MATHEMATICS NOTES FOR FORM THREE – ALL TOPICS

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TOPIC 1 – RELATIONS


TOPIC 2 – FUNCTIONS


TOPIC 3 – STATISTICS


TOPIC 4 – RATES AND VARIATIONS


TOPIC 5 – SEQUENCE AND SERIES


TOPIC 6 – CIRCLES


TOPIC 7 – THE EARTH AS A SPHERE


TOPIC 8 – ACCOUNTS

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IMPORTANCE OF MATHEMATICS IN OUR DAILY LIFE – PART 3 

9. Math is all around us and helps us
understand the world better.

To live in a mathematically-driven world
and not know math is like walking through an art museum with your eyes
closed.  Learning and appreciating math can help you appreciate things
that you would not otherwise notice about the world.  In
reality, math is everywhere! Don’t believe me?  Read on for some
examples of math in nature.

10. Every Career Uses Math

There is no profession in the world that
doesn’t use math. We know that mathematicians and scientists rely on
mathematical principles to perform their basic work. Engineers also use math to
perform their daily tasks. From blue-collar factory workers to managerial-level
white-collar professionals, everyone uses math in their work.

11. Math may boost emotional health

While this research is still in its early
days, what we have seen is promising. The parts of the brain
used to solve math problems seem to work together with the parts of the brain
that regulate emotions. This suggests that math practice can actually help
us cope with difficult situations. In these studies, the better someone
was with numerical calculations, the better they were at regulating fear and
anger. Strong math skills may even be able to help treat anxiety and
depression.

12. Math Helps Us Manage Time

We all depend on time to perform daily
activities, and a clock known for showing time is established as per the math
calculations. With the help of math, we can easily manage our time and predict
the completion time of a particular task. nSuppose we have to go to the office
daily and set our time to leave the house at 9 o’clock sharp so that we can
reach the office at 9:30 am. If we are familiar with math, we can easily assume
the time difference; this way, we can be on time wherever we go.

13. Math Makes Us Sharp Workers

Whatever job we choose requires
mathematical skills to do well in the workplace, such as scientists, finance
managers, teachers, designers, etc. Every field requires the problem-solving
ability to find out solutions to critical issues. Hence, if you know much math,
you can efficiently complete all your office tasks.

…………….

Apollonius

The work of Apollonius of Perga extended the field of geometric constructions far beyond the range in the Elements. For example, Euclid in Book III shows how to draw a circle so as to pass through three given points or to be tangent to three given lines; Apollonius (in a work called Tangencies, which no longer survives) found the circle tangent to three given circles, or tangent to any combination of three points, lines, and circles. (The three-circle tangency construction, one of the most extensively studied geometric problems, has attracted more than 100 different solutions in the modern period.)

conic sections The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

Apollonius is best known for his Conics, a treatise in eight books (Books I–IV survive in Greek, V–VII in a medieval Arabic translation; Book VIII is lost). The conic sections are the curves formed when a plane intersects the surface of a cone (or double cone). It is assumed that the surface of the cone is generated by the rotation of a line through a fixed point around the circumference of a circle which is in a plane not containing that point. (The fixed point is the vertex of the cone, and the rotated line its generator.) There are three basic types: if the cutting plane is parallel to one of the positions of the generator, it produces a parabola; if it meets the cone only on one side of the vertex, it produces an ellipse (of which the circle is a special case); but if it meets both parts of the cone, it produces a hyperbola. Apollonius sets out in detail the properties of these curves. He shows, for example, that for given line segments a and b the parabola corresponds to the relation (in modern notation) y2 = ax, the ellipse to y2 = ax − ax2/b, and the hyperbola to y2 = ax + ax2/b.

Apollonius’s treatise on conics in part consolidated more than a century of work before him and in part presented new findings of his own. As mentioned earlier, Euclid had already issued a textbook on the conics, while even earlier Menaechmus had played a role in their study. The names that Apollonius chose for the curves (the terms may be original with him) indicate yet an earlier connection. In the pre-Euclidean geometry parabolē referred to a specific operation, the “application” of a given area to a given line, in which the line x is sought such that ax = b2 (where a and b are given lines); alternatively, x may be sought such that x(a + x) = b2, or x(a − x) = b2, and in these cases the application is said to be in “excess” (hyperbolē) or “defect” (elleipsis) by the amount of a square figure (namely, x2). These constructions, which amount to a geometric solution of the general quadratic, appear in Books I, II, and VI of the Elements and can be associated in some form with the 5th-century Pythagoreans.

Apollonius presented a comprehensive survey of the properties of these curves. A sample of the topics he covered includes the following: the relations satisfied by the diameters and tangents of conics (Book I); how hyperbolas are related to their “asymptotes,” the lines they approach without ever meeting (Book II); how to draw tangents to given conics (Book II); relations of chords intersecting in conics (Book III); the determination of the number of ways in which conics may intersect (Book IV); how to draw “normal” lines to conics (that is, lines meeting them at right angles; Book V); and the congruence and similarity of conics (Book VI).

By Apollonius’s explicit statement, his results are of principal use as methods for the solution of geometric problems via conics. While he actually solved only a limited set of problems, the solutions of many others can be inferred from his theorems. For instance, the theorems of Book III permit the determination of conics that pass through given points or are tangent to given lines. In another work (now lost) Apollonius solved the problem of cube duplication by conics (a solution related in some way to that given by Menaechmus); further, a solution of the problem of angle trisection given by Pappus may have come from Apollonius or been influenced by his work.

With the advance of the field of geometric problems by Euclid, Apollonius, and their followers, it became appropriate to introduce a classifying scheme: those problems solvable by means of conics were called solid, while those solvable by means of circles and lines only (as assumed in Euclid’s Elements) were called planar. Thus, one can double the square by planar means (as in Elements, Book II, proposition 14), but one cannot double the cube in such a way, although a solid construction is possible (as given above). Similarly, the bisection of any angle is a planar construction (as shown in Elements, Book I, proposition 9), but the general trisection of the angle is of the solid type. It is not known when the classification was first introduced or when the planar methods were assigned canonical status relative to the others, but it seems plausible to date this near Apollonius’s time. Indeed, much of his work—books like the Tangencies, the Vergings (or Inclinations), and the Plane Loci, now lost but amply described by Pappus—turns on the project of setting out the domain of planar constructions in relation to solutions by other means. On the basis of the principles of Greek geometry, it cannot be demonstrated, however, that it is impossible to effect by planar means certain solid constructions (like the cube duplication and angle trisection). These results were established only by algebraists in the 19th century (notably by the French mathematician Pierre Laurent Wantzel in 1837).

angle trisection using a conchoid Nicomedes (3rd century bce) discovered a special curve, known as a conchoid, with which he was able to trisect any acute angle. Given ∠θ, construct a conchoid with its pole at the vertex of the angle (b) and its directrix (n) through one side of the angle and perpendicular to the line (m) containing one of the angle’s sides. Then construct the line (l) through the intersection (c) of the directrix and the remaining side of the angle. The intersection of l and the conchoid at d determines ∠abd = θ/3, as desired.

A third class of problems, called linear, embraced those solvable by means of curves other than the circle and the conics (in Greek the word for “line,” grammē, refers to all lines, whether curved or straight). For instance, one group of curves, the conchoids (from the Greek word for “shell”), are formed by marking off a certain length on a ruler and then pivoting it about a fixed point in such a way that one of the marked points stays on a given line; the other marked point traces out a conchoid. These curves can be used wherever a solution involves the positioning of a marked ruler relative to a given line (in Greek such constructions are called neuses, or “vergings” of a line to a given point). For example, any acute angle (figured as the angle between one side and the diagonal of a rectangle) can be trisected by taking a length equal to twice the diagonal and moving it about until it comes to be inserted between two other sides of the rectangle. If instead the appropriate conchoid relative to either of those sides is introduced, the required position of the line can be determined without the trial and error of a moving ruler. Because the same construction can be effected by means of a hyperbola, however, the problem is not linear but solid. Such uses of the conchoids were presented by Nicomedes (middle or late 3rd century bce), and their replacement by equivalent solid constructions appears to have come soon after, perhaps by Apollonius or his associates.

Some of the curves used for problem solving are not so reducible. For example, the Archimedean spiral couples uniform motion of a point on a half ray with uniform rotation of the ray around a fixed point at its end (see Sidebar: Quadratrix of Hippias). Such curves have their principal interest as means for squaring the circle and trisecting the angle.

Applied geometry

A major activity among geometers in the 3rd century bce was the development of geometric approaches in the study of the physical sciences—specifically, optics, mechanics, and astronomy. In each case the aim was to formulate the basic concepts and principles in terms of geometric and numerical quantities and then to derive the fundamental phenomena of the field by geometric constructions and proofs.

In optics, Euclid’s textbook (called the Optics) set the precedent. Euclid postulated visual rays to be straight lines, and he defined the apparent size of an object in terms of the angle formed by the rays drawn from the top and the bottom of the object to the observer’s eye. He then proved, for example, that nearer objects appear larger and appear to move faster and showed how to measure the height of distant objects from their shadows or reflected images and so on. Other textbooks set out theorems on the phenomena of reflection and refraction (the field called catoptrics). The most extensive survey of optical phenomena is a treatise attributed to the astronomer Ptolemy (2nd century ce), which survives only in the form of an incomplete Latin translation (12th century) based on a lost Arabic translation. It covers the fields of geometric optics and catoptrics, as well as experimental areas, such as binocular vision, and more general philosophical principles (the nature of light, vision, and colour). Of a somewhat different sort are the studies of burning mirrors by Diocles (late 2nd century bce), who proved that the surface that reflects the rays from the Sun to a single point is a paraboloid of revolution. Constructions of such devices remained of interest as late as the 6th century ce, when Anthemius of Tralles, best known for his work as architect of Hagia Sophia at Constantinople, compiled a survey of remarkable mirror configurations.

Mechanics was dominated by the work of Archimedes, who was the first to prove the principle of balance: that two weights are in equilibrium when they are inversely proportional to their distances from the fulcrum. From this principle he developed a theory of the centres of gravity of plane and solid figures. He was also the first to state and prove the principle of buoyancy—that floating bodies displace their equal in weight—and to use it for proving the conditions of stability of segments of spheres and paraboloids, solids formed by rotating a parabolic segment about its axis. Archimedes proved the conditions under which these solids will return to their initial position if tipped, in particular for the positions now called “stable I” and “stable II,” where the vertex faces up and down, respectively.

elliptic paraboloid The figure shows part of the elliptic paraboloid z = x2 + y2, which can be generated by rotating the parabola z = x2 (or z = y2) about the z-axis. Note that cross sections of the surface parallel to the xy plane, as shown by the cutoff at the top of the figure, are ellipses.

ellipsoidAn ellipsoid is a closed surface such that its intersection with any plane will produce an ellipse or a circle. The formula for an ellipsoid is x2/a2 + y2/b2 + z2/c2 = 1. A spheroid, or ellipsoid of revolution, is an ellipsoid generated by rotating an ellipse about one of its axes.

hyperbolic paraboloidThe figure shows part of the hyperbolic paraboloid x2/a2 − y2/b2 = 2cz. Note that cross sections of the surface parallel to the xz- and yz-plane are parabolas, while cross sections parallel to the xy-plane are hyperbolas.

In his work Method Concerning Mechanical Theorems, Archimedes also set out a special “mechanical method” that he used for the discovery of results on volumes and centres of gravity. He employed the bold notion of constituting solids from the plane figures formed as their sections (e.g., the circles that are the plane sections of spheres, cones, cylinders, and other solids of revolution), assigning to such figures a weight proportional to their area. For example, to measure the volume of a sphere, he imagined a balance beam, one of whose arms is a diameter of the sphere with the fulcrum at one endpoint of this diameter and the other arm an extension of the diameter to the other side of the fulcrum by a length equal to the diameter. Archimedes showed that the three circular cross sections made by a plane cutting the sphere and the associated cone and cylinder will be in balance (the circle in the cylinder with the circles in the sphere and cone) if the circle in the cylinder is kept in its original place while the circles in the sphere and cone are placed with their centres of gravity at the opposite end of the balance. Doing this for all the sets of circles formed as cross sections of these solids by planes, he concluded that the solids themselves are in balance—the cylinder with the sphere and the cone together—if the cylinder is left where it is while the sphere and cone are placed with their centres of gravity at the opposite end of the balance. Since the centre of gravity of the cylinder is the midpoint of its axis, it follows that (sphere + cone):cylinder = 1:2 (by the inverse proportion of weights and distances). Since the volume of the cone is one-third that of the cylinder, however, the volume of the sphere is found to be one-sixth that of the cylinder. In similar manner, Archimedes worked out the volumes and centres of gravity of spherical segments and segments of the solids of revolution of conic sections—paraboloids, ellipsoids, and hyperboloids. The critical notions—constituting solids out of their plane sections and assigning weights to geometric figures—were not formally valid within the standard conceptions of Greek geometry, and Archimedes admitted this. But he maintained that, although his arguments were not “demonstrations” (i.e., proofs), they had value for the discovery of results about these figures.

Ptolemaic systemIn Ptolemy’s geocentric model of the universe, the Sun, the Moon, and each planet orbit a stationary Earth. For the Greeks, heavenly bodies must move in the most perfect possible fashion—hence, in perfect circles. In order to retain such motion and still explain the erratic apparent paths of the bodies, Ptolemy shifted the centre of each body’s orbit (deferent) from Earth—accounting for the body’s apogee and perigee—and added a second orbital motion (epicycle) to explain retrograde motion. The equant is the point from which each body sweeps out equal angles along the deferent in equal times. The centre of the deferent is midway between the equant and Earth.

The geometric study of astronomy has pre-Euclidean roots, Eudoxus having developed a model for planetary motions around a stationary Earth. Accepting the principle—which, according to Eudemus, was first proposed by Plato—that only combinations of uniform circular motions are to be used, Eudoxus represented the path of a planet as the result of superimposing rotations of three or more concentric spheres whose axes are set at different angles. Although the fit with the phenomena was unsatisfactory, the curves thus generated (the hippopede, or “horse-fetter”) continued to be of interest for their geometric properties, as is known through remarks by Proclus. Later geometers continued the search for geometric patterns satisfying the Platonic conditions. The simplest model, a scheme of circular orbits centred on the Sun, was introduced by Aristarchus of Samos (3rd century bce), but this was rejected by others, since a moving Earth was judged to be impossible on physical grounds. But Aristarchus’s scheme could have suggested use of an “eccentric” model, in which the planets rotate about the Sun and the Sun in turn rotates about the Earth. Apollonius introduced an alternative “epicyclic” model, in which the planet turns about a point that itself orbits in a circle (the “deferent”) centred at or near Earth. As Apollonius knew, his epicyclic model is geometrically equivalent to an eccentric. These models were well adapted for explaining other phenomena of planetary motion. For instance, if Earth is displaced from the centre of a circular orbit (as in the eccentric scheme), the orbiting body will appear to vary in speed (appearing faster when nearer the observer, slower when farther away), as is in fact observed for the Sun, Moon, and planets. By varying the relative sizes and rotation rates of the epicycle and deferent, in combination with the eccentric, a flexible device may be obtained for representing planetary motion. (See Ptolemy’s model.)

Later trends in geometry and arithmetic

Greek trigonometry and mensuration

After the 3rd century bce, mathematical research shifted increasingly away from the pure forms of constructive geometry toward areas related to the applied disciplines, in particular to astronomy. The necessary theorems on the geometry of the sphere (called spherics) were compiled into textbooks, such as the one by Theodosius (3rd or 2nd century bce) that consolidated the earlier work by Euclid and the work of Autolycus of Pitane (flourished c. 300 bce) on spherical astronomy. More significant, in the 2nd century bce the Greeks first came into contact with the fully developed Mesopotamian astronomical systems and took from them many of their observations and parameters (for example, values for the average periods of astronomical phenomena). While retaining their own commitment to geometric models rather than adopting the arithmetic schemes of the Mesopotamians, the Greeks nevertheless followed the Mesopotamians’ lead in seeking a predictive astronomy based on a combination of mathematical theory and observational parameters. They thus made it their goal not merely to describe but to calculate the angular positions of the planets on the basis of the numerical and geometric content of the theory. This major restructuring of Greek astronomy, in both its theoretical and practical respects, was primarily due to Hipparchus (2nd century bce), whose work was consolidated and further advanced by Ptolemy.

To facilitate their astronomical researches, the Greeks developed techniques for the numerical measurement of angles, a precursor of trigonometry, and produced tables suitable for practical computation. Early efforts to measure the numerical ratios in triangles were made by Archimedes and Aristarchus. Their results were soon extended, and comprehensive treatises on the measurement of chords (in effect, a construction of a table of values equivalent to the trigonometric sine) were produced by Hipparchus and by Menelaus of Alexandria (1st century ce). These works are now lost, but the essential theorems and tables are preserved in Ptolemy’s Almagest (Book I, chapter 10). For computing with angles, the Greeks adopted the Mesopotamian sexagesimal method in arithmetic, whence it survives in the standard units for angles and time employed to this day.

Number theory

Polygonal numbersThe ancient Greeks generally thought of numbers in concrete terms, particularly as measurements and geometric dimensions. Thus, they often arranged pebbles in various patterns to discern arithmetical, as well as mystical, relationships between numbers. A few such patterns are indicated in the figure.

Although Euclid handed down a precedent for number theory in Books VII–IX of the Elements, later writers made no further effort to extend the field of theoretical arithmetic in his demonstrative manner. Beginning with Nicomachus of Gerasa (flourished c. 100 ce), several writers produced collections expounding a much simpler form of number theory. A favourite result is the representation of arithmetic progressions in the form of “polygonal numbers.” For instance, if the numbers 1, 2, 3, 4,…are added successively, the “triangular” numbers 1, 3, 6, 10,…are obtained; similarly, the odd numbers 1, 3, 5, 7,…sum to the “square” numbers 1, 4, 9, 16,…, while the sequence 1, 4, 7, 10,…, with a constant difference of 3, sums to the “pentagonal” numbers 1, 5, 12, 22,…. In general, these results can be expressed in the form of geometric shapes formed by lining up dots in the appropriate two-dimensional configurations (see figure). In the ancient arithmetics such results are invariably presented as particular cases, without any general notational method or general proof. The writers in this tradition are called neo-Pythagoreans, since they viewed themselves as continuing the Pythagorean school of the 5th century bce, and, in the spirit of ancient Pythagoreanism, they tied their numerical interests to a philosophical theory that was an amalgam of Platonic metaphysical and theological doctrines. With its exponent Iamblichus of Chalcis (4th century ce), neo-Pythagoreans became a prominent part of the revival of pagan religion in opposition to Christianity in late antiquity.

An interesting concept of this school of thought, which Iamblichus attributes to Pythagoras himself, is that of “amicable numbers”: two numbers are amicable if each is equal to the sum of the proper divisors of the other (for example, 220 and 284). Attributing virtues such as friendship and justice to numbers was characteristic of the Pythagoreans at all times.

Of much greater mathematical significance is the arithmetic work of Diophantus of Alexandria (c. 3rd century ce). His writing, the Arithmetica, originally in 13 books (six survive in Greek, another four in medieval Arabic translation), sets out hundreds of arithmetic problems with their solutions. For example, Book II, problem 8, seeks to express a given square number as the sum of two square numbers (here and throughout, the “numbers” are rational). Like those of the neo-Pythagoreans, his treatments are always of particular cases rather than general solutions; thus, in this problem the given number is taken to be 16, and the solutions worked out are 256/25 and 144/25. In this example, as is often the case, the solutions are not unique; indeed, in the very next problem Diophantus shows how a number given as the sum of two squares (e.g., 13 = 4 + 9) can be expressed differently as the sum of two other squares (for example, 13 = 324/25 + 1/25).

To find his solutions, Diophantus adopted an arithmetic form of the method of analysis. He first reformulated the problem in terms of one of the unknowns, and he then manipulated it as if it were known until an explicit value for the unknown emerged. He even adopted an abbreviated notational scheme to facilitate such operations, where, for example, the unknown is symbolized by a figure somewhat resembling the Roman letter S. (This is a standard abbreviation for the word number in ancient Greek manuscripts.) Thus, in the first problem discussed above, if S is one of the unknown solutions, then 16 − S2 is a square; supposing the other unknown to be 2S − 4 (where the 2 is arbitrary but the 4 chosen because it is the square root of the given number 16), Diophantus found from summing the two unknowns ([2S − 4]2 and S2) that 4S2 − 16S + 16 + S2 = 16, or 5S2 = 16S; that is, S = 16/5. So one solution is S2 = 256/25, while the other solution is 16 − S2, or 144/25.

Survival and influence of Greek mathematics

Robert Trewick Bone: Hypatia Teaching at AlexandriaHypatia Teaching at Alexandria, watercolour and brown ink on paper by Robert Trewick Bone; in the Yale Center for British Art, New Haven, Connecticut.

Notable in the closing phase of Greek mathematics were Pappus (early 4th century ce), Theon (late 4th century), and Theon’s daughter Hypatia. All were active in Alexandria as professors of mathematics and astronomy, and they produced extensive commentaries on the major authorities—Pappus and Theon on Ptolemy, Hypatia on Diophantus and Apollonius. Later, Eutocius of Ascalon (early 6th century) produced commentaries on Archimedes and Apollonius. While much of their output has since been lost, much survives. They proved themselves reasonably competent in technical matters but little inclined toward significant insights (their aim was usually to fill in minor steps assumed in the proofs, to append alternative proofs, and the like), and their level of originality was very low. But these scholars frequently preserved fragments of older works that are now lost, and their teaching and editorial efforts assured the survival of the works of Euclid, Archimedes, Apollonius, Diophantus, Ptolemy, and others that now do exist, either in Greek manuscripts or in medieval translations (Arabic, Hebrew, and Latin) derived from them.

The legacy of Greek mathematics, particularly in the fields of geometry and geometric science, was enormous. From an early period the Greeks formulated the objectives of mathematics not in terms of practical procedures but as a theoretical discipline committed to the development of general propositions and formal demonstrations. The range and diversity of their findings, especially those of the masters of the 3rd century bce, supplied geometers with subject matter for centuries thereafter, even though the tradition that was transmitted into the Middle Ages and Renaissance was incomplete and defective.

The rapid rise of mathematics in the 17th century was based in part on the conscious imitation of the ancient classics and on competition with them. In the geometric mechanics of Galileo and the infinitesimal researches of Johannes Kepler and Bonaventura Cavalieri, it is possible to perceive a direct inspiration from Archimedes. The study of the advanced geometry of Apollonius and Pappus stimulated new approaches in geometry—for example, the analytic methods of René Descartes and the projective theory of Girard Desargues. Purists like Christiaan Huygens and Isaac Newton insisted on the Greek geometric style as a model of rigour, just as others sought to escape its forbidding demands of completely worked-out proofs. The full impact of Diophantus’s work is evident particularly with Pierre de Fermat in his researches in algebra and number theory. Although mathematics has today gone far beyond the ancient achievements, the leading figures of antiquity, like Archimedes, Apollonius, and Ptolemy, can still be rewarding reading for the ingenuity of their insights.

Wilbur R. Knorr

Mathematics in the Islamic world (8th–15th century)

Origins

mathematicians of the Islamic world This map spans more than 600 years of prominent Islamic mathematicians, from al-Khwārizmī (c. 800 ce) to al-Kāshī (c. 1400 ce). Their names—located on the map under their cities of birth—can be clicked to access their biographies.

In Hellenistic times and in late antiquity, scientific learning in the eastern part of the Roman world was spread over a variety of centres, and Justinian’s closing of the pagan academies in Athens in 529 gave further impetus to this diffusion. An additional factor was the translation and study of Greek scientific and philosophical texts sponsored both by monastic centres of the various Christian churches in the Levant, Egypt, and Mesopotamia and by enlightened rulers of the Sāsānian dynasty in places like the medical school at Gondeshapur.

Also important were developments in India in the first few centuries ce. Although the decimal system for whole numbers was apparently not known to the Indian astronomer Aryabhata (born 476), it was used by his pupil Bhaskara I in 620, and by 670 the system had reached northern Mesopotamia, where the Nestorian bishop Severus Sebokht praised its Hindu inventors as discoverers of things more ingenious than those of the Greeks. Earlier, in the late 4th or early 5th century, the anonymous Hindu author of an astronomical handbook, the Surya Siddhanta, had tabulated the sine function (unknown in Greece) for every 33/4° of arc from 33/4° to 90°. (See South Asian mathematics.)

Within this intellectual context the rapid expansion of Islam took place between the time of Muḥammad’s return to Mecca in 630 from his exile in Medina and the Muslim conquest of lands extending from Spain to the borders of China by 715. Not long afterward, Muslims began the acquisition of foreign learning, and, by the time of the caliph al-Manṣūr (died 775), such Indian and Persian astronomical material as the Brahma-sphuta-siddhanta and the Shah’s Tables had been translated into Arabic. The subsequent acquisition of Greek material was greatly advanced when the caliph al-Maʾmūn constructed a translation and research centre, the House of Wisdom, in Baghdad during his reign (813–833). Most of the translations were done from Greek and Syriac by Christian scholars, but the impetus and support for this activity came from Muslim patrons. These included not only the caliph but also wealthy individuals such as the three brothers known as the Banū Mūsā, whose treatises on geometry and mechanics formed an important part of the works studied in the Islamic world.

Of Euclid’s works the Elements, the Data, the Optics, the Phaenomena, and On Divisions were translated. Of Archimedes’ works only two—Sphere and Cylinder and Measurement of the Circle—are known to have been translated, but these were sufficient to stimulate independent researches from the 9th to the 15th century. On the other hand, virtually all of Apollonius’s works were translated, and of Diophantus and Menelaus one book each, the Arithmetica and the Sphaerica, respectively, were translated into Arabic. Finally, the translation of Ptolemy’s Almagest furnished important astronomical material.

Of the minor writings, Diocles’ treatise on mirrors, Theodosius’s Spherics, Pappus’s work on mechanics, Ptolemy’s Planisphaerium, and Hypsicles’ treatises on regular polyhedra (the so-called Books XIV and XV of Euclid’s Elements) were among those translated.

Mathematics in the 9th century

Thābit ibn Qurrah (836–901), a Sabian from Ḥarrān in northern Mesopotamia, was an important translator and reviser of these Greek works. In addition to translating works of the major Greek mathematicians (for the Banū Mūsā, among others), he was a court physician. He also translated Nicomachus of Gerasa’s Arithmetic and discovered a beautiful rule for finding amicable numbers, a pair of numbers such that each number is the sum of the set of proper divisors of the other number. The investigation of such numbers formed a continuing tradition in Islam. Kamāl al-Dīn al-Fārisī (died c. 1320) gave the pair 17,926 and 18,416 as an example of Thābit’s rule, and in the 17th century Muḥammad Bāqir Yazdī gave the pair 9,363,584 and 9,437,056.

One scientist typical of the 9th century was Muḥammad ibn Mūsā al-Khwārizmī. Working in the House of Wisdom, he introduced Indian material in his astronomical works and also wrote an early book explaining Hindu arithmetic, the Book of Addition and Subtraction According to the Hindu Calculation. In another work, the Book of Restoring and Balancing, he provided a systematic introduction to algebra, including a theory of quadratic equations. Both works had important consequences for Islamic mathematics. Hindu Calculation began a tradition of arithmetic books that, by the middle of the next century, led to the invention of decimal fractions (complete with a decimal point), and Restoring and Balancing became the point of departure and model for later writers such as the Egyptian Abū Kāmil. Both books were translated into Latin, and Restoring and Balancing was the origin of the word algebra, from the Arabic word for “restoring” in its title (al-jabr). The Hindu Calculation, from a Latin form of the author’s name, algorismi, yielded the word algorithm.

Al-Khwārizmī’s algebra also served as a model for later writers in its application of arithmetic and algebra to the distribution of inheritances according to the complex requirements of Muslim religious law. This tradition of service to the Islamic faith was an enduring feature of mathematical work in Islam and one that, in the eyes of many, justified the study of secular learning. In the same category are al-Khwārizmī’s method of calculating the time of visibility of the new moon (which signals the beginning of the Muslim month) and the expositions by astronomers of methods for finding the direction to Mecca for the five daily prayers.

Mathematics in the 10th century

Islamic scientists in the 10th century were involved in three major mathematical projects: the completion of arithmetic algorithms, the development of algebra, and the extension of geometry.

The first of these projects led to the appearance of three complete numeration systems, one of which was the finger arithmetic used by the scribes and treasury officials. This ancient arithmetic system, which became known throughout the East and Europe, employed mental arithmetic and a system of storing intermediate results on the fingers as an aid to memory. (Its use of unit fractions recalls the Egyptian system.) During the 10th and 11th centuries capable mathematicians, such as Abūʾl-Wafāʾ (940–997/998), wrote on this system, but it was eventually replaced by the decimal system.

A second common system was the base-60 numeration inherited from the Babylonians via the Greeks and known as the arithmetic of the astronomers. Although astronomers used this system for their tables, they usually converted numbers to the decimal system for complicated calculations and then converted the answer back to sexagesimals.

The third system was Indian arithmetic, whose basic numeral forms, complete with the zero, eastern Islam took over from the Hindus. (Different forms of the numerals, whose origins are not entirely clear, were used in western Islam.) The basic algorithms also came from India, but these were adapted by al-Uqlīdisī (c. 950) to pen and paper instead of the traditional dust board, a move that helped to popularize this system. Also, the arithmetic algorithms were completed in two ways: by the extension of root-extraction procedures, known to Hindus and Greeks only for square and cube roots, to roots of higher degree and by the extension of the Hindu decimal system for whole numbers to include decimal fractions. These fractions appear simply as computational devices in the work of both al-Uqlīdisī and al-Baghdādī (c. 1000), but in subsequent centuries they received systematic treatment as a general method. As for extraction of roots, Abūʾl-Wafāʾ wrote a treatise (now lost) on the topic, and Omar Khayyam (1048–1131) solved the general problem of extracting roots of any desired degree. Omar’s treatise too is lost, but the method is known from other writers, and it appears that a major step in its development was al-Karajī’s 10th-century derivation by means of mathematical induction of the binomial theorem for whole-number exponents—i.e., his discovery that

During the 10th century Islamic algebraists progressed from al-Khwārizmī’s quadratic polynomials to the mastery of the algebra of expressions involving arbitrary positive or negative integral powers of the unknown. Several algebraists explicitly stressed the analogy between the rules for working with powers of the unknown in algebra and those for working with powers of 10 in arithmetic, and there was interaction between the development of arithmetic and algebra from the 10th to the 12th century. A 12th-century student of al-Karajī’s works, al-Samawʿal, was able to approximate the quotient (20×2 + 30x)/(6×2 + 12) asand also gave a rule for finding the coefficients of the successive powers of 1/x. Although none of this employed symbolic algebra, algebraic symbolism was in use by the 14th century in the western part of the Islamic world. The context for this well-developed symbolism was, it seems, commentaries that were destined for teaching purposes, such as that of Ibn Qunfūdh (1330–1407) of Algeria on the algebra of Ibn al-Bannāʿ (1256–1321) of Morocco.

Other parts of algebra developed as well. Both Greeks and Hindus had studied indeterminate equations, and the translation of this material and the application of the newly developed algebra led to the investigation of Diophantine equations by writers like Abū Kāmil, al-Karajī, and Abū Jaʿfar al-Khāzin (first half of 10th century), as well as to attempts to prove a special case of what is now known as Fermat’s last theorem—namely, that there are no rational solutions to x3 + y3 = z3. The great scientist Ibn al-Haytham (965–1040) solved problems involving congruences by what is now called Wilson’s theorem, which states that, if p is a prime, then p divides (p − 1) × (p − 2)⋯× 2 × 1 + 1, and al-Baghdādī gave a variant of the idea of amicable numbers by defining two numbers to “balance” if the sums of their divisors are equal.

However, not only arithmetic and algebra but geometry too underwent extensive development. Thābit ibn Qurrah, his grandson Ibrāhīm ibn Sinān (909–946), Abū Sahl al-Kūhī (died c. 995), and Ibn al-Haytham solved problems involving the pure geometry of conic sections, including the areas and volumes of plane and solid figures formed from them, and also investigated the optical properties of mirrors made from conic sections. Ibrāhīm ibn Sinān, Abu Sahl al-Kūhī, and Ibn al-Haytham used the ancient technique of analysis to reduce the solution of problems to constructions involving conic sections. (Ibn al-Haytham, for example, used this method to find the point on a convex spherical mirror at which a given object is seen by a given observer.) Thābit and Ibrāhīm showed how to design the curves needed for sundials. Abūʾl-Wafāʾ, whose book on the arithmetic of the scribes is mentioned above, also wrote on geometric methods needed by artisans.

In addition, in the late 10th century Abūʾl-Wafāʾ and the prince Abū Naṣr Manṣur stated and proved theorems of plane and spherical geometry that could be applied by astronomers and geographers, including the laws of sines and tangents. Abū Naṣr’s pupil al-Bīrūnī (973–1048), who produced a vast amount of high-quality work, was one of the masters in applying these theorems to astronomy and to such problems in mathematical geography as the determination of latitudes and longitudes, the distances between cities, and the direction from one city to another.

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WHO NEEDS MATHS?

Many professions are highly dependent on maths. Student scientists, engineers, and accountants may work with maths in every subject. Students in other disciplines may encounter one or more maths courses or assessments. Nursing and paramedics students will encounter medication calculations for example. Similarly, many students who wish to work in people-helping professions will need to be able to interpret statistics to ensure their approaches to problems and interventions are sound. This will include students of psychology, human services, and education. No matter your discipline, this chapter will help you approach the maths content in your courses with successful strategies and a positive attitude.

MATHS ANXIETY AND ITS IMPACT ON STUDYING MATHS

When studying maths, students often become anxious and start to overanalyse the maths content. Overanalysing can further elevate their anxiety and create difficulty in understanding the underpinning maths concepts.

The Australian Council of Educational Research (ACER) defines maths anxiety as “feelings of unease and worry experienced when thinking about mathematics or completing mathematical tasks” (Buckley, Reid, Goos, Lipp, & Thomson, 2016, p. 158). Maths anxiety causes people to have significant self-doubt in their ability to do maths, causing them great distress.

When a person has maths anxiety, their brain continually thinks about the anxiety rather than the actual maths problem. The brain allocates the working memory and other resources that it would normally use with computations of the maths problem to the anxiety itself, making it very difficult to learn or retain the relevant skills or information (Marshall, Staddon, Wilson, & Mann, 2017). Maths anxiety is highly prevalent. Sadly, between 25% to 80% of the college population in the United States of America has some form of maths anxiety and the percentage is likely to be similar in Australia, given the similarities in culture (Koch, 2018).

Symptoms and causes of maths anxiety

Maths anxiety can easily be identified. The symptoms of maths anxiety range from simple low confidence problems to more complex physical symptoms. If you experience the following, you may have maths anxiety:

low confidence and negative thoughts such as “I am no good at maths”, “I won’t be able to do this”, “I am never going to understand this maths concept” (Department of Education, 2020).

physical symptoms ranging from increased heart rate, increased breathing to a panic attack when thinking about or doing maths (Department of Education, 2020).

The symptoms of maths anxiety are triggered when doing maths or from the thought (anticipation) of doing maths. The level of anxiety will vary from person to person (Department of Education, 2020). While maths anxiety is common, it can be managed or resolved allowing you to succeed in your maths learning journey.

Strategies to reduce maths anxiety

You will be able to recognise when you are starting to feel stressed or anxious and having difficulty trying to complete maths problems. This could include avoiding maths classes, revision and assessment.  At this point, you could develop some methods to help you to relax and unwind. Some different strategies you may be able to use distraction techniques including reflecting on how you feel; leaving the room to do another activity for short periods of time; mindful breathing techniques (such as breathing in for a count of 5, and breathing out for a count of 7); or any other techniques you may already use for reducing anxiety.

It is also helpful to remind yourself of what you can do by returning to a problem that you can do before attempting the problem which caused the stress. More information about mental health resources (including sections on stress and anxiety) can be found in the chapter Successful Connections.

Maths anxiety can be managed in a positive way using six strategies. The strategies will need to be employed over time to see the results. Addressing your maths anxiety using these six strategies also allows you to study maths effectively.

Six strategies you can use to help reduce maths anxiety

Strategy 1: Create a safe, calm and comfortable study environment. When you are in a comfortable environment, you have more scope to use your working memory to understand the maths concepts as it is not occupied with the distractions of a busy or stressful environment.

Strategy 2: Check your self-talk and beliefs about your ability to do maths. Self-efficacy is the belief that we are capable of successfully performing a task, for example studying maths. Self-efficacy influences your confidence and likelihood of success. Changing any negative thoughts about maths to positive thoughts will greatly increase the likelihood of succeeding in maths. For example, if you catch yourself thinking, “I can’t do this”, try to tell yourself “I can do this!” Similarly, if you tell yourself “I am no good at maths”, remind yourself that, “I can improve at maths”.

Strategy 3: Keep up with your coursework. Maths courses tend to build on concepts over the course, so skipping classes or homework makes it very difficult to learn work presented later in the term. Completing your work in order each week will also give your brain the time it needs to make mental maps of the concepts, and store these in your working memory. Memorising or “cramming” does not help you with learning maths effectively, and thus should be avoided.

Strategy 4: Show all the processes (the ‘full working’) as you practice maths problems. When practising the maths concepts with the full working, you are storing these processes in your working memory. This allows you to make a mental map of the concept and increases your understanding of that concept. It also allows your brain to form the connection of where and how to apply the maths concept. This allows for easier recall and application of the process. Your setting out will become automatic, for example, aligning your equal signs throughout the problem using the correct symbols and notations, and adding text to explain what you are doing (See Figure 5.3). When these become automatic, you won’t have to worry about them in your assessment. Practising with all the process will also help you to identify what you don’t understand and know when to seek help.

Strategy 5: Seek help as soon as the need arises. Asking for help can be difficult. However, to succeed with maths (and overcome maths anxiety), it is important that you approach your teaching team (or other maths support services) to seek assistance. Your teaching or university maths support team has extensive experience and can help you by breaking down the concepts into smaller and simpler processes that are easier to understand. You may also have access to peer-facilitated study groups and these can be an excellent source of practical help and encouragement.

Strategy 6: Use timed practice. Timed practice models what you will need to do during a timed assessment item, such as an exam. You will collect or create some problems, then set a timer and work through as many of the problems as you can in this time. Practising in a similar, but less pressured environment than an exam can help you to overcome your anxiety of doing maths in timed situations. Using timed practice can build your confidence in completing different maths questions and build your speed in applying the concepts.

Figure 6.3 This example shows poorly or well laid out maths. In (a), the equal signs do not align, the correct symbol is not used (for the multiplication sign) and units appear out of nowhere. In (b) the example is well laid out. The equal signs align, the maths is centred on the page and the answer is given in context with units.

The use of the six strategies will increase your confidence and help you to form good mental maps within your working memory. Over time, this will reduce your maths anxiety and break down the associated barriers that make studying maths more difficult than it needs to be.

FOCUS ON UNDERSTANDING THE PROCESS

The strategies presented to help reduce maths anxiety are good practice for any student learning maths. To be successful in maths, you will also need to understand the process used for solving maths problems. That means you need to understand why the process works. Understanding the process will help you to remember how to do the maths. To develop understanding, revise and rewrite calculations that you are shown in lectures or tutorials. Things always look easier when someone else is showing you how to do it, compared to when you try at home on your own! Rewriting the steps will make it easier to complete different questions and you will have good notes for revision.

Understanding the process also gives you some flexibility when approaching maths problems. Sometimes there may be more than one method for coming to the right answer. If you understand the processes, you will be able to identify the most effective method to complete the question and then apply it. If a question doesn’t specify a particular method to use to solve the problem, you can also choose a method that suits you best.

APPROACHES FOR STUDYING EACH MODULE

Now that you have some strategies for combatting maths anxiety, and studying maths effectively, you can begin your maths journey at university with a positive mindset. We will now discuss how to study maths, module by module. But, what are modules? Some degrees will have entire courses that focus on maths, such as Fundamental Statistics, Foundation Mathematics, or Algebra and Calculus. Within these courses, the maths concepts are broken up into smaller segments for you to study, often known as modules. Modules allow you to look at one new concept at a time and gradually build your knowledge, experience and confidence. When beginning a new module, try these approaches to make your maths study more manageable:

Work out what the module is about and what you are expected to learn. Before you start any exercises, scan the entire module and check the learning objectives for a summary of what to expect.

Start at the beginning of the module, reading through the text and examples. When you come to an activity, attempt the questions yourself. This will help you to learn the formulae and when and how to apply them, thus developing your problem-solving techniques. It will also give you an idea of what you know and where you need to focus.

Do not skip over any of the study materials. Maths is an iterative process, you will need to develop strong foundations and repeatedly revisit and build upon these foundations.

Summarise the module as you work through it. List any new formulae and problem-solving techniques and take note of anything that you do not understand so you can seek assistance.

Talk about your maths. It is amazing how problems can be clarified by talking with somebody. You can do this with friends, work colleagues, at tutorials (in person or online) or through course discussion forums or groups.

Ensure that you have a complete understanding of the topic that you are studying. If you cannot understand a topic, look for alternative resources that may explain it in a different way, contact the teaching team, or university maths support team.

Contact your teaching team or university maths support team for help if you get stuck. Do this as soon as you have a problem so that you can move on with your studies, and not get behind. Check to see if your university has any maths support services, for example, learning advisors, tutors or peer mentoring programs. Knowing in advance what help is available and how to access it can save precious time and help prevent a small issue from becoming something worse over the course of your studies.

STRATEGIES FOR PROBLEM SOLVING

In many ways, maths is like solving a puzzle, where a question is posed, and you must find the answer. At the heart of this process is what we call problem-solving skills. Problem-solving questions are typically the worded questions you find in the application sections of your materials, or in your assignments. Problem-solving skills are something that can be practised and developed to make you more confident and capable with your maths. Most people find problem-solving difficult, and as such, it is an area they need to spend time developing. Here are some tips to help develop your skills:

Read the question or problem carefully and identify what you are expected to find.

Determine whether any of the information is not needed for solving the problem.

Express the relevant information in mathematical terms, defining any variables that you are given and noting any special conditions.

Break down the problem into smaller parts.

Estimate the answer to the part of the problem that you cannot solve yet and proceed from there.

Decide which of the skills or techniques you have learnt in the course could be applied to solve the problem.

Apply the technique that you think will solve this problem. Try a different technique if the first did not work.

Check that your answer makes sense to the problem.

Even if things haven’t gone quite right, there are problem-solving strategies you can use to help put yourself back on the right path. You can:

Check that you copied down everything correctly.

Scan for errors in your calculations.

Look back at your working and answers to similar questions.

Start with a fresh page where you cannot see what you have done previously.

Read the question aloud and slowly.

Leave the problem for tomorrow (but don’t leave it too long).

Ask for help from your teaching team, university maths support team, study group, or whatever other maths support might be available to you.

MAKING THE MOST OF HELP

Sometimes students need assistance with maths. As discussed, getting help is both a successful strategy for managing maths anxiety and a problem-solving strategy. Here are some suggestions to maximise the benefits of the help you have available:

Be specific as to what you don’t understand — you do not want the tutor to cover areas where you do not need help. Being specific about what you need will likely save you some time.

Attempt to solve the problem(s) yourself first and have your working available so that the tutor can discuss it with you. This will develop your problem-solving skills because you will have thought through the problem. It will also show where your understanding is lacking and where you became stuck.

Attempt similar problems from the study materials or other textbooks/websites that have answers provided, so that you can discuss your problems with the tutor rather than requiring tuition in the basic concepts. This can help tailor the support to your specific needs.

Be organised and specific. Make a written list of problems that need clarifying, including page numbers in the text, along with your working.

TIPS FOR MATHS ASSESSMENT

After you have studied the modules, you will need to complete some assessment. This section provides hints and tips for your assignments and exams.

Maths Assignments

When studying maths, it is essential to develop regular study patterns. Often your tutorial questions will help you to develop the skills needed in your assignments, so do not leave your maths study until just before an assignment is due.

When undertaking a maths assignment, you must express yourself clearly both in English and mathematics. Many students think that doing maths just involves ‘doing the sums.’ However, ‘doing the sums’ is only one part of doing and being involved in maths. In fact, it doesn’t matter how good you are at doing these sums if you cannot communicate your answers or solutions with others. Remember – in your career, you must be able to convince your colleagues or clients that your answer is the appropriate one. Therefore, communicating is just as important in maths as it is in all other subject areas.

Finally, you need to allow adequate time to present your assignment. Just like your other assignments, you need to complete a rough draft and then prepare a final ‘good’ copy. Your markers are looking for assignments that are neat, tidy, with the maths formatted correctly, and with logical, well documented communication (mathematical and English). This could be as simple as following the guides for best practice for maths notation, such as aligning equal signs, centring equations on the page, defining any variables you have used including their units. Your textbooks and study modules are a good guide to how your lecturers are expecting you to format your assignments.

Maths Exams

The best preparation for exams is to work consistently through the semester (or other study period) and keep up to date with the recommended study schedules provided to you in your courses. This includes working through suggested questions and tutorial questions. Practising maths regularly will develop your skills, confidence, and fluency. Practising your setting out will help you to automatically set your work out neatly in the exam.

In the weeks and days prior to the exam, you might like to use these preparation techniques for maths exams:

Review the information about spaced practice in the chapter Preparing for Exams to maximise your exam preparation.

As you have been practising your maths throughout the course, you won’t need to cram the night before the exam. See additional information on cramming in the chapter Preparing for Exams.

Review your notes (and worked examples) and make a concise list of key concepts and formulae. Make sure you know these formulae and more importantly, how to use them.

Work through your tutorial problems again (without looking at the solutions). Don’t just read over them. Working through problems will help you to remember how to do them.

Work through any practice or past exams which have been provided to you. You can also make your own practice exam by finding problems from your course materials. See the Practice Testing section in the chapter Preparing for Exams for more information.

When working through practice exams, give yourself a time limit. Don’t use your notes or books. Treat it like the real exam.

For those who suffer maths anxiety—practice any breathing or other techniques that help you to reduce or manage the anxiety.

Finally, try to get a good night’s sleep before the exam so you are well rested and can concentrate when you take the exam.

Further details about preparing for a maths exam can be found in the chapter Types of Exams.

During your exams, remember to set out and communicate your maths in a way that the marker can follow. Normally the marker is not looking for perfection, but that you have used the correct methods (processes). Once again, communicating what you are doing is just as important as completing the actual calculations.

If you experience maths anxiety, be aware that it may be heightened during timed exams so you will need to remember your strategies for managing it.

CONCLUSION

Maths is an integral part of university study, regardless of which discipline you are studying. This chapter identified the value of studying maths and provided strategies to help students manage maths anxiety. It also presented methods for approaching studying maths in general, how to study single modules, hints for success in problem solving and concluded with tips for how to approach maths assessments. Equipped with these tips and strategies, you are ready to learn and work successfully with maths in your university studies.

Key points

Maths is an important part of the learning journey.

The study of maths trains your brain to think logically, accurately, and carefully.

Maths anxiety is something everyone may experience at different stages in their university studies. The six strategies to help you manage it are: Creating a safe, calm and comfortable environment in which to study, developing positive self-efficacy for maths, practising maths (including all working), seeking help when required, timed practice and understanding the process.

When seeking help, show your tutor your attempt to solve the problem(s) (with your working) so that they can discuss it with you. This will give you the most tailored support.

Develop your problem-solving skills to help with applying the concepts in different situations, including assessments.

Present your maths logically with full working and communication in your assignments.