The Root of the Problem: A Brief History of Equation Solving Alison Ramage Department of Mathematics and Statistics University of Strathclyde [email protected] http://www.mathstats.strath.ac.uk/ Mathematical Association 2009 – p.1/30 Background and References Great Moments in Mathematics H. Eves • Mathematics and Mathematicians P.Dedron & J. Hard • The Mathematics of Great Amateurs J.L. Coolidge • A History of Mathematics C. Boyer & U.C. Merzbach • MacTutor History of Mathematics Archive http://www-groups.dcs.st-and.ac.uk:80/• history/ ∼ Mathematical Association 2009 – p.2/30 Background and References Great Moments in Mathematics H. Eves • Mathematics and Mathematicians P.Dedron & J. Hard • The Mathematics of Great Amateurs J.L. Coolidge • A History of Mathematics C. Boyer & U.C. Merzbach • MacTutor History of Mathematics Archive http://www-groups.dcs.st-and.ac.uk:80/• history/ ∼ Bluff Your Way in Maths R. Ainsley • Mathematical Association 2009 – p.2/30 Background and References Great Moments in Mathematics H. Eves • Mathematics and Mathematicians P.Dedron & J. Hard • The Mathematics of Great Amateurs J.L. Coolidge • A History of Mathematics C. Boyer & U.C. Merzbach • MacTutor History of Mathematics Archive http://www-groups.dcs.st-and.ac.uk:80/• history/ ∼ Bluff Your Way in Maths R. Ainsley • “Mathematics is a unique aspect of human thought, and its history differs in essence from all other histories.” Asimov (1920-1992) Mathematical Association 2009 – p.2/30 Notation RHETORICAL Rhind Papyrus c. 1650 BC • Demochares has lived a fourth of his life as a boy, a fifth as a youth, a third as a man and has spent thirteen years in his dotage. How old is he? Mathematical Association 2009 – p.3/30 Notation RHETORICAL Rhind Papyrus c. 1650 BC • Demochares has lived a fourth of his life as a boy, a fifth as a youth, a third as a man and has spent thirteen years in his dotage. How old is he? SYNCOPATED Diophantus c. 250 AD • 1 2345678910 αβγδǫζξηθ ι ς: unknown ∆γ: unknown squared κγ : unknown cubed 3 2 κγα∆γιγςη x + 13x + 8x unknown cubed 1, unknown squared 13, unknown 8 Mathematical Association 2009 – p.3/30 Notation cont. SYMBOLIC • +, Widman 1489 − √ Rudolf 1525 = Recorde 1557 unknowns=vowels Viète 1570 knowns=consonants unknowns=lateletters Descartes 1630 knowns=early letters >,< Harriot 1631 Mathematical Association 2009 – p.4/30 Notation cont. , , π Oughtred 1640 × ∼ Wallis 1650 ∞ f(x) Euler 1750 n! Kramp 1800 Mathematical Association 2009 – p.5/30 Notation cont. , , π Oughtred 1640 × ∼ Wallis 1650 ∞ f(x) Euler 1750 n! Kramp 1800 “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” Hilbert (1862-1943) Mathematical Association 2009 – p.5/30 Notation cont. , , π Oughtred 1640 × ∼ Wallis 1650 ∞ f(x) Euler 1750 n! Kramp 1800 “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” Hilbert (1862-1943) “A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.” Eves (1911-2004) Mathematical Association 2009 – p.5/30 Archimedes (287 BC - 212 BC) Greek mathematician and • astronomer lived and worked in Syracuse • invented war machines • On Floating Bodies • slain by a Roman soldier • On the Measurement of the Circle Geometric methods for calculating square roots using circles with circumscribed hexagons (similar to the Babylonians) Mathematical Association 2009 – p.6/30 Archimedes (287 BC - 212 BC) Greek mathematician and • astronomer lived and worked in Syracuse • invented war machines • On Floating Bodies • slain by a Roman soldier • On the Measurement of the Circle Geometric methods for calculating square roots using circles with circumscribed hexagons (similar to the Babylonians) “ǫνρηκα!” Mathematical Association 2009 – p.6/30 Diophantus (c. 200 BC - 284 BC) Greek mathematician • lived and worked in Alexandria • allowed positive rationals as • solutions and coefficients Arithmetica Collection of 130 problems in solving equations (although only 6 of the original 13 books survive). Introduced algebraic symbolism and Diophantine equations. Mathematical Association 2009 – p.7/30 How old was Diophantus? Here lies Diophantus, the wonder behold. Through art algebraic, the stone tells how old: ’God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father’s life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life. Mathematical Association 2009 – p.8/30 How old was Diophantus? Here lies Diophantus, the wonder behold. Through art algebraic, the stone tells how old: ’God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father’s life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life. 1 1 1 1 3 L + L + L + 5 + L + 4 = L L = 9 L = 84 6 12 7 2 ⇔ 28 ⇔ Mathematical Association 2009 – p.8/30 Marignal notes in Arithmetica Fermat’s Last Theorem If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c. Mathematical Association 2009 – p.9/30 Marignal notes in Arithmetica Fermat’s Last Theorem If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c. “ I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.” Pierre de Fermat (1601-1665) Mathematical Association 2009 – p.9/30 Marignal notes in Arithmetica Fermat’s Last Theorem If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c. “ I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.” Pierre de Fermat (1601-1665) “ Thy soul, Diophantus, be with Satan because of the difficulty of your theorem.” Maximus Planudes (1260-1330) Mathematical Association 2009 – p.9/30 Mohamed ibn-Muso al-Khwarizmi (c.790-840) Arabian librarian • lived and worked in Howarizmi • calculated latitudes and • longitudes for 2402 localities as a basis for a world map wrote about sundials and the • Jewish calendar Hisâb al-jabr w’almuqâbalah Mathematical Association 2009 – p.10/30 Mohamed ibn-Muso al-Khwarizmi (c.790-840) Arabian librarian • lived and worked in Howarizmi • calculated latitudes and • longitudes for 2402 localities as a basis for a world map wrote about sundials and the • Jewish calendar Hisâb al-jabr w’almuqâbalah al-Khwarizmi algorithm, al-jabr algebra ≡ ≡ Mathematical Association 2009 – p.10/30 Science of Reunion and Opposition squares equal to roots x2 = 5x squares equal to numbers x2 = 4 roots equal to numbers 5x = 15 squares and roots equal to numbers x2 + 10x = 39 squares and numbers equal to roots x2 + 21 = 10x roots and numbers equal to squares 3x + 4 = x2 Mathematical Association 2009 – p.11/30 Science of Reunion and Opposition squares equal to roots x2 = 5x squares equal to numbers x2 = 4 roots equal to numbers 5x = 15 squares and roots equal to numbers x2 + 10x = 39 squares and numbers equal to roots x2 + 21 = 10x roots and numbers equal to squares 3x + 4 = x2 “ With a name like this under your belt, you can bluff your way past even a bona fide mathematician.” Bluffer’s Guide to Maths Mathematical Association 2009 – p.11/30 Completing the Square x2 + 10x = 39 e h s x f 5/2 g Mathematical Association 2009 – p.12/30 Completing the Square 1. x2 term: centre square s x2 + 10x = 39 e h s x f 5/2 g Mathematical Association 2009 – p.12/30 Completing the Square 1. x2 term: centre square s 2. 10x: add e, f, g, h, each 5/2 units wide x2 + 10x = 39 (so area of each is 10x/4 units) e h s x f 5/2 g Mathematical Association 2009 – p.12/30 Completing the Square 1. x2 term: centre square s 2. 10x: add e, f, g, h, each 5/2 units wide x2 + 10x = 39 (so area of each is 10x/4 units) 3. total area of solid figure is 39 units e h s x f 5/2 g Mathematical Association 2009 – p.12/30 Completing the Square 1. x2 term: centre square s 2. 10x: add e, f, g, h, each 5/2 units wide x2 + 10x = 39 (so area of each is 10x/4 units) 3. total area of solid figure is 39 units e 4. complete the square: add four squares, each of area 25/4 units h s x f 5/2 g Mathematical Association 2009 – p.12/30 Completing the Square 1. x2 term: centre square s 2. 10x: add e, f, g, h, each 5/2 units wide x2 + 10x = 39 (so area of each is 10x/4 units) 3. total area of solid figure is 39 units e 4. complete the square: add four squares, each of area 25/4 units h s x f 5. area of large square is 39 + 25 = 64 units 5/2 g Mathematical Association 2009 – p.12/30 Completing the Square 1. x2 term: centre square s 2. 10x: add e, f, g, h, each 5/2 units wide x2 + 10x = 39 (so area of each is 10x/4 units) 3.