by DK
Newton made significant discoveries in the fields of gravitation, motion, and optics, where he developed a rivalry with eminent English scientist Robert Hooke. One of several government positions he held was Master of the Royal Mint, where he oversaw the switch of the British currency from the silver to the gold standard. He was also President of the Royal Society. Newton died in 1727.
Key work
1687 Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy)
See also: Syllogistic logic • The problem of maxima • Calculus • Emmy Noether and abstract algebra
IN CONTEXT
KEY FIGURE
Jacob Bernoulli (1655–1705)
FIELD
Probability
BEFORE
c. 1564 Gerolamo Cardano writes Liber de ludo aleae (The Book on Games of Chance), the first work on probability.
1654 Pierre de Fermat and Blaise Pascal develop probability theory.
AFTER
1733 Abraham de Moivre proposes what becomes the central limit theorem—as a sample size increases, the results will more closely match normal distribution, or the bell curve.
1763 Thomas Bayes develops a way of predicting the chance of an outcome by taking into account the starting conditions related to that outcome.
The law of large numbers is one of the foundations of probability theory and statistics. It guarantees that, over the long term, the outcomes of future events can be predicted with reasonable accuracy. This, for example, gives financial companies the confidence to set prices for insurance and pension products, knowing their chances of having to pay out, and ensures that casinos will always make a profit from their gambling customers—eventually.
According to the law, as you make more observations of an event occurring, the measured probability (or chance) of that outcome gets ever closer to the theoretical chance as calculated before any observations began. In other words, the average result from a large number of trials will be a close match to the expected value as calculated using probability theory—and increasing the number of trials will result in that average becoming an even closer match.
The law was named by French mathematician Siméon Poisson in 1835, but its origin is credited to Swiss mathematician Jacob Bernoulli. His breakthrough, which he called the “golden theorem,” was published by his nephew in 1713 in the book Ars Conjectandi (The Art of Conjecturing).
Although not the first person to recognize the relationship between collecting data and predicting results, Bernoulli developed the first proof of this relationship by considering a game with two possible outcomes—a win or a loss. The theoretical chance of winning the game is W, and Bernoulli suspected that the fraction of games (f) that resulted in a win would converge on W as the number of games increased. He proved this by showing that the probability of f being greater or less than W by a specified amount approached 0 (meaning impossible) as the game was repeated.
We define the art of conjecture… as the art of evaluating… the probabilities of things, so that in our judgments and actions we can always base ourselves on what has been found to be the best.
Jacob Bernoulli
The false probability
A coin toss is an example of the law of large numbers. Assuming that the chance of a heads or tails result is equal, the law dictates that after many tosses, half (or very near it) will have landed on heads, and half on tails. However, in the early stages, heads and tails are likely to be more unbalanced. For example, the first 10 tosses could be seven heads and three tails. It might then seem most likely that the next toss will produce a tail. That, however, is the “gambler’s fallacy”—where a person assumes that the outcomes of each game (toss) are connected. A gambler might assume that toss number 11 is likely to be a tail because the number of heads and tails must balance out, but the probability of heads or tails is the same in every toss, and the outcome of one toss occurs independently of any other. This is the starting point of all probability theory. After 1,000 tosses, the imbalance apparent in those first 10 tosses becomes negligible.
When a referee flips a coin, there is no advantage, according to the law of large numbers, in a team captain basing a heads or tails choice on what has been called in previous games.
JACOB BERNOULLI
Born in Basel, Switzerland, in 1655, Jacob Bernoulli studied theology, but developed an interest in mathematics. In 1687, he became a professor of mathematics at the University of Basel, a position he held for the rest of his life.
In addition to his work on probability, Bernoulli is remembered for discovering the mathematical constant e by calculating the growth of funds that received compound interest continuously in infinitesimal increments. He was also involved in the development of calculus, taking the side of Gottfried Leibniz against Isaac Newton in their rival claims to have invented a new mathematical field. Bernoulli worked on calculus with his younger brother Johann. However, Johann became jealous of his brother’s achievements and their relationship broke down several years before Jacob died in 1705.
Key works
1713 Ars Conjectandi (The Art of Conjecturing)
1744 Opera (Collected Works)
See also: Probability • Normal distribution • Bayes’ theorem • The Poisson distribution • The birth of modern statistics
IN CONTEXT
KEY FIGURE
Leonhard Euler (1707–83)
FIELD
Number theory
BEFORE
1618 Logarithms calculated from the number now known as e are listed in an appendix to a book on logarithms by John Napier.
1683 Jacob Bernoulli uses e in his work on compound interest.
1733 Abraham de Moivre discovers “normal distribution”: the way that values for most data cluster at a central point and taper off at the extremes. Its equation involves e.
AFTER
1815 Joseph Fourier’s proof that e is irrational is published.
1873 French mathematician Charles Hermite proves that e is transcendental.
The mathematical constant that became known as e, or Euler’s number—2.718… to an infinite number of decimal places—first appeared in the early 1600s, when logarithms were invented to help simplify complex calculations. Scottish mathematician John Napier compiled tables of logarithms to base 2.718…, which worked particularly well for calculations involving exponential growth. These were later dubbed “natural logarithms” because they can be used to mathematically describe many processes in nature, but with algebraic notation still in its infancy, Napier saw logarithms only as an aid to calculation involving the ratio of distances covered by moving points.
In the late 1600s, Swiss mathematician Jacob Bernoulli used 2.718… to calculate compound interest, but it was Leonhard Euler, a student of Bernoulli’s brother Johann, who first called the number e. Euler calculated e to 18 decimal places, writing his first work on e, the Meditatio (Meditation), in 1727. However, it was not published until 1862. Euler explored e further in his 1748 Introductio (Introduction).
LEONHARD EULER
Born in 1707, in Basel, Switzerland, Euler grew up in nearby Riehen. Taught initially by his father, a Protestant minister who had some mathematical training and was also a friend of the Bernoulli family, Euler developed a passion for mathematics. Although he entered university to study for the ministry, he switched to mathematics with the support of Johann Bernoulli. Euler went on to work in Switzerland and Russia, and became the most prolific mathematician of all time, contributing greatly to calculus, geometry, and trigonometry, among other fields. This was despite steadily losing his sight from 1738 and becoming blind in 1771. Working to the very end, he died in 1783 in St. Petersburg.
Key works
1748 Introductio in analysin infinitorium (Introduction to Analysis of the Infinite)
1862 Meditatio in experimenta explosione tormentorum nuper instituta (Meditation upon experiments made recently on the firing of Cannon)
Com
pound interest
One of the earliest appearances of e was in calculating compound interest—where the interest on a savings account, for example, is paid into the account to increase the amount saved, rather than being paid out to the investor. If the interest is calculated on a yearly basis, an investment of $100 at an interest rate of 3% per year would produce $100 × 1.03 = $103 after one year. After two years, it would be 100 × 1.03 × 1.03 = $106.09, and after 10 years it would be $100 × 1.0310 = $134.39. The formula for this is A = P (1 + r)t, where A is the final amount, P is the original investment (principal), r is the interest rate (as a decimal), and t is the number of years.
If interest is calculated more often than annually, the calculation changes. For example, if interest is calculated monthly, the monthly rate is 1⁄12 of the yearly rate. 3 ÷ 12 = 0.25, so the investment after a year would be $100 × 1.002512 = $103.04. If interest is calculated daily, the rate is 3 ÷ 365 = 0.008… and the amount after one year is $100 × 1.00008…365 = $103.05. The formula for this is A = P(1 + r⁄n)nt, where n is the number of times the interest is calculated in each year. As the time intervals at which interest is calculated get smaller, the amount of interest yielded at the end of a year approaches A = Per. Bernoulli came close to working this out in his calculations, when he identified e as the limit of (1 + 1⁄n)n as n approaches infinity (n → ∞). The formula (1 + 1⁄n)n gives closer values for e as n increases. For example, n = 1 gives a value for e of 2, n = 10 gives a value for e of 2.5937… and n = 100 gives a value for e of 2.7048….
When Euler calculated a value for e correct to 18 decimal places, he probably used the sequence e = 1 + 1 + 1⁄2 + 1⁄6 + 1⁄24 + 1⁄120 + 1⁄720, going up to 20 terms. He arrived at these denominators by using the factorial for each integer. The factorial of an integer is the product of the integer and all the integers below it: 2 (2 × 1), 3 (3 × 2 × 1), 4 (4 × 3 × 2 × 1), 5 (5 × 4 × 3 × 2 × 1) and so on, adding one more term in the product each time. This can be shown as e = 1 + 1 + 1⁄2! + 1⁄3! +1⁄4! in factorial notation.
Euler calculated e to 18 decimal places, but noted that the decimals continued indefinitely. This means that e is irrational. In 1873, French mathematician Charles Hermite proved that e is also non-algebraic—it is not a number with a terminating decimal that can be used in a regular polynomial equation. This makes it a “transcendental” number—a real number that cannot be computed by solving an equation.
Compounding interest yields a bigger total sum. The examples below show how a $10 principal investment accrues interest if the yearly interest rate is 100 percent, versus compound interest paid at shorter intervals.
The exponential function can be used to calculate compound interest. The function produces the curve y = ex, which cuts the y axis at (0,1), and gets exponentially steeper. This graph also shows the tangent to the curve.
The growth curve
Compound interest is an example of exponential growth. Such growth can be plotted on a graph and will appear as a curve. In the 1600s, English cleric Thomas Malthus posited that population also increases exponentially if there are no checks on its growth, such as war, famine, or food shortages. This means that the population continues to grow at the same rate, leading to ever-larger totals. Constant population growth can be calculated with the formula P = P0ert where P0 is the original population number, r is the growth rate, and t is time.
Plotted on a graph, e shows other special properties. The graph of y = ex (the exponential function) is a curve whose tangent (the straight line that touches but does not intersect the curve) at the coordinates (0,1) also has a gradient (steepness) of precisely 1. This is because the derivative (rate of change) of ex is, in fact, ex, and the derivative is used to find the tangent. The tangent is used to calculate the rate of change at a specific point on a curve. Because the derivative is ex, the slope (a measure of direction and steepness) of the tangent line will always be the same as the y value.
For the sake of brevity, we will always represent this number, 2.718281828… by the letter e.
Leonhard Euler
Derangements
The various ways in which a set of items can be ordered are called permutations. For example, the set 1, 2, 3 can be arranged as 1, 3, 2, or 2, 1, 3, or 2, 3, 1, or 3, 1, 2, or 3, 2, 1. There are six total ways, including the original, as the number of permutations in a set is equal to the factorial of the highest integer, in this case 3! (short for 3 × 2 × 1). Euler’s number is also significant in a type of permutation called a derangement. In a derangement, none of the items can remain in their original position. For four items, the number of possible permutations is 24, but to find the derangements of 1, 2, 3, 4, all other arrangements beginning with 1 must first be eliminated. There are three derangements starting with 2: 2, 1, 4, 3; 2, 3, 4, 1; and 2, 4, 1, 3. There are also three derangements starting with 3 and three starting with 4, making nine in total. With five items, the total number of permutations is 120, and with six it is 720, making the task of finding all derangements a substantial one.
Euler’s number makes it possible to calculate the number of derangements in any set. This number equals the number of permutations divided by e, rounded to the nearest whole number. For example, for the set of 1, 2, 3, where there are six permutations, 6 ÷ e = 2.207… or 2, to the nearest whole number. Euler analyzed derangements of 10 numbers for Frederick the Great of Prussia, who hoped to create a lottery to pay off his debts. For 10 numbers, Euler found that the probability of getting a derangement is 1⁄e to an accuracy of six decimal places.
[Frederick the Great is] always at war; in summer with the Austrians, in winter with mathematicians.
Jean le Rond d’Alembert
French mathematician
Other uses
Euler’s number is relevant in many other calculations—for example, in splitting up (partitioning) a number to discover which numbers in the partition have the largest product. With the number 10, partitions include 3 and 7, with a product of 21; or 6 and 4 to produce 24; or 5 and 5 to give 25, which is the maximum product for a partition of 10 using two numbers. With three numbers, 3, 3, 4 has a product of 36, but moving into fractional numbers, 31⁄3 × 31⁄3 × 31⁄3 = 1000⁄27 = 37.037… the largest for three numbers. For a four-way partition, 21⁄2 × 21⁄2 × 21⁄2 × 21⁄2 = 39.0625, but in a five-way split, 2 × 2 × 2 × 2 × 2 = 32. In short, (10⁄2)2 = 25, (10⁄3)3 = 37.037..., (10⁄4)4 = 39.0625, and (10⁄5)5 = 32. This smaller result for a five-way. partition suggests that the optimal number of splits for 10 is between 3 and 4. Euler’s number can help to find both the maximum product, as e(10⁄e) = 39.598…, and number of partitions: 10⁄e = 3.678….
To carbon-date organic material, researchers test a sample—here from an ancient human bone—and use Euler’s number to calculate its age from the rate of radioactive decay.
The catenary
The Gateway Arch in St. Louis, Missouri is a flattened catenary arch, designed by Finnish-American architect Eero Saarinen in 1947.
Sometimes defined as the shape a hanging chain takes if it is only supported at its ends, a catenary is a curve with the formula y = 1⁄2 × (ex + e-x). Catenaries are often found in nature and in technology. For example, a square sail under pressure from the wind takes the form of a catenary. Arches in the shape of an inverted catenary are often used in architecture and construction due to their strength.
For a long time, the catenary’s shape was believed to be the same as that of a parabola. Dutch mathematician Christiaan Huygens—who coined the name catenary from the Latin catena (“chain”) in 1690—showed that, unlike a parabola, a catenary curve could not be given by a polynomial equation. Three mathematicians—Huygens, Gottfried Leibniz, and Johann Bernoulli—calculated a formula for the catenary, coming to the same conclusion. Their results were published together in 1691. In 1744, Euler described a catenoid—shaped like a waisted cylinder and produced by rotating a catenary around an axis.
See also: Positional numbers • Irrational numbers
• Calculating pi • Decimals • Logarithms • Probability • The law of large numbers • Euler’s identity
IN CONTEXT
KEY FIGURES
Abraham de Moivre (1667–1754), Carl Friedrich Gauss (1777–1855)
FIELDS
Statistics, probability
BEFORE
1710 British physician John Arbuthnot publishes a statistical proof of divine providence in relation to the number of men and women in a population.
AFTER
1920 Karl Pearson, a British statistician, expresses regret about describing the Gaussian curve as the “normal curve” because it gives the impression that all other probability distributions were “abnormal.”
1922 In the US, the New York Stock Exchange introduces the use of normal distribution to model the risks of investments.
In the 18th century, French mathematician Abraham de Moivre made an important step forward in statistics; building on Jacob Bernoulli’s discovery of binomial distribution, de Moivre showed that events cluster around the mean (b on graph below). This phenomenon is known as normal distribution.
Binomial distribution (used to describe outcomes based on one of two possibilities) was first shown by Bernoulli in Ars Conjectandi (The Art of Conjecturing), published in 1743. When a coin is flipped, there are two possible outcomes: “success” and “failure.” This type of test, with two equally likely outcomes, is called a Bernoulli trial. Binomial probabilities arise when a fixed number, n, of such Bernoulli trials, each with the same success probability, p, are carried out and the total number of successes is counted. The resulting distribution is written as b(n, p). Binomial distribution b(n, p) can take values from 0 to n, centered on a mean of np.