number 0. The number 0 is the least non-negative integer. It may or may not be considered a natural number, but it is a whole number and hence a rationl number and a real number. It is neither a prime nor a composite number, nor is it a unit. In set theory, it is defined as an empty set.

The earlist certain use of zero as a decimal positional digit dates to the 5th century. The Hindu-Arabic numeral system (base 10) reached Europe in the 11th century.Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians.

The importance of the creation of number 0 can never be exaggrated. It seems like a wonderful magic crystal ball crystallized by the love between the divine spirit and the unmatched beauty in the human world.

2.      Number π.It is defined as ratio of the circumference of a circle to its diameter or ratio of the area of a circle to power of its diameter, or more simply, the circumference of a circle of diameter 1. It is an irrational number . In Greek,the original meaning of  irrational is it can not be expressed by the ratio of two integers. π is also a transcendental number so that it is impossible to change a circle into a square by use a compasses and a square.

In history, Zuchongzhi〔AD 429-500〕 first searched out the seven decimals of π in the world.He also had discovered the approximate ratio 22/7 and close ratio 355/113. In modern, 103993/33102 are obtained from the simple continued fraction expansion of π.The current record is 2576980370000 decimals, set by Daisuke Takahashi. It was also known that the first 38 decimals was the largest prime in π some years ago.

She ascend to heaven from a flower ring and become an immortal. She seems like an elf  flutter about all over the universe. Pi appears routinely in physics describing fundamemtal priciple of the universe. Following are some examples: 。The cosmological constant: Λ＝8πGρ/3c²　　。Heisenberg uncertainty principle:ΔxΔp≥h/4π 　。Coulomb law for the electric force: F＝〔Q₁Q₂〕/4πε r²     。Magnetic permeability of free space: μ₀＝4π·10^-7N/A²   。Kepler third law constant: P²/a³＝〔2π〕²/G〔M＋m〕.　

3.The golden ratio φ: The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. Indeed, the length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. It is approximately equal to 1.618.Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavours. In these phenomena he saw the golden ratio operating as a universal law. Zeising wrote in 1854:

[The Golden Ratio is a universal law] in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

Beginning in the Renaissance, a body of literature on the aesthetics of the golden ratio has developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works.

Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.　

The negative root of the equantion for φ is 1－φ ≈－0.618. Its absolute value is sometimes refered to as the golden ratio conjugate.It is denoted here by Φ ≈0.618.　

The golden ratio is the link spread through the heaven,earth and human. All things on earth have their various charms to form colourful and complicated scenery. There are much embodiment of golden ratio in human anatony, physiology theory of chinese medical science. There is a unchaged, simplest law -golden ratio-behind the phenomenon of numenous changes.The human and nature achieve harmonious unity on the basis of this law.　

4．Euler identity: e^iπ＋1＝0, Euler's identity is considered by many to be remarkable for its mathematic beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

• The number 0.
• The number 1.
• The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis (π ≈ 3.14159).
• The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
• The number i, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.

Stanford mathematics professor Keith Devlin says, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's identity reaches down into the very depths of existence.

5.Some wonderful primes: 。The prime consisting of all of the numbers 1:｛I_n｝express 111…sequences consisting n numbers 1. It was discovered that I ₂, I₁₉, I₂₃ , I_317, are prime. American H.C.Williams discovered and proved I₁₀₃₁ in 1986. 。 The prime consisting of only the number 0, 1: 101 is only one in 10000 and the other larger prime is 11…1☆00…0★1.　☆:1 is 2700 digits.★: 0 is 3155 digits. 。The prime include the most 0: Engineer H. Dubner had discovered prime 1340488× 10¹⁵⁰³⁷＋1＝13408800……0☆1. ☆: 0 is 15036 digits. 。 The prime of form N！±1:  Some examples, when n ＝340, 399, 427, N！＋ 1 is prime; when n ＝324, 379 ,469, N ！－1 is prime. 。 Palindromic prime: Some examples, 787, 797, 919, etc. palindromic prime pair: There are 13 pairs in 3 digits, 102 pairs in 4 digits, and 684 pairs in 5 digits.

6.The integer solution of  indeterminate equation: In 1972,Mr. Wuziqian discovered 1827⁸ ＝ 1067⁸ ＋1066⁸＋ 1065⁸＋……There are 127 numbers altogether in the expansion, in them 1067 to 960 are 108 continued numbers. In 1976, he discovered 9339636⁹ ＝8445344⁹＋8441982⁹＋…… There are 90 numbers altogether in the expansion , in them 7092 to 6939 are 52 series of equal difference, the common difference is 3 .

7. Betti number: In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces.

Each Betti number is a natural number or infinity. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes〕, the sequence of Betti numbers is 0 from some points onwards (Betti numbers vanish about the dimension of a space), and they are all finite.

The (rational) Betti numbers b_k(X) do not take into account any torsion in the homology groups, but they are very useful basic topological invariants. In the most intuitive terms, they allow one to count the number of holes of different dimensions. For a circle, the first Betti number is 1. For a general pretzel, the first Betti number is twice the number of holes. In topological graph theory the first Betti number of a graph G with N vertices, M edges and K connected components equals M－N＋K.

8. Ordinal and cardinal: Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order-type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal.  Here we may meet some concepts such as limit ordinal,weakly compact cardinal, measurable cardinal, and Woodin cardinal, etc .They are beyond our understanding. I seem like to have been lost in the foggy precipice that caused by these concepts which hold the astonishing beauty. I may well be in danger to fall over the cliff at any time so I have to pull back before it is too late.

Ah,God！You give me curiosity without talent for resolving problems. When I cannot help but carefully view the pretty numbers, in my heart well up a sense of exhiloration similar to aesthetic appeal comes from hearing an ecstatic melody. Then I would involuntarily say something in which I seem never know what I am talking about, nor whether what I am saying is true. Those mystical numbers who like seductive fox spirit or lovely elves flashed frequently before my eyes. How I wonder what you exactly are！Note: Some contents are extracted from Wikipedia and a few refer to Chinese 〈science〉 and 〈nature  magazine〉.                      24.  Nov. 2009