“It is easy to create an interstellar radio message which can be recognized as emanating unambiguously from intelligent beings. A modulated signal (‘beep,’ ‘beep-beep,’. . . ) comprising the numbers 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, for example, consists exclusively of the first 12 prime numbers—that is, numbers that can be divided only by 1, or by themselves. A signal of this kind, based on a simple mathematical concept, could only have a biological origin.. . . But by far the most promising method is to send pictures.”

— Carl Sagan

Smithsonian magazine, May 1978, pp. 43, 44. Quoted in Awake! magazine, 1978, 8/22.

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Carl Sagan 365

American astrophysicist, cosmologist, author and science ed… 1934–1996

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“I would teach the world how the Greeks proved, more than 2,000 years ago, that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics – when humankind realised that, by pure thought alone, it could prove eternal truths of the universe.
Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers.”

Marcus du Sautoy (1965) British professor of mathematics

In "Life lessons" http://www.theguardian.com/science/2005/apr/07/science.highereducation?fb_ref=desktop The Guardian (7 April 2005)

“1. Zero is a number.
2. The successor of any number is another number.
3. There are no two numbers with the same successor.
4. Zero is not the successor of a number.
5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers.”

Giuseppe Peano (1858–1932) Italian mathematician

As expressed in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
Peano axioms

“It is picked out from numbers progressing in continuous proportion. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical one which advances by unequal and proportionally increasing or decreasing intervals. Arithmetical progressions: 1, 2, 3, 4, 5, 6, 7, &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c, Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.”

John Napier (1550–1617) Scottish mathematician

The Construction of the Wonderful Canon of Logarithms (1889)

“His work, called Aryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is an alphabetical system in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, …., 100 etc. The other mathematical parts of Aryabhatiya consists of rules without examples. Another alphabetic system prevailed in Southern India, the numbers 1-19 being designated by consonants, etc.”

Aryabhata (476–550) Indian mathematician-astronomer

Florian Cajori in: A History of Mathematical Notations http://books.google.co.in/books?id=_byqAAAAQBAJ&pg=PT961&dq=Notations&hl=en&sa=X&ei=Wz65U5WYDIKulAW1qIGYDA&ved=0CBwQ6AEwAA#v=onepage&q=Notation&f=false, Courier Dover Publications, 26 September 2013, p. 47.

“There are 2 rules in life:
Number 1- Never quit
Number2- Never forget rule number 1.”

Duke Ellington (1899–1974) American jazz musician, composer and band leader

“Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c.; The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered?
Alt.1,21) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1.
2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2. Alt.3
3) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6.
Alt.34) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24.
5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c.;
6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please.”

John Wallis (1616–1703) English mathematician

Source: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.II Of Alternations, or the different Change of Order, in any Number of Things proposed.

“Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 180; also cited in Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914) pp. 156-157. https://books.google.com/books?id=G0wtAAAAYAAJ&pg=PA156

“First, the eight rules of tolerance: 1. Stay out of the way. 2. Don't pick a fight. 3. If challenged, walk away. 4. Avoid eye contact. 5. If your enemy is diurnal, learn to be nocturnal. 6. Vice versa. 7. Possess nothing the other wants. 8. Draw the line at hurting kids: I will fight to the death.
After tolerance is achieved comes play. Here are the nine rules of play: 1. Know when to quit. 2. Learn how to handicap. 3. Learn what frightens the other (cat claws). 4. Don't let it get to you. It's just a game; you mustn't take it seriously (cats have trouble with this one—escalation is always a risk in cat games). 5. Don't eat your playmate. 6. Pay attention to the signals on the other side—for instance, “enough” and “quit.” 7. Don't suddenly change the rules. 8. Don't be a sore loser. 9. Remember: It's only a game.
And if play succeeds, we can move on to the eight rules of friendship: 1. Learn the rules of your opposite number. 2. Recognize that danger is no longer relevant. 3. Take your time. 4. One step at a time. 5. Apologize often by learning the other's words, gestures, sounds, or postures for “I'm sorry.””

Jeffrey Moussaieff Masson (1941) American writer and activist

6. Acknowledge mistakes. 7. Make the offer of friendship more than once. 8. Express curiosity about what the other is like.
Source: Raising the Peaceable Kingdom (2005), Ch. 5

“A googolplex is precisely as far from infinity as is the number 1… no matter what number you have in mind, infinity is larger still.”

Carl Sagan book Cosmos

Source: Cosmos (1980), p. 220

“If, in a few months, I’m only number 8 or number 10 in the world, I’ll have to look at what off-the-court work I can do. I will need to do something if I want to be number 1.”

John McEnroe (1959) US tennis player

On losing to Tim Mayotte in the Ebel US Pro Indoor Championships, NY Times (February 9, 1987)

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Carl Sagan 365

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