How Long Is A Spoonful Of Water?

I can't quite remember how the teaspoon question arose except to recall that it came up in conversation with my son, a physics student.

First let me clarify the question. To state it more precisely: if you took all the molecules in a teaspoon of water and laid them end to end, how far would that thin aqueous line stretch? 

The first step is to figure out how many water molecules we have. 

A typical teaspoon holds about 5 millilitres (mL), which weights 5 grams. To find out how many water molecules there are in 5 grams you need to know that the molecular weight of water is 18 — the sum of the weights of one oxygen atom (16) and two hydrogen atoms (1 a piece) in H2O. What that means is 18 grams of water contains one mole of water molecules. Students of chemistry also know the mole to be a defined number of atoms or molecules (which relates the arbitrarily set scale of atomic weights — hydrogen = 1, helium = 2, and so on — to the actual weights of atoms in grams). It's rather big: one mole is 6.022 x 1023 in scientific notation. That's

602,200,000,000,000,000,000,000

So in 5 grams of water there would be 5/18ths of this number which is:

167,300,000,000,000,000,000,000

In other words: a lot.

But water molecules are very small; each one is only about 0.3 nanometers wide. That's 0.0003 micrometres, or 0.0000003 millimetres or 0.0000000003 metres. These are bizarre numbers — we have no real experience of them so it's hard to get much sense of scale. But let's plough on anyway.

If we lay down 167,300,000,000,000,000,000,000 water molecules end to end, the total length of the line is:

167,300,000,000,000,000,000,000 molecules x 0.0000000003 metres per molecule.

Which is 50,190,000,000,000 metres.

Or 50,190,000,000 km (that's 31,368,750,000 miles for older readers).

Which is 50 billion km. (How good was your guess?)

That's over 10 times the width of the solar system. From a teaspoon. 

Just think how far you could go with a bucket of water. 

Stephen Curry is a Professor of Structural Biology at Imperial College.

Arabic Numerals

Arabic numerals are based on angles, established by the great Muslim scientist: Khwarizmi.

The Muslim Scientist: Al-Khwarizmi

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī  (Arabic: عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي‎), earlier transliterated as Algoritmi or Algaurizin, (c. 780, Khwārizm – c. 850) was a Persian mathematician, astronomer and geographer during the Abbasid Empire, a scholar in theHouse of Wisdom in Baghdad.

In the twelfth century, Latin translations of his work on the Indian numerals introduced the decimal positional number system to the Western world.^[4] His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources.^[6] He revised Ptolemy's Geography and wrote on astronomy and astrology.

Some words reflect the importance of al-Khwarizmi's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name.^[7] His name is also the origin of (Spanish) guarismo^[8] and of (Portuguese) algarismo, both meaning digit.

CONTRIBUTIONS

Al-Khwārizmī's contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra and trigonometry. His systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his 830 book on the subject, "The Compendious Book on Calculation by Completion and Balancing" (al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabalaالكتاب المختصر في حساب الجبر والمقابلة).

On the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Indian system of numeration throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum. Al-Khwārizmī, rendered as (Latin) Algoritmi, led to the term "algorithm".

Some of his work was based on Persian and Babylonian astronomy, Indian numbers, and Greek mathematics.

Al-Khwārizmī systematized and corrected Ptolemy's data for Africa and the Middle East. Another major book was Kitab surat al-ard ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for theMediterranean Sea, Asia, and Africa.

He also wrote on mechanical devices like the astrolabe and sundial.

He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers.

When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in Europe. He introduced Arabic numerals into the Latin West, based on a place-value decimal system developed from Indian sources.

Algebra

Main article: The Compendious Book on Calculation by Completion and Balancing

Further information: Latin translations of the 12th century and Islamic science

Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة‎, 'The Compendious Book on Calculation by Completion and Balancing') is a mathematical book written approximately 830 CE. The book was written with the encouragement of the Caliph al-Ma'mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance.^[16] The term algebra is derived from the name of one of the basic operations with equations (al-jabr, meaning completion, or, subtracting a number from both sides of the equation) described in this book. The book was translated in Latin as Liber algebrae et almucabalaby Robert of Chester (Segovia, 1145) hence "algebra", and also by Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.^[17]

It provided an exhaustive account of solving polynomial equations up to the second degree,^[18] and discussed the fundamental methods of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.^[19]

Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and care positive integers)

• squares equal roots (ax² = bx)
• squares equal number (ax² = c)
• roots equal number (bx = c)
• squares and roots equal number (ax² + bx = c)
• squares and number equal roots (ax² + c = bx)
• roots and number equal squares (bx + c = ax²)

by dividing out the coefficient of the square and using the two operations al-jabr (Arabic: الجبر‎ "restoring" or "completion") and al-muqābala ("balancing"). Al-jabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x² = 40x − 4x² is reduced to 5x² = 40x. Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example, x² + 14 = x + 5 is reduced to x² + 9 = x.

The above discussion uses modern mathematical notation for the types of problems which the book discusses. However, in al-Khwārizmī's day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. For example, for one problem he writes, (from an 1831 translation)

"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts."

Several authors have also published texts under the name of Kitāb al-jabr wa-l-muqābala, including |Abū Ḥanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Ṭūsī.

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before."

R. Rashed and Angela Armstrong write:

"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."

Arithmetic

Al-Khwārizmī's second major work was on the subject of arithmetic, which survived in a Latin translation but was lost in the original Arabic. The translation was most likely done in the twelfth century by Adelard of Bath, who had also translated the astronomical tables in 1126.

The Latin manuscripts are untitled, but are commonly referred to by the first two words with which they start: Dixit algorizmi ("So said al-Khwārizmī"), orAlgoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of Reckoning"), a name given to the work by Baldassarre Boncompagni in 1857. The original Arabic title was possibly Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind^[22] ("The Book of Addition and Subtraction According to the Hindu Calculation").^[23]

Al-Khwarizmi's work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu-Arabic numeral system developed in Indian mathematics, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwarizmi's name, Algoritmi andAlgorismi, respectively.

Astronomy

Al-Khwārizmī's Zīj al-Sindhind^[12] (Arabic: زيج "astronomical tables of Sind and Hind") is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is the first of many Arabic Zijes based on the Indian astronomical methods known as the sindhind. The work contains tables for the movements of the sun, themoon and the five planets known at the time. This work marked the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.

The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslamah Ibn Ahmad al-Majriti (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (January 26, 1126).^[25] The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).

Trigonometry

Al-Khwārizmī's Zīj al-Sindhind also contained tables for the trigonometric functions of sines and cosine.^[24] A related treatise on spherical trigonometryis also attributed to him.^[20]

Geography

Hubert Daunicht's reconstruction of al-Khwārizmī's planisphere.

Al-Khwārizmī's third major work is his Kitāb ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض "Book on the appearance of the Earth" or "The image of the Earth" translated as Geography), which was finished in 833. It is a revised and completed version of Ptolemy's Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.

There is only one surviving copy of Kitāb ṣūrat al-Arḍ, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid.^{[citation needed]} The complete title translates as Book of the appearance of the Earth, with its cities, mountains, seas, all the islands and rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwārizmī, according to the geographical treatise written by Ptolemy the Claudian.

The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. As Paul Gallez^{[dubious – discuss]} points out, this excellent system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition as to make it practically illegible.

Neither the Arabic copy nor the Latin translation include the map of the world itself; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduces them from the context where they were not legible. He transferred the points onto graph paper and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He then does the same for the rivers and towns.

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea from the Canary Islands to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of longitude, while al-Khwarizmi almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done." Al-Khwarizmi thus set the Prime Meridian of the Old World at the eastern shore of the Mediterranean, 10–13 degrees to the east of Alexandria (the prime meridian previously set by Ptolemy) and 70 degrees to the west of Baghdad. Most medieval Muslim geographers continued to use al-Khwarizmi's prime meridian.

Jewish Calendar

Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar (Risāla fi istikhrāj taʾrīkh al-yahūd "Extraction of the Jewish Era"). It describes the 19-year intercalation cycle, the rules for determining on what day of the week the first day of the month Tishrī shall fall; calculates the interval between the Jewish era (creation of Adam) and the Seleucid era; and gives rules for determining the mean longitude of the sun and the moon using the Jewish calendar. Similar material is found in the works of al-Bīrūnī and Maimonides.

Other Works

Ibn al-Nadim in his Kitab al-Fihrist (an index of Arabic books) mentions al-Khwārizmī's Kitab al-Tarikh, a book of annals. No direct manuscript survives; however, a copy had reached Nisibis by the 1000s, where its metropolitan, Elias bar Shinaya, found it. Elias's chronicle quotes it from "the death of the Prophet" through to 169 AH, at which point Elias's text itself hits a lacuna.

Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials; the Fihrist credits al-Khwārizmī with Kitāb ar-Rukhāma(t). Other papers, such as one on the determination of the direction of Mecca, are on the spherical astronomy.

Two texts deserve special interest on the morning width (Maʿrifat saʿat al-mashriq fī kull balad) and the determination of the azimuth from a height (Maʿrifat al-samt min qibal al-irtifāʿ).

He also wrote two books on using and constructing astrolabes.

An Idea That Changed The World

Image via: NowPublishers.com

... Jan. 23, 1913, a century ago this week. Mathematician Andrey A. Markov delivered a lecture that day to the Imperial Academy of Sciences in St. Petersburg on a computational technique now called the Markov chain.

Little noticed in its day, his idea for modeling probability is fundamental to all of present-day science, statistics, and scientific computing. Any attempt to simulate probable events based on vast amounts of data — the weather, a Google search, the behavior of liquids — relies on Markov’s idea.

His lecture went on to engender a series of concepts, called Markov chains and Markov proposals, that calculate likely outcomes in complex systems. His technique is still evolving and expanding. “This is a growth industry,” said Boston-area science writer Brian Hayes. “You really can’t turn around in the sciences without running into some kind of Markov process.”

Before Markov, the theory of probability involved observing a series of events that were independent of each another. The classic example is flipping a coin, an activity that makes probability easy to calculate.

Markov added the idea of interdependence to probability, the notion that what happens next is linked to what is happening now. That creates “chain of linked events,” and not just a series of independent acts. The world, Markov posited, is not just a series of random events. It is a complex thing, and mathematics can help reveal its hidden interconnectedness and likely probabilities. 

In an article on Markov that will appear in the next issue of American Scientist, Hayes contrasts the probabilistic simplicity of coin flipping with the complexity of the board game “Monopoly.” Moves rely on a roll of the dice, but where the player ends up — Park Place? Jail? — also depends on where the player begins. Suddenly, probability (where you end up) is linked to a present state (where you start). Events are linked, not independent. That’s a Markov chain.

A “Monopoly” board has 40 possible “states,” the same as the number of squares. But Markov chains now are vastly larger. Google’s PageRank search algorithm, for instance, has as many states as there are pages on the Web, perhaps 40 billion.

Markov in his 1913 lecture, introduced his technique by analyzing the frequency of vowels and consonants in a work of literature. (Markov used the first 20,000 letters of Alexander Pushkin’s 1833 verse novel “Eugene Onegin,” a work that almost every Russian knew).

Such numerical analysis “is about the most primitive and superficial thing you can do to a poem,” said Hayes. But it proved what Markov wanted: that letters in language are interdependent, and that over time they converge into stable patterns. These patterns of behavior in a complex system are at the bottom of what modern scientists want — a simulation of what reality is, whether on the level of a cell or on the level of the entire Web.

Via: "Harvard University"

The Beauty Of Math

With: William “Bud” Abbott and Lou Costello