MERIT Number and Algebra. Algebra. The History of Algebra. Egyptian Algebra. Linear and quadratic polynomials Next sessions:

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1 MERIT Number and Algebra Algebra T polynomials in one variable This session: Linear and quadratic polynomials Next sessions: More on the algebra of polynomials Polynomials and abstract algebra Algebra It is a branch of mathematics which uses symbols to represent numbers and variable quantities. It is used to express relationships between variable quantities. The History of Algebra Algebra may divided into "classical algebra" (equation solving or "find the unknown number" problems) and "abstract algebra", also called "modern algebra" (the study of groups, rings, and fields). Classical algebra has been developed over a period of 4000 years. Abstract algebra has only appeared in the last 200 years. Understanding Algebra: The process involved in learning algebra parallels the historic development of algebra as a symbol system. Rhetorical Stage: (3000 B.C TO A.D. 250) era in which problems and solutions were written out in words. Syncopated Stage (250 to 1591) era in which letters are used to represent unknown quantities and abbrevations were created for operations. Letters represented unknowns for which there was one unique solution. Symbolic Stage: (1591-> present) letters used to represent both the given and unknown quantities. Egyptian Algebra MORE INFO Africa/mad_ancient_egypt_algebra.html#rhind28 1

2 Much of our knowledge of ancient Egyptian mathematics, including algebra, is based on the Rhind papyrus, which was written about 1650 B.C. and is thought to represent the state of Egyptian mathematics of about 1850 B.C. They could solve problems equivalent to a linear equation in one unknown. Their method was what is now called the "method of false position." Their algebra was rhetorical, that is, it used no symbols. Problems were stated and solved verbally. Babylonian Algebra MORE INFO: n.html The mathematics of the Old Babylonian Period ( B.C.) was more advanced than that of Egypt. Their excellent sexagesimal [base 60] led to a highly developed algebra. They had a general procedure equivalent to solving quadratic equations, although they recognized only one root and that root had to be positive. In effect, they had the quadratic formula. They also dealt with the equivalent of systems of two equations in two unknowns. They considered some problems involving more than two unknowns and a few equivalent to solving equations of higher degree They worked out a calendar for the year along with developed units of measurement. As early as 2500 B.C. the Babylonians had established standards for length, weight, and volume. Mathematical and astronomical texts show that the Babylonians came up with the 360 degree circle and also the 60 minute hour. The Babylonians could predict eclipses of the sun and moon. The Babylonian system also had units. They did not have a symbol for zero, but the idea of zero was used. When Babylonians wanted to express zero they just left a blank space in the number they were writing. 2

3 Maya Mathematics MORE INFO: Instead of ten digits like we have today, the Maya used a base number of 20. (Base 20 is vigesimal.) Because the base of the number system was 20, larger numbers were written down in powers of 20. We do that in our decimal system too: for example 32 is 3*10+2. In the Maya system, this would be 1*20+12, because they used 20 as base. It was very easy to add and subtract using this number system, but they did not use fractions. The Mayans devised a counting system that was able to represent very large numbers by using only 3 symbols, a dot, a bar, and a symbol for zero, or completion, usually a shell. Like our numbering system, they used place values to expand this system to allow the statement of very large values. Their system has two significant differences from the system we use: 1) the place values are arranged vertically, and 2) they use a base 20, or vigesimal, system. For example,the number in the third place has a value of 20 2, or 400, times the value of the numeral (whereas the number in the third place in our number system represents 100 (10 2 )). Other Achievements With crude instruments (by today s standards) the Maya were able to calculate the length of the year to be days (the modern value is days). Two further remarkable calculations are of the length of the lunar month. At Copán (now on the border between Honduras and Guatemala) the Mayan astronomers found that 149 lunar months lasted 4400 days, which gives days as the length of the lunar month. At Palenque in Tabasco they calculated that 81 lunar months lasted 2392 days. This gives days as the length of the lunar month. The modern value for the lunar month is days. Roman Algebra MORE INFO: Addition and subtraction were made easy with the concepts of place holders and this was the first real system in base 10. I represents one II means one plus one (two), followed by III (three) IV means five minus one (four). V represents five. VI The V followed by a 'I' means five plus one (six) followed by VII (seven), VIII (eight). IX means ten minus one (nine). X represents ten L represents fifty C represents one hundred D represents five hundred M represents one thousand 3

4 Example 982 = = one thousand minus one hundred = CM 80 = fifty plus ten plus ten plus ten = LXXX 2 = one plus one = II = CM + LXXX + II = CMLXXXII. Hindu Algebra MORE INFO: The Hindu civilization dates back to at least 2000 B.C. Their record in mathematics dates from about 800 B.C. Most Hindu mathematics was motivated by astronomy and astrology. A base ten, positional notation system was standard by 600 A.D. They treated zero as a number and discussed operations involving this number. The Hindus introduced negative numbers to represent debts. The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. Arabic Algebra MORE INFO Abu Ja'far Muhammad ibn Musa Al- Khwarizmi Born: about 780 in Baghdad (now in Iraq) Died: about 850 Known as the father of Algebra He introduced the natural numbers He first reduced an equation (linear or quadratic) to one of six standard forms: 1. Squares equal to roots. x 2 = 2x 2. Squares equal to numbers. x 2 = 2 3. Roots equal to numbers. 10x = 2 4. Squares and roots equal to numbers; x x = Squares and numbers equal to roots; x = 10 x. 6. Roots and numbers equal to squares; 3 x + 4 = x 2. 4

5 Example: x x = 39 Al- Khwarizmi solved this problem as follows:... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square. Symbolic Stage: (1591-> present) letters used to represent both the given and unknown quantities. European Algebra after 1500 At the beginning of this period, zero had been accepted as a number Irrationals were used freely although people still worried about whether they were really numbers. Negative numbers were known but were not fully accepted. MORE INFO Complex numbers were as yet unimagined. Full acceptance of all components of our familiar number system did not come until the 19th century. Algebra in 1500 was still largely rhetorical. Renaissance mathematics was to be characterized by the rise of algebra. In the 16th century there were great advances in technique, notably the solution of the cubic and quadratic equations - achievements called by Boyer perhaps the greatest contribution to algebra since the Babylonians learned to solve quadratic equations almost four millennia earlier. There were also at this time many important improvements in symbolism which made possible a science of algebra. 5

6 The landmark advance in symbolism was made by Viète (French, ) who used letters to represent known constants (parameters). This advance freed algebra from the consideration of particular equations and thus allowed a great increase in generality and opened the possibility for studying the relationship between the coefficients of an equation and the roots of the equation ("theory of equations") Symbolic algebra reached full maturity with the publication of Descartes' La Géométrie in This work also gave the world the wonderfully fruitful marriage of algebra and geometry that we know today as analytic geometry (developed independently by Fermat and Descartes). Polynomial Function Two branches of studying polynomial functions such as: f(x) = a 0 x 0 + a 1 x 1 + a 2 x a n x n Modern Algebra view - theory of solving polynomials by factoring Modeling view solving real world problems modeled by polynomial functions which almost never factor Solving Polynomial Equations- Algebraic Method Set Polynomial Equation equal to zero. Factor the resulting polynomial expression into a product of linear expressions. Reduce the equation to a series of linear equations. This is a classic example of analytic reasoning reducing a more complex problem to one we already know how to solve. Example 2 f ( x) = x + x 10 When does f(x) = 20? Zero Product Rule If a b = 0, then a=0 or b=0 Question: What property of Modern Algebra that we have studied supports this statement? Question: What does this indicate about the replacement set or domain set? 6

7 Linear equation a *x = b a+x = b ax = b a*x=b Linear equation ax+b = cx+d Two operations implicit need a Field Theorem If a, b, c, and d are elements of a field F and a c, then the equation ax+b=cx+d has exactly one solution in F. If a=c then Babylonian problem Find two numbers whose sum is 10 and whose product is

8 ax Quadratic formula 2 + bx + c = 0 b ± x = b 2 4ac 2a 8