by DK
1202 Leonardo of Pisa’s Liber Abaci (The Book of Calculation) brings the Arabic number system to Europe.
AFTER
1799 The metric system is introduced for French currency and measures during the French Revolution.
1971 Britain introduces decimalization, dispensing with pounds, shillings, and pence, which stemmed from the Latin system.
Fractions—so named for the Latin word fractio, meaning “break”—were used from around 1800 BCE in Egypt to express parts of a whole. At first they were limited to unit fractions, which are those with a 1 as the numerator (top number). The ancient Egyptians had symbols for 2⁄3 and 3⁄4, but other fractions were expressed as the sum of unit fractions, for example as 1⁄3 + 1⁄13 + 1⁄17. This system worked well for recording amounts but not for doing calculations. It was not until after Simon Stevin’s De Thiende (The Art of Tenths) was published in 1585 that a decimal system became commonplace.
By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems.
Alfred North Whitehead
British mathematician
The importance of 10
Simon Stevin, a Flemish engineer and mathematician in the late 16th and early 17th century, used many calculations in his work. He simplified these by using fractions with a base system of tenth powers. Stevin correctly predicted that a decimal system would eventually be universal.
Cultures throughout history had used many different bases for expressing parts of a whole. In ancient Rome, fractions were based on a system of twelfths, and written out in words: 1⁄12 was called uncia, 6⁄12 was semis, and 1⁄24 was semiuncia, but this cumbersome system made it difficult for people to do any calculations. In Babylon, fractions were expressed using their base-60 number system, but in writing, it was difficult to distinguish which numbers represented integers and which were part of the whole.
For many centuries, Europeans used Roman numerals to record numbers and to do calculations. Medieval Italian mathematician Leonardo of Pisa (also known as Fibonacci) came across the Indian place-value number system while he was traveling in the Arab world. He quickly realized its usefulness and efficiency for both recording and calculating with whole numbers. His Liber Abaci (1202), which brought many useful Arabic ideas to the west, also introduced a new notation for fractions to Europe that would form the basis of the notation used today. Fibonacci employed a horizontal bar to divide the numerator and denominator (bottom number), but followed the Arabic practice of writing the fraction to the left of the integer, rather than to the right.
SIMON STEVIN
Born in 1548 in Bruges, now in Belgium, Simon Stevin worked as a bookkeeper, cashier, and clerk before entering the University of Leiden in 1583. There he met Prince Maurice, the heir of William of Orange, and they became friends. Stevin tutored the prince in mathematics and also advised him on military strategy, leading to some significant victories over the Spanish. In 1600, Prince Maurice asked Stevin, who was also an outstanding engineer, to found a School of Engineering at the University in 1600. As quarter-master general from 1604, Stevin was responsible for several innovative military and engineering ideas that were adopted across Europe. He authored many books on a variety of subjects, including mathematics. He died in 1620.
Key works
1583 Problemata geometrica (Geometric Problems)
1585 De Thiende (The Art of Tenths)
1585 De Beghinselen der Weeghconst (Principles of the Art of Weighing)
Introducing decimals
Finding that conventional fractions were both time-consuming and prone to errors, Stevin began using a decimal system. The idea of “decimal fractions”—which have powers of 10 as the denominator—had been used five centuries before Stevin, in the Middle East, but it was Stevin who made decimals commonplace in Europe, both for recording and calculating with parts of a whole. He suggested a notation system for decimal fractions, replicating the advantages of the Indian place-value system for whole numbers.
In Stevin’s new notation, numbers that would previously have been written as the sum of fractions—for example, 32 + 5⁄10 + 6⁄100 + 7⁄1,000—could now be written as a single number. Stevin placed circles after each number; these were shorthand for the denominator of the original decimal fraction. The whole 32 would be followed by a 0, because 32 is an integer, whereas the 6⁄100, for example, was expressed as 6 and a 2 inside a circle. This 2 denoted the power of 10 of the original denominator, as 100 is 102. In the same vein, the 7⁄1,000 became a 7 followed by a 3 inside a circle. The entire sum could be written out following this pattern. The symbol that is placed between the whole-number part and the fractional part of a number is called the decimal separator. Stevin’s zero inside a circle later evolved into a dot, now called the decimal point. The dot was positioned on the midline (at a middle height) in Stevin’s notation but has now moved to be on the baseline to avoid confusion with the dot notation sometimes used for multiplication. Stevin’s circled numbers for tenth powers were also done away with, meaning that 32 + 5⁄10 + 6⁄100 + 7⁄1,000 could now be written as 32.567.
Decimals [are] a kind of arithmetic invented by the tenth progression, consisting in characters of cyphers.
Simon Stevin
Stevin’s notation used circles to indicate the power of ten of the denominator of the converted fraction. This represents how Stevin would have written the number now expressed as 32.567.
The decimal system makes it easier to divide and multiply fractions, especially by 10. Shown here with the example of 32.567 (or 32 + 5⁄10 + 6⁄100 + 7⁄1,000), numbers shift one column to the left or right, crossing over the decimal separator.
Different systems
The decimal point has never become universally accepted. Many countries use a comma as the decimal separator instead of a point. There would be no problem with the two common notations if not for the use of delimiters—symbols that separate groups of three digits in the whole-number section of a very large or sometimes very small number. For example, in the UK, the commas in the number 2,500,000 are delimiters and are used to make it easier both to read the number and to recognize its size. The UK uses a point for the decimal separator and a comma as a delimiter. Elsewhere in the world, if a comma is used for the decimal separator, a point is then used as the delimiter. In Vietnam, for example, a price of two hundred thousand Vietnamese dong is often written as 200.000.
Usually, the context is sufficient for people to interpret the notation correctly, but this can go badly wrong. In an attempt to solve this problem, the 22nd General Conference on weights and measures—a meeting of delegates from 60 nations of the International Bureau of Weights and Measures—decided in 2003 that, although either a point or comma on the line could be used as the decimal separator, the delimiter was to be a space rather than either of the previous symbols. This notation is yet to become universal.
In Spain, the decimal separator is a comma, as seen in the prices at this market stall in Catalonia. In handwritten Spanish, an upper comma (similar to an apostrophe) is also common.
Benefits of decimals
The same processes of addition, subtraction, multiplication, and division of whole numbers can be used with decimal numbers, resulting in a far simpler way of performing basic arithmetic than the previous method, which relied on learning a different set of rules for calculations with fractions. When multiplying fractions, for example, the numerators would be multiplied separately from the denominators, and the resulting fraction would then be reduced. With decimal fractions, multiplying and dividing by powers of 10 is straightforward—as in the example of 32.567, the decimal separator can be simply moved left or right.
Stevin believed that the universal introduction of decimal coinage, weights, and measures would only be a matter of time. The introduction of decimal measures for length and weight (using meters and kilograms) arrived in Europe some 200 years later, during the French Revolution. When it introduced the metric
system, France also tried to introduce a decimal system for time; there would be 10 hours in a day, 100 minutes in each hour, and 100 seconds in each minute. The attempt was so unpopular that it was dropped after just one year. The Chinese had introduced various forms of decimal time over some 3,000 years, but finally abandoned it in 1645 CE.
In the US, the use of a decimal system for measurement and coinage was championed by Thomas Jefferson. His 1784 paper persuaded Congress to introduce a decimal system for money using dollars, dimes, and cents. In fact, the name “dime” originates from Disme, the French title of The Art of Tenths. Yet Jefferson’s view did not hold sway for measurement, and inches, feet, and yards are still used today. While many European currencies were decimalized in the 1800s, it was not until 1971 that decimal currency was introduced in the UK.
This marble plaque on the rue de Vaugirard, Paris, is one of 16 original meter markers installed in 1791, after the French Académie des Sciences defined the meter for the first time.
Perhaps the most important event in the history of science… [is] the invention of the decimal system…
Henri Lebesgue
French mathematician
Terminating and recurring decimals
Fractions are converted to decimals by dividing the numerator by the denominator. If the denominator is only divisible by 2 or 5 and no other prime numbers–as is the case for 10—then the decimal will terminate. For example, 3⁄40 can be expressed as 0.075, and this value is exact because 40 is only divisible by the primes 2 and 5.
Other fractions become recurring decimals—meaning that they do not end. For example, 2⁄11 is decimalized as 0.18181818…, denoted as to show that both the 1 and 8 recur. The length of the recurring cycle (two numbers in the case of ) can be predicted as it will be a factor of the denominator minus 1 (so if the denominator of the fraction is 11, the number of digits in the cycle is a factor of 10). These differ from irrational numbers, which do not terminate and have no pattern of recurrence. Irrational numbers cannot be expressed as a fraction of two integers.
See also: Positional numbers • Irrational numbers • Negative numbers • The Fibonacci sequence • Binary numbers
IN CONTEXT
KEY FIGURE
John Napier (1550–1617)
FIELD
Number systems
BEFORE
14th century The Indian mathematician Madhava of Kerala constructs an accurate table of trigonometric sines to aid calculation of angles in right-angled triangles.
1484 In France, Nicolas Chuquet writes an article about calculation using geometric series.
AFTER
1622 English mathematician and clergyman William Oughtred invents the slide rule using logarithmic scales.
1668 In Logarithmo-technia, German mathematician Nicholas Mercator first uses the term “natural logarithms.”
For thousands of years, most calculations were carried out by hand, using devices such as counting boards or the abacus. Multiplication was especially long-winded and much more difficult than addition. In the scientific revolution of the 16th and 17th centuries, the lack of a reliable calculating tool hampered progress in areas such as navigation and astronomy, where the potential for error was greater because of the lengthy calculations involved.
Solving by series
In the 1400s, French mathematician Nicolas Chuquet investigated how the relationships between arithmetic and geometric sequences could aid calculation. In an arithmetic sequence, each number differs from the one preceding it by a constant quantity, such as 1, 2, 3, 4, 5, 6… (going up by 1), or 3, 6, 9, 12… (going up by 3). In a geometric sequence, each number after the first term is determined by multiplying the previous number by a fixed amount, called the “common ratio.” For example, the sequence 1, 2, 4, 8, 16 has a common ratio of 2. Setting down a geometric sequence (such as 1, 2, 4, 8…) and above it an arithmetic sequence (such as 1, 2, 3, 4…), it can be seen that the top numbers are the exponents to which 2 is raised to arrive at the series below. It was a much more sophisticated version of this scheme that lay at the heart of the tables of logarithms developed by Scottish landowner John Napier.
Generating logarithms
Napier was fascinated by numbers and spent much of his time finding ways of making calculations easier. In 1614, he published the first description and table of logarithms; a logarithm of a given number is the exponent or power to which another fixed number (the base) is raised to produce that given number. The use of such tables facilitated complex calculations and advanced the development of trigonometry.
As Napier recognized, the basic principle of calculating was simple enough: he could replace the tedious task of multiplication by the simpler operation of addition. Each number would have its equivalent “artificial number” as he initially termed it. (Napier later settled on the name “logarithm,” derived by combining the Greek words logos, meaning proportion, and arithmos, meaning number.) Adding the two logarithms, and then converting the answer back to an ordinary number, produces the result of multiplying the original numbers. For division, one logarithm is subtracted from another and the result is converted back.
To generate his logarithms, Napier imagined two particles traveling along two parallel lines. The first line was of infinite length, while the second was of fixed length. Each particle left the same starting position at the same time and at the same velocity. The particle on the infinite line traveled with uniform motion, so it covered equal distances in equal times. The velocity of the second particle was proportional to the distance remaining to the end of the line. Halfway between the starting point and the end of the line, the second particle is traveling at half the velocity it started with; at the three-quarter point, it is traveling with a quarter of its initial velocity; and so on. This means that the second particle is never going to reach the end of the line, and equally, the first particle, on its infinite line, will never arrive at the end of its journey. At any instant there is a unique correspondence between the positions of the two particles. The distance the first particle has traveled is the logarithm of the distance the second particle has yet to go. The first particle’s progress can be viewed as an arithmetic progression, while that of the second particle is geometric.
The lower row of this table is a geometric sequence (progressing powers of 2), while the top row is an arithmetic sequence that reveals the exponents (powers) by which 2 is raised to arrive at the numbers in the lower row. (Anything to the power of 0 is 1.) To multiply the numbers 16 and 32 in the lower row, their exponents (4 + 5) can be added together to produce 2 9 (= 512).
JOHN NAPIER
Born into a wealthy family in 1550 at Merchiston Castle, near Edinburgh, John Napier would later become 8th Laird of Merchiston. Aged just 13, he entered St. Andrews University and became passionately interested in theology. Before graduating, however, he left to study in Europe, although few details of this time are known.
Napier returned to Scotland in 1571 and devoted much time to his estates, where he devised new methods of agriculture to improve his land and livestock. A fervent Protestant, he also wrote a prominent book attacking Catholicism. His keen interest in astronomy, and a desire to find simpler ways to perform the calculations that it required, led to his invention of logarithms. He also created Napier’s Bones, a calculation device using numbered rods. Napier died at Merchiston Castle in 1617.
Key works
1614 Mirifici Logarithmorum Canonis Descriptio (A Description of the Marvellous Rule of Logarithms)
1617 Rabdologiae
Improving the method
It took Napier 20 years to complete his calculations and to publish his first logarithm tables as Mirifici Logarithmorum Canonis Descriptio (A Description of the Marvellous Rule of Logarithms). Henry Briggs, professor of mathematics at the University of Oxford, recognized the significance of Napier’s tables but thought they were unwieldy.
Briggs visited Napier in 1616 and again in 1617. Following their discussions, the two
agreed that the logarithm of 1 should be redefined as 0 and the logarithm of 10 as 1. This approach made logarithms much easier to use. Briggs also helped with the calculation of logarithms of ordinary numbers based on the logarithm of 10 being 1 and spent several years recalculating the tables. The results were published in 1624 with the logarithms calculated to 14 decimal places. The base-10 logarithms calculated by Briggs are known as log10 or common logarithms. The earlier table to the power of 2 (see Generating logarithms) can be thought of as a simple base-2, or log2 table.
I found at length some excellent brief rules.
John Napier
The impact of logarithms
Logarithms had an immediate impact on science, and on astronomy in particular. German astronomer Johannes Kepler had published his first two laws of planetary motion in 1605, but only after the invention of log tables was he able to make the breakthrough to discover his third law. This describes how the time it takes for a planet to complete one orbit of the Sun is related to its average orbital distance. When he published this finding in 1620 in his book Ephemerides novae motuum coelestium, Kepler dedicated it to Napier.
Napier’s book describing logarithms was published in 1614, as its title page shows. The principles behind his logarithm tables were published in 1619, two years after his death.
