When and where did the concept of “numbers” and counting first develop?

The human adult (including some of the higher animals) can discern the numbers one through four without any training. After that people must learn to count. To count requires a system of number manipulation skills, a scheme to name the numbers, and some way to record the numbers. Early people began with fingers and toes and progressed to shells and pebbles. In the fourth millennium B.C.E. in Elam (near what is today Iran along the Persian Gulf), accountants began using unbaked clay tokens instead of pebbles. Each represented one order in a numbering system: a stick shape for the number one, a pellet for ten, a ball for 100, and so on. During the same period, another clay-based civilization in Sumer in lower Mesopotamia invented the same system.

When was a symbol for the concept of zero first used?

Surprisingly, the symbol for zero emerged later than the concept for the other numbers. Although the Babylonians (600 B.C.E. and earlier) had a symbol for zero, it was merely a placeholder and not used for computational purposes. The ancient Greeks conceived of logic and geometry, concepts providing the foundation for all mathematics, yet they never had a sym- bol for zero. The Maya also had a symbol for zero as a placeholder in the fourth century, but they also did not use zero in computations. Hindu mathematicians are usually given credit for developing a symbol for the concept “zero.” They recognized zero as representing the absence of quantity and developed its use in mathematical calculations. It appears in an inscription at Gwalior dated 870 C.E. However, it is found even earlier than that in inscriptions dating from the seventh century in Cambodia, Sumatra, and Bangka Island (off the coast of Sumatra). Although there is no documented evidence in printed material for the zero in China before 1247, some historians maintain that there was a blank space on the Chinese counting board, representing zero, as early as the fourth century B.C.E.

What are Roman numerals?

Roman numerals are symbols that stand for numbers. They are written using seven basic symbols: I (1), V (5), X (10), L (50), C (100), D (500), and M (1,000). Sometimes a bar is placed over a numeral to multiply it by 1,000. A smaller numeral appearing before a larger numeral indicates that the smaller numeral is subtracted from the larger one. This notation is generally used for 4s and 9s; for example, 4 is written IV, 9 is IX, 40 is XL, and 90 is XC.

What are Fibonacci numbers?

Fibonacci numbers are a series of numbers where each, after the second term, is the sum of the two preceding numbers—for example, 1, 1, 2, 3, 5, 8, 13, 21, and so on). They were first described by Leonardo Fibonacci (c. 1180–c. 1250), also known as Leonard of Pisa, as part of a thesis on series in his most famous book Liber abaci (The Book of the Calculator), published in 1202 and later revised by him. Fibonacci numbers are used frequently to illustrate natural sequences, such as the spiral organization of a sunflower’s seeds, the chambers of a nautilus shell, or the reproductive capabilities of rabbits.

What is the largest prime number presently known?

A prime number is one that is evenly divisible only by itself and 1. The integers 1, 2, 3, 5, 7, 11, 13, 17, and 19 are prime numbers. Euclid (c. 335–270 B.C.E.) proved that there is no “largest prime number,” because any attempt to define the largest results in a paradox. If there is a largest prime number (P), adding 1 to the product of all primes up to and including P, 1 1 (1 3 2 3 3 3 5 3 … 3 P), yields a number that is itself a prime number, because it cannot be divided evenly by any of the known primes. In 2003, Michael Shafer discovered the largest known (and the fortieth) prime number: 220996011 – 1. This is over six million digits long and would take more than three weeks to write out by hand. In July 2010, double-checking proved this was the fortieth Mersenne prime (named after Marin Mersenne, 1588–1648, a French monk who did the first work in this area). Mersenne primes occur where 2n–1 is prime.

There is no apparent pattern to the sequence of primes. Mathematicians have been trying to find a formula since the days of Euclid, without success. The fortieth prime was discovered on a personal computer as part of the GIMPS effort (the Great Internet Mersenne Prime Search), which was formed in January 1996 to discover new world-record-size prime numbers. GIMPS relies on the computing efforts of thousands of small, personal computers around the world. Interested participants can become involved in the search for primes by going to:

What is a perfect number?

A perfect number is a number equal to the sum of all its proper divisors (divisors small- er than the number) including 1. The number 6 is the smallest perfect number; the sum of its divisors 1, 2, and 3 equals 6. The next three perfect numbers are 28, 496, and 8,126. No odd perfect numbers are known. The largest known perfect number is (23021376)(23021377 – 1) It was discovered in 2001.

What is the Sieve of Eratosthenes?

Eratosthenes (c. 285 –c. 205 B.C.E.) was a Greek mathematician and philosopher who devised a method to identify (or “sift” out) prime numbers from a list of natural num- bers arranged in order. It is a simple method, although it becomes tedious to identify large prime numbers. The steps of the sieve are:

1. Write all natural numbers in order, omitting 1.

2. Circle the number 2 and then cross out every other number. Every second number will be a multiple of 2 and hence is not a prime number.

3. Circle the number 3 and then cross out every third number which will be a multiple of 3 and, therefore, not a prime number.

4. The numbers that are circled are prime and those that are crossed out are composite numbers.

How large is a googol?

A googol is 10100 (the number 1 followed by 100 zeros). Unlike most other names for numbers, it does not relate to any other numbering scale. The American mathematician Edward Kasner (1878–1955) first used the term in 1938; when searching for a term for this large number, Kasner asked his nephew, Milton Sirotta (1911–1981), then about nine years old, to suggest a name. The googolplex is 10 followed by a googol of zeros, represented as 10googol. The popular Web search engine is named after the concept of a googol.

What is an irrational number?

Numbers that cannot be expressed as an exact ratio are called irrational numbers; numbers that can be expressed as an exact ratio are called rational numbers. For instance, 1/2 (one half, or 50 percent of something) is rational; however, 1.61803 (ф), 3.14159 (π), 1.41421 (G¯2¯), are irrational. History claims that Pythagoras in the sixth century B.C.E. first used the term when he discovered that the square root of 2 could not be expressed as a fraction.


What are imaginary numbers?

Imaginary numbers are the square roots of negative numbers. Since the square is the product of two equal numbers with like signs it is always positive. Therefore, no number multiplied by itself can give a negative real number. The symbol “i” is used to indicate an imaginary number.

What is the value of pi out to 30 digits past the decimal point?

Pi (π) represents the ratio of the circumference of a circle to its diameter, used in calculating the area of a circle (πr2) and the volume of a cylinder (πr2h) or cone. It is a “transcendental number,” an irrational number with an exact value that can be measured to any degree of accuracy, but that can’t be expressed as the ratio of two integers. In theory, the decimal extends into infinity, though it is generally rounded to 3.1416. The Welsh-born mathematician William Jones (1675–1749) selected the Greek symbol (π) for pi. Rounded to 30 digits past the decimal point, it equals 3.1415926535 89793238462643383279.

In 1989, Gregory (1952–) and David Chudnovsky (1947–) at Columbia University in New York City calculated the value of pi to 1,011,961,691 decimal places. They performed the calculation twice on an IBM 3090 mainframe and on a CRAY-2 supercomputer with matching results. In 1991, they calculated pi to 2,260,321,336 decimal places.


In 1999, Yasumasa Kanada (1948–) and Daisuke Takahashi of the University of Tokyo calculated pi out to 206,158,430,000 digits. Professor Kanada at the University of Tokyo continues to calculate the value of pi to greater and greater digits. His laboratory’s newest record, achieved in 2002 and subsequently verified, calculated pi to 1.2411 × 1012 digits (more than one trillion). The calculation required more than 600 hours of computing time using a Hitachi SR8000 computer with access to a memory of about 1 terabyte.

Mathematicians have also calculated pi in binary format (i.e., 0s and 1s). The five trillionth binary digit of pi was computed by Colin Percival and 25 others at Simon Fraser University. The computation took over 13,500 hours of computer time.

What are some examples of numbers and mathematical concepts in nature?

The world can be articulated with numbers and mathematics. Some numbers are especially prominent. The number six is ubiquitous: every normal snowflake has six sides; every honeybee colony’s combs are six-sided hexagons. The curved, gradually decreasing chambers of a nautilus shell are propagating spirals of the golden section and the Fibonacci sequence of numbers. Pine cones also rely on the Fibonacci sequence, as do many plants and flowers in their seed and stem arrangements. Fractals are evident in shorelines, blood vessels, and mountains.


How is arithmetic different from mathematics?
Arithmetic is the study of positive integers (i.e., 1, 2, 3, 4, 5) manipulated with addition, subtraction, multiplication, and division, and the use of the results in daily life. Mathematics is the study of shape, arrangement, and quantity. It is traditionally viewed as consisting of three fields: algebra, analysis, and geometry. But any lines of division have evaporated because the fields are now so interrelated.

What is the most enduring mathematical work of all time?
The Elements of Euclid (c. 300 B.C.E.) has been the most enduring and influential mathematical work of all time. In it, the ancient Greek mathematician presented the work of earlier mathematicians and included many of his own innovations. The Elements is divided into thirteen books: the first six cover plane geometry; seven to nine address arithmetic and number theory; ten treat irrational numbers; and eleven to thirteen dis- cuss solid geometry. In presenting his theorems, Euclid used the synthetic approach, in which one proceeds from the known to the unknown by logical steps. This method became the standard procedure for scientific investigation for many centuries, and the Elements probably had a greater influence on scientific thinking than any other work.
Who invented calculus?
The German mathematician Gottfried Wilhelm Leibniz (1646–1716) published the first paper on calculus in 1684. Most historians agree that Isaac Newton invented calculus eight to ten years earlier, but he was typically very late in publishing his works. The invention of calculus marked the beginning of higher mathematics. It provided scientists and mathematicians with a tool to solve problems that had been too complicated to attempt previously.

Is it possible to count to infinity?
No. Very large finite numbers are not the same as infinite numbers. Infinite numbers are defined as being unbounded, or without limit. Any number that can be reached by counting or by the representation of a number followed by billions of zeros is a finite number.

How long has the abacus been used?
The abacus grew out of early counting boards, with hollows in a board holding pebbles or beads used to calculate. It has been documented in Mesopotamia back to around 3500 B.C.E. The current form, with beads sliding on rods, dates back at least to fifteenth-century China. Before the use of decimal number systems, which allowed the familiar paper-and-pencil methods of calculation, the abacus was essential for almost all multiplication and division. Unlike the modern calculator, the abacus does not perform any mathematical computations. The person using the abacus performs calculations in his/her head relying on the abacus as a physical aid to keep track of the sums. It has become a valuable tool for teaching arithmetic to blind students.

What are Napier’s bones?
In the sixteenth century, the Scottish mathematician John Napier (1550–1617), Baron of Merchiston, developed a method of simplifying the processes of multiplication and division, using exponents of 10, which Napier called logarithms (commonly abbreviated as logs). Using this system, multiplication is reduced to addition and division to subtraction. For example, the log of 100 (102) is 2; the log of 1000 (103) is 3; the multiplication of 100 by 1000, 100 × 1000 = 100,000, can be accomplished by adding their logs: log[(100)(1000)] = log(100) + log(1000) = 2 + 3 = 5 = log(100,000). Napier published his methodology in A Description of the Admirable Table of Logarithms in 1614. In 1617 he published a method of using a device, made up of a series of rods in a frame, marked with the digits 1 through 9, to multiply and divide using the principles of logarithms. This device was commonly called “Napier’s bones” or “Napier’s rods.”

What are Cuisenaire rods?
The Cuisenaire method is a teaching system used to help young students independently discover basic mathematical principles. Developed by Emile-Georges Cuisenaire (1891–1976), a Belgian school- teacher, the method uses rods of ten different colors and lengths that are easy to handle. The rods help students understand mathematical principles rather than merely memorize them. They are also used to teach elementary arithmetic properties such as associative, commutative, and distributive properties.


What is a slide rule, and who invented it?
Up until about 1974, most engineering and design calculations for buildings, bridges, automobiles, airplanes, and roads were done on a slide rule. A slide rule is an apparatus with moveable scales based on logarithms, which were invented by John Napier, Baron of Merchiston, and pub- lished in 1614. The slide rule can, among other things, quickly multiply, divide, square root, or find the logarithm of a number. In 1620, Edmund Gunter (1581–1626) of Gresham College, London, England, described an immediate forerunner of the slide rule, his “logarithmic line of numbers.” William Oughtred (1574–1660), rector of Ald- bury, England, made the first rectilinear slide rule in 1621. This slide rule consisted of two logarithmic scales that could be manipulated together for calculation. His former pupil, Richard Delamain, published a description of a circular slide rule in 1630 (and received a patent about that time for it), three years before Oughtred published a description of his invention (at least one source says that Delamain published in 1620). Oughtred accused Delamain of stealing his idea, but evidence indicates that the inventions were probably arrived at independently.

The earliest existing straight slide rule using the modern design of a slider moving in fixed stock dates from 1654. A wide variety of specialized slide rules were developed by the end of the seventeenth century for trades such as masonry, carpentry, and excise tax collecting. Peter Mark Roget (1779–1869), best known for his Thesaurus of English Words and Phrases, invented a log-log slide rule for calculating the roots and powers of numbers in 1814. In 1967, Hewlett-Packard produced the first pocket calculators. Within a decade, slide rules became the subject of science trivia and collector’s books. Interestingly, slide rules were carried on five of the Apollo space missions, including a trip to the moon. They were known to be accurate and efficient in the event of a computer malfunction.

How is casting out nines used to check the results of addition or multiplication?
The method of “casting out nines” is based on the excess of nines in digits of whole numbers (the remainder when a sum of digits is divided by 9). Illustrating this process in the multiplication example below, the method begins by adding the digits in both the multiplicand (one of the terms that are being multiplied) and the multiplier (the other term being multiplied). In the example below, this operation leads to the results of “13” and “12,” respectively. If these results are greater than 9 (>9), then the operation is repeated until the resulting figures are less than 9 (<9). In the example below, the repeated calculation gives the results as “4” and “3,” respectively. Multiply the resulting “excess” from the multiplicand by the excess from the multiplier (4 3 3 below). Add the digits of the result to eventually yield a number equal to or less than 9 (“9). Repeat the process of casting out nines in the multiplication product (the result of the multiplication process). The result must equal the result of the previous set of transactions, in this case, “3.” If the two figures disagree, then the original multiplication procedure was done incorrectly. “Casting out nines” can also be applied to check the accuracy of the results of addition.

What is the difference between a median and a mean?

If a string of numbers is arranged in numerical order, the median is the middle value of the string. If there is an even number of values in the string, the median is found by adding the two middle values and dividing by two. The arithmetic mean, also known as the simple average, is found by taking the sum of the numbers in the string and dividing by the number of items in the string. While easy to calculate for relatively short strings, the arithmetic mean can be misleading, as very large or very small values in the string can distort it. For example, the mean of the salaries of a professional football team would be skewed if one of the players was a high-earning superstar; it could be well above the salaries of any of the other players thus making the mean higher. The mode is the number in a string that appears most often.

For the string 111222234455667, for example, the median is the middle number of the series: 3. The arithmetic mean is the sum of numbers divided by the number of numbers in the series, 51 / 15 = 3.4. The mode is the number that occurs most often, 2.

When did the concept of square root originate?

A square root of a number is a number that, when multiplied by itself, equals the given number. For instance, the square root of 25 is 5 (5 × 5 = 25). The concept of the square root has been in existence for many thousands of years. Exactly how it was discovered is not known, but several different methods of exacting square roots were used by early mathematicians. Babylonian clay tablets from 1900 to 1600 B.C.E. contain the squares and cubes of integers 1 through 30. The early Egyptians used square roots around 1700 B.C.E., and during the Greek Classical Period (600 to 300 B.C.E.) better arithmetic methods improved square root operations. In the sixteenth century, French mathematician René Descartes (1596–1650) was the first to use the square root symbol, called “the radical sign,” G

What are Venn diagrams?

Venn diagrams are graphical representations of set theory, which use circles to show the logical relationships of the elements of different sets, using the logical operators (also called in computer parlance “Boolean Operators”) and, or, and not. John Venn (1834–1923) first used them in his 1881 Symbolic Logic, in which he interpreted and corrected the work of George Boole (1815–1864) and Augustus de Morgan (1806– 1871). While his attempts to clarify perceived inconsistencies and ambiguities in Boole’s work are not widely accepted, the new method of the diagram is considered to be an improvement. Venn used shading to better illustrate inclusion and exclusion. Charles Dodgson (1832–1898), better known by his pseudonym Lewis Carroll, refined Venn’s system, in particular by enclosing the diagram to represent the universal set.

What does the expression “tiling the plane” mean?
It is a mathematical expression describing the process of forming a mosaic pattern (a “tessellation”) by fitting together with an infinite number of polygons so that they cover an entire plane. Tessellations are the familiar patterns that can be seen in designs for quilts, floor coverings, and bathroom tile work.

What is a golden section?
The Golden section, also called the divine proportion, is the division of a line segment so that the ratio of the whole segment to the larger part is equal to the ratio of the larger part to the smaller part. The ratio is approximately 1.61803 to 1. The number 1.61803 is called the golden number (also called Phi [with a capital P]). The golden number is the limit of the ratios of consecutive Fibonacci numbers, such as, for instance, 21/13 and 34/21. A golden rectangle is one whose length and width correspond to this ratio. The ancient Greeks thought this shape had the most pleasing proportions. Many famous painters have used the golden rectangle in their paintings, and architects have used it in their design of buildings, the most famous example being the Greek Parthenon.

What is a Möbius strip?
A Möbius strip is a surface with only one side, usually made by connecting the two ends of a rectangular strip of paper after putting a half-twist (180 degrees relative to the opposite side) in the strip. Cutting a Möbius strip in half down the center of the length of the strip results in a single band with four half-twists. Devised by the German mathematician August Ferdin and Möbius (1790–1868) to illustrate the properties of one-sided surfaces, it was presented in a paper that was not dis- covered or published until after his death. Another nineteenth-century German mathematician, Johann Benedict Listing (1808–1882), developed the idea independently at the same time.

How is the rule of 70 used?
This rule is a quick way of estimating the period of time it will take a quantity to double given the percentage of increase. Divide the percentage of increase into 70. For example, if a sum of money is invested at six percent interest, the money will double in value in 70/6 = 11.7 years.

How is the percent of increase calculated?
To find the percent of the increase, divide the amount of increase by the base amount. Multiply the result by 100 percent. For example, a raise in salary from $10,000 to $12,000 would have percent of increase = (2,000/10,000) × 100% = 20%.

How many different bridge games are possible?
Roughly 54 octillion different bridge games are possible.

What are fractals?
A fractal is a set of points that are too irregular to be described by traditional geometric terms, but that often possess some degree of self-similarity; that is, are made of parts that resemble the whole. They are used in image processing to compress data and to depict apparently chaotic objects in nature such as mountains or coastlines. Scientists also use fractals to better comprehend rainfall trends, patterns formed by clouds and waves, and the distribution of vegetation. Fractals are also used to create computer-generated art.

What is the difference between simple interest and compound interest?
Simple interest is calculated on the amount of principal only. Compound interest is calculated on the amount of principal plus any previous interest already earned. For example, $100 invested at a rate of five percent for one year will earn $5.00 after one year earning simple interest. The same $100 will earn $5.12 if compounded monthly.

What is the probability of a triple play occurring in a single baseball game?
The odds against a triple play in a game of baseball are 1,400 to 1.

What is the law of very large numbers?
Formulated by Persi Diaconis (1945–) and Frederick Mosteller (1916–2006) of Harvard University, this long-understood law of statistics states that “with a large enough sample, any outrageous thing is apt to happen.” Therefore, seemingly amazing coincidences can actually be expected if given sufficient time or a large enough pool of sub-jects. For example, when a New Jersey woman won the lottery twice in four months, the media publicized it as an incredible long shot of 1 in 17 trillion. However, when statisticians looked beyond this individual’s chances and asked what were the odds of the same happening to any person buying a lottery ticket in the United States over a six-month period, the number dropped dramatically to 1 in 30. According to researchers, coincidences arise often in statistical work, but some have hidden causes and therefore are not coincidences at all. Many are simply chance events reflecting the luck of the draw.

What is the Königsberg Bridge Problem?
The city of Königsberg was located in Prussia on the Pregel River. Two islands in the river were connected by seven bridges. By the eighteenth century, it had become a tradition for the citizens of Königsberg to go for a walk through the town trying to cross each bridge only once. No one was able to succeed, and the question was asked whether it was possible to do so. In 1736, Leonhard Euler (1707–1783) proved that it was not possible to cross the Königsberg bridges only once. Euler’s solution led to the development of two new areas of mathematics: graph theory, which deals with questions about networks of points that are connected by lines; and topology, which is the study of those aspects of the shape of an object that does not depend on length measurements.

How long did it take to prove the four-color map theorem?
The four-color map problem was first posed by Francis Guthrie (1831–1899) in 1852. While coloring a map of the English counties, Guthrie discovered he could do it with only four colors and no two adjacent counties would be the same color. He extrapolated the question to whether every map, no matter how complicated and how many countries are on the map, could be colored using only four colors with no two adjacent countries being the same color. The theorem was not proved until 1976, 124 years after the question had been raised, by Kenneth Appel (1932–) and Wolfgang Haken (1928–). Their proof is considered correct although it relies on computers for the calculations. There is no known simple way to check the proof by hand.

What is the science of chaos?
Chaos or chaotic behavior is the behavior of a system whose final state depends very sensitively on the initial conditions. The behavior is unpredictable and cannot be distinguished from a random process, even though it is strictly determinate in a mathematical sense. Chaos studies the complex and irregular behavior of many systems in nature, such as changing weather patterns, the flow of turbulent fluids, and swinging pendulums. Scientists once thought they could make exact predictions about such systems, but found that a tiny difference in starting conditions can lead to greatly different results. Chaotic systems do obey certain rules, which mathematicians have described with equations, but the science of chaos demonstrates the difficulty of predicting the long-range behavior of chaotic systems.

What is Zeno’s paradox?
Zeno of Elea (c. 490–c. 425 B.C.E.), a Greek philosopher and mathematician, is famous for his paradoxes, which deal with the continuity of motion. One form of the paradox is: If an object moves with constant speed along a straight line from point 0 to point 1, the object must first cover half the distance (1/2), then half the remaining distance (1/4), then half the remaining distance (1/8), and so on without end. The conclusion is that the object never reaches point 1. Because there is always some distance to be covered, motion is impossible. In another approach to this paradox, Zeno used an allegory telling of a race between a tortoise and Achilles (who could run 100 times as fast), where the tortoise started running 10 rods (165 feet) in front of Achilles. Because the tortoise always advanced 1/100 of the distance that Achilles advanced in the same time period, it was theoretically impossible for Achilles to pass him. The English mathematician and writer Charles Dodgson, better known as Lewis Carroll, used the characters of Achilles and the tortoise to illustrate his paradox of infinity.

Are there any unsolved problems in mathematics?
The earliest challenges and contests to solve important problems in mathematics date back to the sixteenth and seventeenth centuries. Some of these problems have continued to challenge mathematicians until modern times. For example, Pierre de Fermat (1601–1665) issued a set of mathematical challenges in 1657, many on prime numbers and divisibility. The solution to what is now known as Fermat’s Last Theorem was not established until the late 1990s by Andrew Wiles (1953–). David Hilbert (1862– 1943), a German mathematician, identified 23 unsolved problems in 1900 with the hope that these problems would be solved in the twenty-first century. Although some of the problems were solved, others remain unsolved to this day. More recently, in 2000 the Clay Mathematics Institute named seven mathematical problems that had not been solved with the hope that they could be solved in the twenty-first century. A $1 million prize will be awarded for solving each of these seven problems.

What are the seven Millennium Prize Problems?
The seven Millennium Prize Problems are:
1. Birch and Swinnerton-Dyer Conjecture
2. Hodge Conjecture
3. Poincaré Conjecture
4. Riemann Hypothesis
5. Solution of the Navier-Stokes equations
6. Formulation of the Yang-Mills theory
7. P vs NP

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