NUMBERS & MATHEMATICS SCIENCE HISTORY QUESTION ANSWERS

**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: http://www.mersenne.org/default.php.

**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 Google.com 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.

**MATHEMATICS**

**How is arithmetic different from mathematics?**

**What is the most enduring mathematical work of all time?**

**Who invented calculus?**

**Is it possible to count to infinity?**

**How long has the abacus been used?**

**What are Napier’s bones?**

**What are Cuisenaire rods?**

**What is a slide rule, and who invented it?**

**How is casting out nines used to check the results of addition or multiplication?**

**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?**

**What is a golden section?**

**What is a Möbius strip?**

**How is the rule of 70 used?**

**How is the percent of increase calculated?**

**How many different bridge games are possible?**

**What are fractals?**

**What is the difference between simple interest and compound interest?**

**What is the probability of a triple play occurring in a single baseball game?**

**What is the law of very large numbers?**

**What is the Königsberg Bridge Problem?**

**How long did it take to prove the four-color map theorem?**

**What is the science of chaos?**

**What is Zeno’s paradox?**

**Are there any unsolved problems in mathematics?**

**What are the seven Millennium Prize Problems?**

__The seven Millennium Prize Problems are:__