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Pi

Pi


Pi is represented by the symbol π (the sixteenth letter of the Greek alphabet) and appears in a number of measurement formulas. These formulas include c = 2πr and A = πr 2, where c is a circle's circumference or the distance around its outer edge, r is its radius or the distance from the center of the circle to its outer edge, and A is the circle's area. What is pi and why does π appear in these formulas?

What Is Pi?

What do the following items have in common with each other and all other circular figures: a quarter, approximately 1 inch across; a compact disc, 12 centimeters wide; a circular patio pool, spanning about 23 feet; and Earth's equator, roughly 8,000 miles in diameter . These estimated diameters are summarized in the table below, along with the estimated circumferences of the circular items.

In each case in the table, the circumference is about three times the diameter. The table also lists more precise measurements for the dimensions for our four examples. To determine the ratio of circumference (c ) to its diameter (d ), divide c by d. The resulting ratio of c:d is, in each case, a decimal slightly larger than 3.14. In fact, the circumference of any circle is about 3.14 times larger than its diameter. The relation, then, between the diameter and circumference of a circle can be summarized algebraically as c 3.14 × d. One may recognize that 3.14 is the approximate (rounded off) value of π. Thus, the exact relation between the diameter of a circle and its circumference can be summarized by the equation c = πd.

So, if one knows the diameter of a circle, it is possible to determine the distance around the circle without measuring it. For instance, if a car tire is 25 inches in diameter, its circumference can be calculated by multiplying

DIMENSIONS FOR FOUR COMMON CIRCULAR OBJECTS
Diameter Circumference
Item Estimated Precise Estimated Precise
Quarter 1 in 15/16 or 0.9375 in 3 in 2.95 in
CD 12 cm 12.0 cm 38 cm 37.7 cm
Pool 23 ft 22.9 ft 72 ft 72.0 ft
Earth's equator 8,000 mi 7,927 mi 25,000 mi 24,901 mi

this measure by 3.14, yielding 78.5 inches. Conversely, if a tire is about 78.5 inches around, then the diameter must be about 3.14 times smaller, or 25.0 inches.

What then is pi? Recall from the previous discussion that if the circumference of any circle is divided by its diameter, the result is always 3.14 or π. This fixed relation can be summarized by the equation . So, π is a constant; namely, the constant quotient or result of dividing a circle's circumference by its diameter. Put in more formal terms, the constant pi is the ratio of a circle's circumference to its distance ().

A History of Pi

The person or persons who discovered the constant relationship between a circle's circumference and diameter are unknown. What history does show is that people's understanding of pi has developed gradually, and that gauging its value has involved a range of methods, from educated guesses based on measurements to using highly theoretical mathematics.

The number represented by pi appears in the earliest historical records of mathematics. Babylonian clay tablets (1800 b.c.e.1650 b.c.e.) mention how a circle's area could be determined from its circumference by using a constant, which in effect is equivalent to 3 or 3.125. The Rhind papyrus (c. 1650 b.c.e.) includes mention of a pi-like constant, equivalent to or about 3.16, in the context of geometrical problems that involved finding a square equal in area to a given circle. A biblical account of the construction of King Solomon's temple (c. 1000 b.c.e.) suggests that the ancient Hebrews were aware that c = 3 d : "And he made a molten sea, ten cubits from one brim to the other. It was round all aboutand a line of thirty cubits did compass it about" (I Kings 7:23). Archimedes of Syracuse (287 b.c.e.212 b.c.e.) explicitly discussed the ratio of a circle's circumference and diameter and may have been the first to provide a theoretical calculation of pi. By using a series of polygons inscribed in a circle and another circumscribing it, he was able to determine pi was less than but greater than or about 3.1418.

Until the sixteenth century, it was commonly thought that pi, which embodies a fundamental truth about the most perfect of geometric forms (the circle) had to be a special number. Efforts by mathematicians to determine the value of pi since have led to the conclusion that it does not have an exact valuethat its decimal value extends on infinitely without repeating a pattern (3.14159265). As Carl Sagan noted in his novel Contact (1985), mathematicians can use calculus to prove formulas for π that would permit calculation of its value. As of 1999, computers have been used to do such calculations to at least 200 billion places without a repeating pattern.

Characteristics of Pi

As an infinitely nonrepeating decimal, pi is not a unique number. Most people are familiar with rational numbers, which can be represented by a ratio or common fraction. Some rational numbers are represented by terminating decimals. For example, , and 5 (which can be put in the form ) have decimal equivalents of 0.5, 0.375, 1.5, and 5.0, respectively. Other rational numbers have as a decimal equivalent an infinitely repeating decimal (i.e., a set of digits that repeats forever). For example, are represented by 0.6666 (), 0.8333 (0.83), 1.142857142857 respectively.

Unlike rational numbers, pi cannot be represented by a common fraction, although , which is equal to the infinitely repeating decimal , is a reasonable approximation of its value for many everyday uses. In other words, is an irrational number. Although it is probably the best known number in this category, there are, in fact, an infinite number of other irrational numbers (e.g., , which can represent the diagonal of a square with a side of 3 linear units).

Why Does π Appear in Measurement Formulas Involving Circles?

Consider the formula for calculating the circumference of a circle. Recall that we established that c = π d. The diameter of a circle can be thought of as two radii stuck together to form a straight line, d = 2r. Substituting 2r for d in the equation gives us c = π 2r, which can be rewritten as the familiar formula for the circumference of a circle c = 2πr.

The formula for calculating the area can be derived by considering how a circle can be transformed into a shape for which we already know the area formula. As suggested by the following figure, a circle can, in theory, be divided up into an infinite number of infinitely small "pie wedges" and these pie wedges can then be put "head to toe." The result of rearranging the infinitely small wedges in this manner is a rectangle, the area of which is length (l ) times width (w ). Because the circumference of the circle forms the top and bottom of the rectangle, a length of the rectangle is one-half the circumference of the original circle (). The radius of the original circle forms the width of the rectangle, w = r. Substituting for l and r for w in the area formula for a rectangle (A = l w ) yields . Recall that c = πd and that d = 2r, c = π 2r or 2πr. Substituting 2πr for c in the formula , produces the equation , which can be simplified by canceling the 2's in the numerator and denominator. This equation can then be rewritten as the familiar formula for the area of a circle: A = πr 2.

Note that rearranging the equation A = πr 2 can yield . In other words, pi can also be thought of as the ratio of a circle's area and the square of its radius.

see also Decimals; Fractions; Numbers, Irrational; Numbers, Rational.

Arthur J. Baroody

Bibliography

Beckmann, Petr. History of Pi. New York: St. Martin's Press, 1976.

Blatner, David. The Joy of Pi. New York: Walker & Co., 1999.

Bunt, Lucas N. H., Philip S. Jones, and Jack D. Bedient. The Historical Roots of Elementary Mathematics. Englewood Cliffs, NJ: Prentice-Hall, 1976.

Sagan, Carl. Contact. New York: Simon and Schuster, 1983.

Internet Resources

Kanada Laboratory Homepage. Dr. Yasumasa Kanada, University of Tokyo. <http://www.super-computing.org>


A TEENAGER'S QUEST

After five months of human time and one-and-a-half years of computer time, Colin Percival discovered the five trillionth binary digit of pi: 0. This 1998 accomplishment was significant because, for the first time, the calculations were distributed among twenty-five computers around the world.

Percival, who was 17 years old and a high school senior at the time, had concurrently been attending Simon Fraser University in Canada since he was 13.


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pi

pi, in mathematics, the ratio of the circumference of a circle to its diameter. The symbol for pi is π. The ratio is the same for all circles and is approximately 3.1416. It is of great importance in mathematics not only in the measurement of the circle but also in more advanced mathematics in connection with such topics as continued fractions, logarithms of imaginary numbers, and periodic functions. Throughout the ages progressively more accurate values have been found for π; an early value was the Greek approximation 31/7, found by considering the circle as the limit of a series of regular polygons with an increasing number of sides inscribed in the circle. About the mid-19th cent. its value was figured to 707 decimal places and by the mid-20th cent. an electronic computer had calculated it to 100,000 digits. Although it has now been calculated to some 2.6 trillion digits, the exact value of π cannot be computed. It was shown by the German mathematician Johann Lambert in 1770 that π is irrational and by Ferdinand Lindemann in 1882 that π is transcendental; i.e., cannot be the root of any algebraic equation with rational coefficients. The important connection between π and e, the base of natural logarithms, was found by Leonhard Euler in the famous formula e=-1, where i=-1.

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pi

pi / / • n. the sixteenth letter of the Greek alphabet (Π, π), transliterated as ‘p.’ ∎  the numerical value of the ratio of the circumference of a circle to its diameter (approx. 3.14159). ∎  (Pi) [followed by Latin genitive] Astron. the sixteenth star in a constellation: Pi Herculis. ∎  Chem. & Physics relating to or denoting an electron or orbital with one unit of angular momentum about an internuclear axis. • symb. ∎  (π) the numerical value of pi. ∎  (Π) osmotic pressure. ∎  (Π) mathematical product.

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pi

pi (π) Symbol used for the ratio of the circumference of a circle to its diameter. It is an irrational number, and an approximation to five decimal places is 3.14159. The ratio 22/7 is often used as a rougher approximation.

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pi

pi the sixteenth letter of the Greek alphabet (Π, π), transliterated as ‘p’. Also, the numerical value of the ratio of the circumference of a circle to its diameter (approximately 3.14159).

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pi

pi (math.) ratio of the circumference of a circle to the diameter. XIX. name of the Gr. letter Π π (P p), initial letter of periphéreia periphery and perímetros perimeter.

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PI

PI • abbr. private investigator

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pi

pially, Altai, apply, assai, awry, ay, aye, Baha'i, belie, bi, Bligh, buy, by, bye, bye-bye, chi, Chiangmai, Ciskei, comply, cry, Cy, Dai, defy, deny, Di, die, do-or-die, dry, Dubai, dye, espy, eye, fie, fly, forbye, fry, Frye, goodbye (US goodby), guy, hereby, hi, hie, high, I, imply, I-spy, July, kai, lie, lye, Mackay, misapply, my, nearby, nigh, Nye, outfly, passer-by, phi, pi, pie, ply, pry, psi, Qinghai, rai, rely, rocaille, rye, scry, serai, shanghai, shy, sigh, sky, Skye, sky-high, sly, spin-dry, spry, spy, sty, Sukhotai, supply, Tai, Thai, thereby, thigh, thy, tie, Transkei, try, tumble-dry, underlie, Versailles, Vi, vie, whereby, why, wry, Wye, xi, Xingtai, Yantai

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