MATHEMATICS NOTES FOR FORM FOUR – ALL TOPICS

TOPIC 1 – COORDINATE GEOMETRY


TOPIC 2 – AREA AND PERIMETER


TOPIC 3 – THREE DIMENSIONAL FIGURES


TOPIC 4 – PROBABILITY


TOPIC 5 – TRIGONOMETRY


TOPIC 6 – VECTORS


TOPIC 7 – MATRICES AND TRANSFORMATIONS


TOPIC 8 – LINEAR PROGRAMMING

IMPORTANCE OF MATHEMATICS IN OUR DAILY LIFE – PART 4 

13. Math Helps You with Your Finances

Math is also helpful with your finance.
With the help of math, you can easily make your financial budget. You can
calculate how much money you have and how you can spend your money. Almost
every single human in the world uses math for their finance. The salaried
person uses math to calculate their expenses and salaries.

On the other hand, businessmen use math to
calculate their profits and loss. They also use it to calculate their loans and
many more. It highlights the importance of business mathematics and also plays
a crucial role in business accounting.

14. Math develops flexible thinking and
creativity

Practicing math has been shown to
improve investigative skills, resourcefulness and creativity. This is
because math problems often require us to bend our thinking and approach
problems in more than one way. The first process we try might not work. We need
flexibility and creativity to think of new pathways to the solution. And just
like anything else, this way of thinking is strengthened with practice.

15. Math Helps in Budgeting Our Life

Daily, we must pay various expenses to get
our essentials to survive. Without budgeting, we might become extravagant
and spend entire pocket money on useless matters. Therefore, it becomes
necessary to make a monthly budget to save money, and, of course, it is
possible only with the help of math. We can calculate things and estimate them
reasonably if we know math formulas. Hence math is essential to escape from
spending excessive money and save it for the future.

16. Helps in Effective Decision Making

Math is the only subject that makes
students confident to face the world. Math contains problems, and the correct
equation solves these problems. Students have to decide which formula is
accurate to solve the problems. This approach makes the students better
decision-makers by adopting the right decision at the right time. In addition,
this practice develops intellectual ability and a better sense of humour in the
students.

…………..

Omar Khayyam

The mathematician and poet Omar Khayyam was born in Neyshābūr (in Iran) only a few years before al-Bīrūnī’s death. He later lived in Samarkand and Eṣfahān, and his brilliant work there continued many of the main lines of development in 10th-century mathematics. Not only did he discover a general method of extracting roots of arbitrary high degree, but his Algebra contains the first complete treatment of the solution of cubic equations. Omar did this by means of conic sections, but he declared his hope that his successors would succeed where he had failed in finding an algebraic formula for the roots.

quadrilateral of Omar Khayyam Omar Khayyam constructed the quadrilateral shown in the figure in an effort to prove that Euclid’s fifth postulate, concerning parallel lines, is superfluous. He began by constructing line segments AD and BC of equal length perpendicular to the line segment AB. Omar recognized that if he could prove that the internal angles at the top of the quadrilateral, formed by connecting C and D, are right angles, then he would have proved that DC is parallel to AB. Although Omar showed that the internal angles at the top are equal (as shown by the proof demonstrated in the figure), he could not prove that they are right angles.

Omar was also a part of an Islamic tradition, which included Thābit and Ibn al-Haytham, of investigating Euclid’s parallel postulate. To this tradition Omar contributed the idea of a quadrilateral with two congruent sides perpendicular to the base, as shown in the figure. The parallel postulate would be proved, Omar recognized, if he could show that the remaining two angles were right angles. In this he failed, but his question about the quadrilateral became the standard way of discussing the parallel postulate.

That postulate, however, was only one of the questions on the foundations of mathematics that interested Islamic scientists. Another was the definition of ratios. Omar Khayyam, along with others before him, felt that the theory in Book V of Euclid’s Elements was logically satisfactory but intuitively unappealing, so he proved that a definition known to Aristotle was equivalent to that given in Euclid. In fact, Omar argued that ratios should be regarded as “ideal numbers,” and so he conceived of a much broader system of numbers than that used since Greek antiquity, that of the positive real numbers.

Islamic mathematics to the 15th century

In the 12th century the physician al-Samawʿal continued and completed the work of al-Karajī in algebra and also provided a systematic treatment of decimal fractions as a means of approximating irrational quantities. In his method of finding roots of pure equations, xn = N, he used what is now known as Horner’s method to expand the binomial (a + y)n. His contemporary Sharaf al-Dīn al-Ṭūsī late in the 12th century provided a method of approximating the positive roots of arbitrary equations, based on an approach virtually identical to that discovered by François Viète in 16th-century France. The important step here was less the general idea than the development of the numerical algorithms necessary to effect it.

Sharaf al-Dīn was the discoverer of a device, called the linear astrolabe, that places him in another important Islamic mathematical tradition, one that centred on the design of new forms of the ancient astronomical instrument known as the astrolabe. The astrolabe, whose mathematical theory is based on the stereographic projection of the sphere, was invented in late antiquity, but its extensive development in Islam made it the pocket watch of the medievals. In its original form it required a different plate of horizon coordinates for each latitude, but in the 11th century the Spanish Muslim astronomer al-Zarqallu invented a single plate that worked for all latitudes. Slightly earlier, astronomers in the East had experimented with plane projections of the sphere, and al-Bīrūnī invented such a projection that could be used to produce a map of a hemisphere. The culminating masterpiece was the astrolabe of the Syrian Ibn al-Shāṭir (1305–75), a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways.

On the other hand, Muslim astronomers had developed other methods for solving these problems using the highly accurate trigonometry tables and the new trigonometry theorems they had developed. Out of these developments came the creation of trigonometry as a mathematical discipline, separate from its astronomical applications, by Naṣīr al-Dīn al-Ṭūsī at his observatory in Marāgheh in the 13th century. (It was there too that al-Ṭūsī’s pupil Quṭb al-Dīn al-Shīrāzī [1236–1311] and his pupil Kamāl al-Dīn Fārisī, using Ibn al-Haytham’s great work, the Optics, were able to give the first mathematically satisfactory explanation of the rainbow.)

Al-Ṭūsī’s observatory was supported by a grandson of Genghis Khan, Hülegü, who sacked Baghdad in 1258. Ulūgh Beg, the grandson of the Mongol conqueror Timur, founded an observatory at Samarkand in the early years of the 15th century. Ulūgh Beg was himself a good astronomer, and his tables of sines and tangents for every minute of arc (accurate to five sexagesimal places) were one of the great achievements in numerical mathematics up to his time. He was also the patron of Jamshīd al-Kāshī (died 1429), whose work The Reckoners’ Key summarizes most of the arithmetic of his time and includes sections on algebra and practical geometry as well. Among al-Kāshī’s works is a masterful computation of the value of 2π, which, when expressed in decimal fractions, is accurate to 16 places, as well as the application of a numerical method, now known as fixed-point iteration, for solving the cubic equation with sin 1° as a root. His work was indeed of a quality deserving Ulūgh Beg’s description as “known among the famous of the world.”

Al-Kāshī lived almost five centuries after the first translations of Arabic material into Latin, and by his time the Islamic mathematical tradition had given the West not only its first versions of many of the Greek classics but also a complete set of algorithms for Hindu-Arabic arithmetic, plane and spherical trigonometry, and the powerful tool of algebra. Although mathematical inquiry continued in Islam in the centuries after al-Kāshī’s time, the mathematical centre of gravity was shifting to the West. That this was so is, of course, in no small measure due to what the Western mathematicians had learned from their Islamic predecessors during the preceding centuries.

John L. Berggren

European mathematics during the Middle Ages and Renaissance

Until the 11th century only a small part of the Greek mathematical corpus was known in the West. Because almost no one could read Greek, what little was available came from the poor texts written in Latin in the Roman Empire, together with the very few Latin translations of Greek works. Of these the most important were the treatises by Boethius, who about 500 ce made Latin redactions of a number of Greek scientific and logical writings. His Arithmetic, which was based on Nicomachus, was well known and was the means by which medieval scholars learned of Pythagorean number theory. Boethius and Cassiodorus provided the material for the part of the monastic education called the quadrivium: arithmetic, geometry, astronomy, and music theory. Together with the trivium (grammar, logic, rhetoric), these subjects formed the seven liberal arts, which were taught in the monasteries, cathedral schools, and, from the 12th century on, universities and which constituted the principal university instruction until modern times.

For monastic life it sufficed to know how to calculate with Roman numerals. The principal application of arithmetic was a method for determining the date of Easter, the computus, that was based on the lunar cycle of 19 solar years (i.e., 235 lunar revolutions) and the 28-year solar cycle. Between the time of Bede (died 735), when the system was fully developed, and about 1500, the computus was reduced to a series of verses that were learned by rote. Until the 12th century, geometry was largely concerned with approximate formulas for measuring areas and volumes in the tradition of the Roman surveyors. About 1000 ce the French scholar Gerbert of Aurillac, later Pope Sylvester II, introduced a type of abacus in which numbers were represented by stones bearing Arabic numerals. Such novelties were known to very few.

The transmission of Greek and Arabic learning

In the 11th century a new phase of mathematics began with the translations from Arabic. Scholars throughout Europe went to Toledo, Córdoba, and elsewhere in Spain to translate into Latin the accumulated learning of the Muslims. Along with philosophy, astronomy, astrology, and medicine, important mathematical achievements of the Greek, Indian, and Islamic civilizations became available in the West. Particularly important were Euclid’s Elements, the works of Archimedes, and al-Khwārizmī’s treatises on arithmetic and algebra. Western texts called algorismus (a Latin form of the name al-Khwārizmī) introduced the Hindu-Arabic numerals and applied them in calculations. Thus, modern numerals first came into use in universities and then became common among merchants and other laymen. It should be noted that, up to the 15th century, calculations were often performed with board and counters. Reckoning with Hindu-Arabic numerals was used by merchants at least from the time of Leonardo of Pisa (beginning of the 13th century), first in Italy and then in the trading cities of southern Germany and France, where maestri d’abbaco or Rechenmeister taught commercial arithmetic in the various vernaculars. Some schools were private, while others were run by the community.

The universities

Mathematics was studied from a theoretical standpoint in the universities. The Universities of Paris and Oxford, which were founded relatively early (c. 1200), were centres for mathematics and philosophy. Of particular importance in these universities were the Arabic-based versions of Euclid, of which there were at least four by the 12th century. Of the numerous redactions and compendia which were made, that of Johannes Campanus (c. 1250; first printed in 1482) was easily the most popular, serving as a textbook for many generations. Such redactions of the Elements were made to help students not only to understand Euclid’s textbook but also to handle other, particularly philosophical, questions suggested by passages in Aristotle. The ratio theory of the Elements provided a means of expressing the various relations of the quantities associated with moving bodies, relations that now would be expressed by formulas. Also in Euclid were to be found methods of analyzing infinity and continuity (paradoxically, because Euclid always avoided infinity).

Studies of such questions led not only to new results but also to a new approach to what is now called physics. Thomas Bradwardine, who was active in Merton College, Oxford, in the first half of the 14th century, was one of the first medieval scholars to ask whether the continuum can be divided infinitely or whether there are smallest parts (indivisibles). Among other topics, he compared different geometric shapes in terms of the multitude of points that were assumed to compose them, and from such an approach paradoxes were generated that were not to be solved for centuries. Another fertile question stemming from Euclid concerned the angle between a circle and a line tangent to it (called the horn angle): if this angle is not zero, a contradiction quickly ensues, but, if it is zero, then, by definition, there can be no angle. For the relation of force, resistance, and the speed of the body moved by this force, Bradwardine suggested an exponential law. Nicholas Oresme (died 1382) extended Bradwardine’s ideas to fractional exponents.

uniformly accelerated motionUniformly accelerated motion; s = speed, a = acceleration, t = time, and v = velocity.

Another question having to do with the quantification of qualities, the so-called latitude of forms, began to be discussed at about this time in Paris and in Merton College. Various Aristotelian qualities (e.g., heat, density, and velocity) were assigned an intensity and extension, which were sometimes represented by the height and bases (respectively) of a geometric figure. The area of the figure was then considered to represent the quantity of the quality. In the important case in which the quality is the motion of a body, the intensity its speed, and the extension its time, the area of the figure was taken to represent the distance covered by the body. Uniformly accelerated motion starting at zero velocity gives rise to a triangular figure (see the figure). It was proved by the Merton school that the quantity of motion in such a case is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion; in modern formulation, s = 1/2at2 (Merton rule). Discussions like this certainly influenced Galileo indirectly and may have influenced the founding of coordinate geometry in the 17th century. Another important development in the scholastic “calculations” was the summation of infinite series.

Basing his work on translated Greek sources, about 1464 the German mathematician and astronomer Regiomontanus wrote the first book (printed in 1533) in the West on plane and spherical trigonometry independent of astronomy. He also published tables of sines and tangents that were in constant use for more than two centuries.

The Renaissance

Italian artists and merchants influenced the mathematics of the late Middle Ages and the Renaissance in several ways. In the 15th century a group of Tuscan artists, including Filippo Brunelleschi, Leon Battista Alberti, and Leonardo da Vinci, incorporated linear perspective into their practice and teaching, about a century before the subject was formally treated by mathematicians. Italian maestri d’abbaco tried, albeit unsuccessfully, to solve nontrivial cubic equations. In fact, the first general solution was found by Scipione del Ferro at the beginning of the 16th century and rediscovered by Niccolò Tartaglia several years later. The solution was published by Gerolamo Cardano in his Ars magna (Ars Magna or the Rules of Algebra) in 1545, together with Lodovico Ferrari’s solution of the quartic equation.

By 1380 an algebraic symbolism had been developed in Italy in which letters were used for the unknown, for its square, and for constants. The symbols used today for the unknown (for example, x), the square root sign, and the signs + and − came into general use in southern Germany beginning about 1450. They were used by Regiomontanus and by Fridericus Gerhart and received an impetus about 1486 at the University of Leipzig from Johann Widman. The idea of distinguishing between known and unknown quantities in algebra was first consistently applied by François Viète, with vowels for unknown and consonants for known quantities. Viète found some relations between the coefficients of an equation and its roots. This was suggestive of the idea, explicitly stated by Albert Girard in 1629 and proved by Carl Friedrich Gauss in 1799, that an equation of degree n has n roots. Complex numbers, which are implicit in such ideas, were gradually accepted about the time of Rafael Bombelli (died 1572), who used them in connection with the cubic.

Apollonius’s Conics and the investigations of areas (quadratures) and of volumes (cubatures) by Archimedes formed part of the humanistic learning of the 16th century. These studies strongly influenced the later developments of analytic geometry, the infinitesimal calculus, and the theory of functions, subjects that were developed in the 17th century.

Menso Folkerts

Mathematics in the 17th and 18th centuries

The 17th century

The 17th century, the period of the scientific revolution, witnessed the consolidation of Copernican heliocentric astronomy and the establishment of inertial physics in the work of Johannes Kepler, Galileo, René Descartes, and Isaac Newton. This period was also one of intense activity and innovation in mathematics. Advances in numerical calculation, the development of symbolic algebra and analytic geometry, and the invention of the differential and integral calculus resulted in a major expansion of the subject areas of mathematics. By the end of the 17th century, a program of research based in analysis had replaced classical Greek geometry at the centre of advanced mathematics. In the next century this program would continue to develop in close association with physics, more particularly mechanics and theoretical astronomy. The extensive use of analytic methods, the incorporation of applied subjects, and the adoption of a pragmatic attitude to questions of logical rigour distinguished the new mathematics from traditional geometry.

Institutional background

Until the middle of the 17th century, mathematicians worked alone or in small groups, publishing their work in books or communicating with other researchers by letter. At a time when people were often slow to publish, “invisible colleges,” networks of scientists who corresponded privately, played an important role in coordinating and stimulating mathematical research. Marin Mersenne in Paris acted as a clearinghouse for new results, informing his many correspondents—including Pierre de Fermat, Descartes, Blaise Pascal, Gilles Personne de Roberval, and Galileo—of challenge problems and novel solutions. Later in the century John Collins, librarian of London’s Royal Society, performed a similar function among British mathematicians.

In 1660 the Royal Society of London was founded, to be followed in 1666 by the French Academy of Sciences, in 1700 by the Berlin Academy, and in 1724 by the St. Petersburg Academy. The official publications sponsored by the academies, as well as independent journals such as the Acta Eruditorum (founded in 1682), made possible the open and prompt communication of research findings. Although universities in the 17th century provided some support for mathematics, they became increasingly ineffective as state-supported academies assumed direction of advanced research.

Numerical calculation

The development of new methods of numerical calculation was a response to the increased practical demands of numerical computation, particularly in trigonometry, navigation, and astronomy. New ideas spread quickly across Europe and resulted by 1630 in a major revolution in numerical practice.

Simon Stevin of Holland, in his short pamphlet La Disme (1585), introduced decimal fractions to Europe and showed how to extend the principles of Hindu-Arabic arithmetic to calculation with these numbers. Stevin emphasized the utility of decimal arithmetic “for all accounts that are encountered in the affairs of men,” and he explained in an appendix how it could be applied to surveying, stereometry, astronomy, and mensuration. His idea was to extend the base-10 positional principle to numbers with fractional parts, with a corresponding extension of notation to cover these cases. In his system the number 237.578 was denotedin which the digits to the left of the zero are the integral part of the number. To the right of the zero are the digits of the fractional part, with each digit succeeded by a circled number that indicates the negative power to which 10 is raised. Stevin showed how the usual arithmetic of whole numbers could be extended to decimal fractions, using rules that determined the positioning of the negative powers of 10.

In addition to its practical utility, La Disme was significant for the way it undermined the dominant style of classical Greek geometry in theoretical mathematics. Stevin’s proposal required a rejection of the distinction in Euclidean geometry between magnitude, which is continuous, and number, which is a multitude of indivisible units. For Euclid, unity, or one, was a special sort of thing, not number but the origin, or principle, of number. The introduction of decimal fractions seemed to imply that the unit could be subdivided and that arbitrary continuous magnitude could be represented numerically; it implicitly supposed the concept of a general positive real number.

Tables of logarithms were first published in 1614 by the Scottish laird John Napier in his treatise Description of the Marvelous Canon of Logarithms. This work was followed (posthumously) five years later by another in which Napier set forth the principles used in the construction of his tables. The basic idea behind logarithms is that addition and subtraction are easier to perform than multiplication and division, which, as Napier observed, require a “tedious expenditure of time” and are subject to “slippery errors.” By the law of exponents, anam = an + m; that is, in the multiplication of numbers, the exponents are related additively. By correlating the geometric sequence of numbers a, a2, a3,…(a is called the base) and the arithmetic sequence 1, 2, 3,…and interpolating to fractional values, it is possible to reduce the problem of multiplication and division to one of addition and subtraction. To do this Napier chose a base that was very close to 1, differing from it by only 1/107. The resulting geometric sequence therefore yielded a dense set of values, suitable for constructing a table.

In his work of 1619 Napier presented an interesting kinematic model to generate the geometric and arithmetic sequences used in the construction of his tables. Assume two particles move along separate lines from given initial points. The particles begin moving at the same instant with the same velocity. The first particle continues to move with a speed that is decreasing, proportional at each instant to the distance remaining between it and some given fixed point on the line. The second particle moves with a constant speed equal to its initial velocity. Given any increment of time, the distances traveled by the first particle in successive increments form a geometrically decreasing sequence. The corresponding distances traveled by the second particle form an arithmetically increasing sequence. Napier was able to use this model to derive theorems yielding precise limits to approximate values in the two sequences.

Napier’s kinematic model indicated how skilled mathematicians had become by the early 17th century in analyzing nonuniform motion. Kinematic ideas, which appeared frequently in mathematics of the period, provided a clear and visualizable means for the generation of geometric magnitude. The conception of a curve traced by a particle moving through space later played a significant role in the development of the calculus.

Napier’s ideas were taken up and revised by the English mathematician Henry Briggs, the first Savilian Professor of Geometry at Oxford. In 1624 Briggs published an extensive table of common logarithms, or logarithms to the base 10. Because the base was no longer close to 1, the table could not be obtained as simply as Napier’s, and Briggs therefore devised techniques involving the calculus of finite differences to facilitate calculation of the entries. He also devised interpolation procedures of great computational efficiency to obtain intermediate values.

In Switzerland the instrument maker Joost Bürgi arrived at the idea for logarithms independently of Napier, although he did not publish his results until 1620. Four years later a table of logarithms prepared by Kepler appeared in Marburg. Both Bürgi and Kepler were astronomical observers, and Kepler included logarithmic tables in his famous Tabulae Rudolphinae (1627; “Rudolphine Tables”), astronomical tabulations of planetary motion derived by using the assumption of elliptical orbits about the Sun.

Analytic geometry

The invention of analytic geometry was, next to the differential and integral calculus, the most important mathematical development of the 17th century. Originating in the work of the French mathematicians Viète, Fermat, and Descartes, it had by the middle of the century established itself as a major program of mathematical research.

Two tendencies in contemporary mathematics stimulated the rise of analytic geometry. The first was an increased interest in curves, resulting in part from the recovery and Latin translation of the classical treatises of Apollonius, Archimedes, and Pappus, and in part from the increasing importance of curves in such applied fields as astronomy, mechanics, optics, and stereometry. The second was the emergence a century earlier of an established algebraic practice in the work of the Italian and German algebraists and its subsequent shaping by Viète into a powerful mathematical tool at the end of the century.

Viète was a prominent representative of the humanist movement in mathematics that set itself the project of restoring and furthering the achievements of the Classical Greek geometers. In his In artem analyticem isagoge (1591; “Introduction to the Analytic Arts”), Viète, as part of his program of rediscovering the method of analysis used by the ancient Greek mathematicians, proposed new algebraic methods that employed variables, constants, and equations, but he saw this as an advancement over the ancient method, a view he arrived at by comparing the geometric analysis contained in Book VII of Pappus’s Collection with the arithmetic analysis of Diophantus’s Arithmetica. Pappus had employed an analytic method for the discovery of theorems and the construction of problems; in analysis, by contrast to synthesis, one proceeds from what is sought until one arrives at something known. In approaching an arithmetic problem by laying down an equation among known and unknown magnitudes and then solving for the unknown, one was, Viète reasoned, following an “analytic” procedure.

Viète introduced the concept of algebraic variable, which he denoted using a capital vowel (A, E, I, O, U), as well as the concept of parameter (an unspecified constant quantity), denoted by a capital consonant (B, C, D, and so on). In his system the equation 5BA2 − 2CA + A3 = D would appear as B5 in A quad − C plano 2 in A + A cub aequatur D solido.

Viète retained the classical principle of homogeneity, according to which terms added together must all be of the same dimension. In the above equation, for example, each of the terms has the dimension of a solid or cube; thus, the constant C, which denotes a plane, is combined with A to form a quantity having the dimension of a solid.

It should be noted that in Viète’s scheme the symbol A is part of the expression for the object obtained by operating on the magnitude denoted by A. Thus, operations on the quantities denoted by the variables are reflected in the algebraic notation itself. This innovation, considered by historians of mathematics to be a major conceptual advance in algebra, facilitated the study of the symbolic solution of algebraic equations and led to the creation of the first conscious theory of equations.

After Viète’s death the analytic art was applied to the study of curves by his countrymen Fermat and Descartes. Both men were motivated by the same goal, to apply the new algebraic techniques to Apollonius’s theory of loci as preserved in Pappus’s Collection. The most celebrated of these problems consisted of finding the curve or locus traced by a point whose distances from several fixed lines satisfied a given relation.

Fermat adopted Viète’s notation in his paper “Ad Locos Planos et Solidos Isagoge” (1636; “Introduction to Plane and Solid Loci”). The title of the paper refers to the ancient classification of curves as plane (straight lines, circles), solid (ellipses, parabolas, and hyperbolas), or linear (curves defined kinematically or by a locus condition). Fermat considered an equation among two variables. One of the variables represented a line measured horizontally from a given initial point, while the other represented a second line positioned at the end of the first line and inclined at a fixed angle to the horizontal. As the first variable varied in magnitude, the second took on a value determined by the equation, and the endpoint of the second line traced out a curve in space. By means of this construction Fermat was able to formulate the fundamental principle of analytic geometry:

Whenever two unknown quantities are found in final equality, there results a locus fixed in place, and the endpoint of one of these unknown quantities describes a straight line or a curve.

The principle implied a correspondence between two different classes of mathematical objects: geometric curves and algebraic equations. In the paper of 1636 Fermat showed that, if the equation is a quadratic, then the curve is a conic section—that is, an ellipse, parabola, or hyperbola. He also showed that the determination of the curve given by an equation is simplified by a transformation involving a change of variables to an equation in standard form.

Descartes’s La Géométrie appeared in 1637 as an appendix to his famous Discourse on Method, the treatise that presented the foundation of his philosophical system. Although supposedly an example from mathematics of his rational method, La Géométrie was a technical treatise understandable independently of philosophy. It was destined to become one of the most influential books in the history of mathematics.

In the opening sections of La Géométrie, Descartes introduced two innovations. In place of Viète’s notation he initiated the modern practice of denoting variables by letters at the end of the alphabet (x, y, z) and parameters by letters at the beginning of the alphabet (a, b, c) and of using exponential notation to indicate powers of x (x2, x3,…). More significant conceptually, he set aside Viète’s principle of homogeneity, showing by means of a simple construction how to represent multiplication and division of lines by lines; thus, all magnitudes (lines, areas, and volumes) could be represented independently of their dimension in the same way.

Descartes’s goal in La Géométrie was to achieve the construction of solutions to geometric problems by means of instruments that were acceptable generalizations of ruler and compass. Algebra was a tool to be used in this program:

If, then, we wish to solve any problem, we first suppose the solution already effected, and give names to all the lines that seem necessary for its construction—to those that are unknown as well as to those that are known. Then, making no distinction in any way between known and unknown lines, we must unravel the difficulty in any way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other.

In the problem of Apollonius, for example, one sought to find the locus of points whose distances from a collection of fixed lines satisfied a given relation. One used this relation to derive an equation, and then, using a geometric procedure involving acceptable instruments of construction, one obtained points on the curve given by the roots of the equation.

Descartes described instruments more general than the compass for drawing “geometric” curves. He stipulated that the parts of the instrument be linked together so that the ratio of the motions of the parts could be knowable. This restriction excluded “mechanical” curves generated by kinematic processes. The Archimedean spiral, for example, was generated by a point moving on a line as the line rotated uniformly about the origin. The ratio of the circumference to the diameter did not permit exact determination:

the ratios between straight and curved lines are not known, and I even believe cannot be discovered by men, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact.

Descartes concluded that a geometric or nonmechanical curve was one whose equation f(x, y) = 0 was a polynomial of finite degree in two variables. He wished to restrict mathematics to the consideration of such curves.

Descartes’s emphasis on construction reflected his classical orientation. His conservatism with respect to what curves were acceptable in mathematics further distinguished him as a traditional thinker. At the time of his death, in 1650, he had been overtaken by events, as research moved away from questions of construction to problems of finding areas (then called problems of quadrature) and tangents. The geometric objects that were then of growing interest were precisely the mechanical curves that Descartes had wished to banish from mathematics.

Following the important results achieved in the 16th century by Gerolamo Cardano and the Italian algebraists, the theory of algebraic equations reached an impasse. The ideas needed to investigate equations of degree higher than four were slow to develop. The immediate historical influence of Viète, Fermat, and Descartes was to furnish algebraic methods for the investigation of curves. A vigorous school of research became established in Leiden around Frans van Schooten, a Dutch mathematician who edited and published in 1649 a Latin translation of La Géométrie. Van Schooten published a second two-volume translation of the same work in 1659–1661 that also contained mathematical appendixes by three of his disciples, Johan de Witt, Johan Hudde, and Hendrick van Heuraet. The Leiden group of mathematicians, which also included Christiaan Huygens, was in large part responsible for the rapid development of Cartesian geometry in the middle of the century.

The calculus

The historian Carl Boyer called the calculus “the most effective instrument for scientific investigation that mathematics has ever produced.” As the mathematics of variability and change, the calculus was the characteristic product of the scientific revolution. The subject was properly the invention of two mathematicians, the German Gottfried Wilhelm Leibniz and the Englishman Isaac Newton. Both men published their researches in the 1680s, Leibniz in 1684 in the recently founded journal Acta Eruditorum and Newton in 1687 in his great treatise, the Principia. Although a bitter dispute over priority developed later between followers of the two men, it is now clear that they each arrived at the calculus independently.

The calculus developed from techniques to solve two types of problems, the determination of areas and volumes and the calculation of tangents to curves. In classical geometry Archimedes had advanced farthest in this part of mathematics, having used the method of exhaustion to establish rigorously various results on areas and volumes and having derived for some curves (e.g., the spiral) significant results concerning tangents. In the early 17th century there was a sharp revival of interest in both classes of problems. The decades between 1610 and 1670, referred to in the history of mathematics as “the precalculus period,” were a time of remarkable activity in which researchers throughout Europe contributed novel solutions and competed with each other to arrive at important new methods.

The precalculus period

In his treatise Geometria Indivisibilibus Continuorum (1635; “Geometry of Continuous Indivisibles”), Bonaventura Cavalieri, a professor of mathematics at the University of Bologna, formulated a systematic method for the determination of areas and volumes. As had Archimedes, Cavalieri regarded a plane figure as being composed of a collection of indivisible lines, “all the lines” of the plane figure. The collection was generated by a fixed line moving through space parallel to itself. Cavalieri showed that these collections could be interpreted as magnitudes obeying the rules of Euclidean ratio theory. In proposition 4 of Book II, he derived the result that is written today as

Let there be given a parallelogram in which a diagonal is drawn; then “all the squares” of the parallelogram will be triple “all the squares” of each of the triangles determined by the diagonal.

Cavalieri’s principle Bonaventura Cavalieri observed that figures (solids) of equal height and in which all corresponding cross sections match in length (area) are of equal area (volume). For example, take a regular polygon equal in area to an equilateral triangle; erect a pyramid on the triangle and a conelike figure of the same height on the polygon; cross sections of both figures taken at the same height above the bases are equal; therefore, by Cavalieri’s theorem, so are the volumes of the solids.

Cavalieri showed that this proposition could be interpreted in different ways—as asserting, for example, that the volume of a cone is one-third the volume of the circumscribed cylinder (see the figure) or that the area under a segment of a parabola is one-third the area of the associated rectangle. In a later treatise he generalized the result by proving

for n = 3 to n = 9. To establish these results, he introduced transformations among the variables of the problem, using a result equivalent to the binomial theorem for integral exponents. The ideas involved went beyond anything that had appeared in the classical Archimedean theory of content.

Although Cavalieri was successful in formulating a systematic method based on general concepts, his ideas were not easy to apply. The derivation of very simple results required intricate geometric considerations, and the turgid style of the Geometria Indivisibilibus was a barrier to its reception.

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How I Rewired My Brain to Become Fluent in Math

was a wayward kid who grew up on the literary side of life, treating math and science as if they were pustules from the plague. So it’s a little strange how I’ve ended up now—someone who dances daily with triple integrals, Fourier transforms, and that crown jewel of mathematics, Euler’s equation. It’s hard to believe I’ve flipped from a virtually congenital math-phobe to a professor of engineering.

One day, one of my students asked me how I did it—how I changed my brain. I wanted to answer Hell—with lots of difficulty! After all, I’d flunked my way through elementary, middle, and high school math and science. In fact, I didn’t start studying remedial math until I left the Army at age 26. If there were a textbook example of the potential for adult neural plasticity, I’d be Exhibit A.

Learning math and then science as an adult gave me passage into the empowering world of engineering. But these hard-won, adult-age changes in my brain have also given me an insider’s perspective on the neuroplasticity that underlies adult learning. Fortunately, my doctoral training in systems engineering—tying together the big picture of different STEM (Science, Technology, Engineering, Math) disciplines—and then my later research and writing focusing on how humans think have helped me make sense of recent advances in neuroscience and cognitive psychology related to learning.

In the years since I received my doctorate, thousands of students have swept through my classrooms—students who have been reared in elementary school and high school to believe that understanding math through active discussion is the talisman of learning. If you can explain what you’ve learned to others, perhaps drawing them a picture, the thinking goes, you must
understand it.

Japan has become seen as a much-admired and emulated exemplar of these active, “understanding-centered” teaching methods. But what’s often missing from the discussion is the rest of the story: Japan is also home of the Kumon method of teaching mathematics, which emphasizes memorization, repetition, and rote learning hand-in-hand with developing the child’s mastery over the material. This intense afterschool program, and others like it, is embraced by millions of parents in Japan and around the world who supplement their child’s participatory education with plenty of practice, repetition, and yes, intelligently designed rote learning, to allow them to gain hard-won fluency with the material.

Teachers can inadvertently set their students up for failure as those students blunder in illusions of competence.

In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods that scientists are—and have been—telling us work with the brain’s natural process to learn complex subjects like math and science.

The latest wave in educational reform in mathematics involves the Common Core—an attempt to set strong, uniform standards across the U.S., although critics are weighing in to say the standards fail by comparison with high-achieving countries. At least superficially, the standards seem to show a sensible perspective. They propose that in mathematics, students should gain equal facility in conceptual understanding, procedural skills and fluency, and application.

The devil, of course, lies in the details of implementation. In the current educational climate, memorization and repetition in the STEM disciplines (as opposed to in the study of language or music), are often seen as demeaning and a waste of time for students and teachers alike. Many teachers have long been taught that conceptual understanding in STEM trumps everything else. And indeed, it’s easier for teachers to induce students to discuss a mathematical subject (which, if done properly, can do much to help promote understanding) than it is for that teacher to tediously grade math homework. What this all means is that, despite the fact that procedural skills and fluency, along with application, are supposed to be given equal emphasis with conceptual understanding, all too often it doesn’t happen. Imparting a conceptual understanding reigns supreme—especially during precious class time.

The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence. As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.

There is an interesting connection between learning math and science, and learning a sport. When you learn how to swing a golf club, you perfect that swing from lots of repetition over a period of years. Your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.

In the same way, once you understand why you do something in math and science, you don’t have to keep re-explaining the how to yourself every time you do it. It’s not necessary to go around with 25 marbles in your pocket and lay out 5 rows of 5 marbles again and again so that you get that 5 x 5 = 25. At some point, you just know it fluently from memory. You memorize the idea that you simply add exponents—those little superscript numbers—when multiplying numbers that have the same base (10⁴ x 10⁵ = 10⁹). If you use the procedure a lot, by doing many different types of problems, you will find that you understand both the why and the how behind the procedure very well indeed. The greater understanding results from the fact that your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way.

…………….

I learned these things about math and the process of learning not in the K-12 classroom but in the course of my life, as a kid who grew up reading Madeleine L’Engle and Dostoyevsky, who went on to study language at one of the world’s leading language institutes, and then to make the dramatic shift to become a professor of engineering.

As a young woman with a yen for learning language and no money or skills to speak of, I couldn’t afford to go to college (college loans weren’t then in the picture). So I launched directly from high school into the Army. I had loved learning new languages in high school, and the Army seemed to be a place where people could actually get paid for their language study, even as they attended the top-ranked Defense Language Institute—a place that had made language- learning a science. I chose Russian because it was very different from English, but not so difficult that I could study it for a lifetime only to perhaps gain the fluency of a 4-year-old. Besides, the Iron Curtain was mysteriously appealing—could I somehow use my knowledge of Russian to peer behind it?

After leaving the service, I became a translator for the Russians on Soviet trawlers on the Bering Sea. Working for the Russians was fun and engrossing—but it was also a superficially glamorous form of migrant work. You go to sea during fishing season, make a decent salary while getting drunk all the time, then go back to port when the season’s over and hope they’ll rehire you next year. There was pretty much only one other alternative for a Russian language speaker—working for the National Security Agency. (My Army contacts kept pointing me that way, but it wasn’t for me.)

I began to realize that while knowing another language was nice, it was also a skill with limited opportunities and potential. People weren’t pounding down my door looking for my Russian declension abilities. Unless, that is, I was willing to put up with seasickness and sporadic malnutrition out on stinking trawlers in the middle of the Bering Sea. I couldn’t help but reflect back on the West Point-trained engineers I’d worked with in the Army. Their mathematically and scientifically based approach to problem-solving was clearly useful for the real world—far more useful than my youthful misadventures with math had been able to imagine.

So, at age 26, as I was leaving the Army and casting about for fresh opportunities, it occurred to me: If I really wanted to try something new, why not tackle something that could open a whole world of new perspectives for me? Something like engineering? That meant I would be trying to learn another very different language—the language of calculus.

You go to sea during fishing season, make a decent salary while getting drunk all the time, then go back to port when the season’s over.

With my poor understanding of even the simplest math, my post-Army retraining efforts began with not-for-credit remedial algebra and trigonometry. This was way below mathematical ground zero for most college students. Trying to reprogram my brain sometimes seemed like a ridiculous idea—especially when I looked at the fresh young faces of my younger classmates and realized that many of them had already dropped their hard math and science classes—and here I was heading right for them. But in my case, from my experience becoming fluent in Russian as an adult, I suspected—or maybe I just hoped—that there might be aspects to language learning that I might apply to learning in math and science.

What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop. Where my language classmates had often been content to concentrate on simply understanding Russian they heard or read, I instead tried to gain an internalized, deep-rooted fluency with the words and language structure. I wouldn’t just be satisfied to know that понимать meant “to understand.” I’d practice with the verb—putting it through its paces by conjugating it repeatedly with all sorts of tenses, and then moving on to putting it into sentences, and then finally to understanding not only when to use this form of the verb, but also when not to use it. I practiced recalling all these aspects and variations quickly. After all, through practice, you can understand and translate dozens—even thousands— of words in another language. But if you aren’t fluent, when someone throws a bunch of words at you quickly, as with normal speaking (which always sounds horrifically fast when you’re learning a new language), you have no idea what they’re actually saying, even though technically you understand all the component words and structure. And you certainly can’t speak quickly enough yourself for native speakers to find it enjoyable to listen to you.

This approach—which focused on fluency instead of simple understanding—put me at the top of the class. And I didn’t realize it then, but this approach to learning language had given me an intuitive understanding of a fundamental core of learning and the development of expertise—chunking.

Chunking was originally conceptualized in the groundbreaking work of Herbert Simon in his analysis of chess—chunks were envisioned as the varying neural counterparts of different chess patterns. Gradually, neuroscientists came to realize that experts such as chess grand masters are experts because they have stored thousands of chunks of knowledge about their area of expertise in their long-term memory. Chess masters, for example, can recall tens of thousands of different chess patterns. Whatever the discipline, experts can call up to consciousness one or several of these well-knit-together, chunked neural subroutines to analyze and react to a new learning situation. This level of true understanding, and ability to use that understanding in new situations, comes only with the kind of rigor and familiarity that repetition, memorization, and practice can foster.

As studies of chess masters, emergency room physicians, and fighter pilots have shown, in times of critical stress, conscious analysis of a situation is replaced by quick, subconscious processing as these experts rapidly draw on their deeply ingrained repertoire of neural subroutines—chunks. At some point, self-consciously “understanding” why you do what you do just slows you down and interrupts flow, resulting in worse decisions. When I felt intuitively that there might be a connection between learning a new language and learning mathematics, I was right. Day-by-day, sustained practice of Russian fired and wired together my neural circuits, and I gradually began to knit together chunks of Slavic insight that I could call into working memory with ease. By interleaving my learning—in other words, practicing so that I knew not only when to use that word, but when not to use it, or to use a different variant of it—I was actually using the same approaches that expert practitioners use to learn in math and science.

When learning math and engineering as an adult, I began by using the same strategy I’d used to learn language. I’d look at an equation, to take a very simple example, Newton’s second law of f = ma. I practiced feeling what each of the letters meant—f for force was a push, m for mass was a kind of weighty resistance to my push, and a was the exhilarating feeling of acceleration. (The equivalent in Russian was learning to physically sound out the letters of the Cyrillic alphabet.) I memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side? Playing with the equation was like conjugating a verb. I was beginning to intuit that the sparse outlines of the equation were like a metaphorical poem, with all sorts of beautiful symbolic representations embedded within it. Although I wouldn’t have put it that way at the time, the truth was that to learn math and science well, I had to slowly, day by day, build solid neural “chunked” subroutines—such as surrounding the simple equation f = ma—that I could easily call to mind from long term memory, much as I’d done with Russian.

Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

In other words, in science and math education in particular, it’s easy to slip into teaching methods that emphasize understanding and that avoid the sometimes painful repetition and practice that underlie fluency. I learned Russian not just by understanding it—understanding, after all, is facile, and can easily slip away. (What did that word понимать mean?) I learned Russian by gaining fluency through practice, repetition, and rote learning—but rote learning that emphasized the ability to think flexibly and quickly. I learned math and science by applying precisely those same ideas. Language, math, and science, as with almost all areas of human expertise, draw on the same reservoir of brain mechanisms.

As I forayed into a new life, becoming an electrical engineer and, eventually, a professor of engineering, I left the Russian language behind. But 25 years after I’d last raised an inebriated glass on the Soviet trawlers, my family and I decided to take the trans-Siberian railway across Russia. Although I was excited to take the long-dreamed-of trip, I was also worried. I’d barely uttered a word of Russian in all that time. What if I’d lost it all? What had those years of gaining fluency really bought me?

Sure enough, when we first got on the train, I spoke Russian like a 2-year-old. I’d grasp for words, my declensions and conjugations were all wrong, and my formerly near-perfect accent sounded dreadful. But the foundation was there, and day by day, my Russian improved. And even with my rudimentary Russian, I could handle the day-to-day needs of our traveling. Soon, tour guides were coming to me for help translating for the other passengers. When we finally arrived in Moscow, we hopped in a taxi. The driver, I soon discovered, was intent on ripping us off—heading directly the wrong way and trapping us in a logjam of cars, where he expected us ignorant foreigners to quietly acquiesce to an unnecessary extra hour of meter time. Suddenly, Russian words I hadn’t spoken for decades flew from my mouth. I hadn’t even consciously known I knew those words.

Underneath it all, when it was needed, the fluency was there—and it quickly got us out of trouble (and into another taxi). Fluency allows understanding to become embedded, emerging when needed.

As I look today at the shortage of science and math majors in this country, and our current trend in how we teach people to learn, and as I reflect on my own pathway, knowing what I know now about the brain, it occurs to me that we can do better. As parents and teachers, we can use simple, accessible methods for deepening understanding and making it useful and flexible. We can encourage others and ourselves to try new disciplines that we thought were too hard—math, dance, physics, language, chemistry, music—opening new worlds for ourselves and others.

As I discovered, having a basic, deep-seated fluency in math and science—not just an “understanding,” is critical. It opens doors for many of life’s most intriguing jobs. Looking back, I realize that I didn’t have to just blindly follow my initial inclinations and passions. The “fluency” part of me that loved literature and language was also the same part of me that ultimately fell in love with math and science—and transformed and enriched my life.