Hamad Ameen, Abdulqader Othman (2025) FINITE MATHEMATICS. koya university.
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Abstract
This book is based on lectures developed by the author for B.SC and M.SC. Students at Koya University, Department of Mathematics, and on the lectures taught by the author in postgraduate studies to master’s and doctoral students at Salahaddin-Erbil and Sulaymaniyah Universities, Department of Mathematics, as well the book is also the product of the accumulated notes from teaching under the author’s postgraduate studies. In mathematical structures and mathematics education, finite Mathematics is a syllabus and method that depends on foundations of mathematics, while it is independent of calculus. Thereby, since we devoted ourselves to writing the book Foundations of Mathematics (Hamadameen, 2022), it was necessary and inevitable for us to write this book in order to help the student and reader apply mathematical problem solving and logical thinking to real world phenomena, making it an important field of knowledge for students who are pursuing careers in the field of business and its branches (Applied mathematics, social sciences, computer sciences, and applied sciences in the fields of statistics, probabilities, medicine, physics, chemistry, biology, most engineering branches, and other practical professional specializations). The book takes into consideration the necessity of the contents of this book for students to study mathematics as well as the other applied sciences mentioned above. The contents of this book and its academic goals aim at: (i) Helping students to be fully familiar with the mathematical induction and its principals. When can mathematical induction be used, what are the conditions for it, what is its application, and where can it be used in real life? (ii) Motivating students to learn about complex numbers, their properties, their representation in the Cartesian plane, and algebraic operations on them. Polar form of complex numbers, and applications of De Moivre’s theorem in the field of complex numbers. What conjugate numbers, their properties, and what are the relations between complex numbers and their conjugates? In addition to absolute value inequalities of complex numbers, their square roots, and roots of unity. (iii) What are polynomials, and their properties? Quotient of polynomials, long division algorithm of polynomials, their roots, and duplicate roots. Greatest common factor of polynomials, Cardian’s method to solve cubic equations, quartic equations, and method to solve them. (iv) Helping students to be familiar with numerical solutions to nonlinear equations and when to applies to such solutions. Also, helping the students to find approximate values for the roots of nonlinear equations using some practical methods, including Descartes’ sign rule and Horner’s method. In addition, presenting numerical methods to the student to find the approximate values of the roots of equations, including the bisection method, Newton- Raphson’s method, secant method, Birge-Vita method, and Graeffe’s root-squaring method. And making the student understand that all of these methods are practical, applicable, and have a solid algorithm for application on the computer. (v) Considering matrices to students and expressing the system of linear equations as matrices. Types of matrices and their properties, operations on matrices in addition to matrices and linear vectors as matrices, as well as identifying dependent and independent linear vectors and how to find solutions to the linear systems in the form of matrices via operations on rows and columns of the matrices. Moreover, considering determinants as an inevitable result of the matrices, and the related concepts to them. Types of determinants, how to find them, and the general formula for finding determinants. Permutations and determinants, and the relationship between them, and the main property of determinants. Furthermore, Inverse of matrix, elementary transformations of the matrix, and inverse transformations. Equivalence, norm, form, and rank of matrices. Besides, inserting matrix inverse methods. And, how getting inverse of complex matrix, and what are the methods to find the inverse of complex matrix? (vi) Encouraging students to turn to the system of linear equations and their role in solving real life problems and how to formulate them. Provide some methods like; Cramer’s method, Gauss’s method, and some other methods. In addition to those methods and their operations on the computer. (vii) Explaining the concepts of eigenvalues and eigenvectors to the student and explaining their advantages. What are eigenvalues and eigenvectors through changing direction transformations and eigenvalues? Applications of eigenvalues and eigenvectors, and their properties of matrices. Moreover, introducing two different methods for finding eigenvalues. (viii) Considering each of permutations and combinations and how to formulate them. What is the basic principle in arithmetic? The basic properties of permutations and combinations and the difference between them. Embarking on the binomial theorem and the polynomial theorem. It is noteworthy that, most of the theorems, corollaries, and exercises in this book are adapted from the references (Balfour and Beveridge, 1972; Britton and Snively, 1954; Brown and Churchill, 2009; Conte and De Boor, 2017; Fraser, 1958; Froberg, 1965; Goult, 1974; Hoffman and Kunze, 1967; Hohn, 1972; Knopp, 1952; MacDuffee, 1954; Parsonson, 1970; Ralston and Rabinowitz, 2001; Ralston, 1965; Strang, 2006; Uspensky, 1948; Wilkinson, 1971). The contents of this book are organized as follows: chapter 1 is dedicated to discussing to the mathematical induction, the basic concepts, and the principal of it. Chapter 2, deals with the complex numbers, properties, Cartesian representations, polar form of a complex numbers. In addition to De Moivre’s theorem, complex numbers and their conjugates and roots of complex numbers. Chapter 3 is devoted to polynomials in which it defines the concept of polynomials, the properties of polynomials, and the long division algorithm for polynomials. It also shows the relationship between roots and polynomial equations, repeated roots, the greatest common factor of polynomials, solving cubic equations using Cardan’s method and solving quadratic equations. Numerical solution of nonlinear equations, finding the differential via Horner’s method, and Numerical methods for finding approximate values of them took their place in chapter 4. Chapter 5 deals with the matrices, types of matrices, operations on matrices, matrices partition, vector expression, linearly dependent vectors, and linearly independent vectors. Chapter 6 deals with the determinants, types of determinants, algorithm for finding determinant of a matrix of third order or higher, General methods for finding determinants, permutations and the determinant, and properties of determinants. Chapter 7 is about inverse of a matrix, matrix inverse methods like; the method of adjoint matrix, and the method of elementary transformations. In addition to the method of transformations on rows including; Jacobian method, the method of Triangularization, and the method of Escalator. Moreover, considered inverse of a complex matrix, and method to find the inverse of it. Chapter 8, deals with numerical solution of a system of linear equations, mathematical formulation of linear system, solutions for systems with equal equations and variables including; Cramer’s method, Gauss’s method, Gauss’s method and row echelon form, coefficient matrix partition method, and matrix inverse method. Furthermore, the methods and their operations on the computer. eigenvalues and eigenvectors, and their features. Chapter 9 is about eigenvalue and eigenvector through changes direction, transformations and eigenvalues, polynomial equation of degree n in eigenvalue, eigenvalues and eigenvectors of matrices. Conclusions from eigenvalues, eigenvectors and traces. Methods for finding eigenvalues like; LU method, and Jacobi method. Finally, chapter 10 considered permutations and combinations, and their formulations. Basic principle of the arithmetic of of permutations, and combinations. In addition to difference between combination and permutations, binomial theorem, multinomial theorem, and harmonic series with its properties. And, summation by fragmentation. It is worth to be mentioned that theorems and their corollaries are printed in italics, while, the end of the proofs to theorems and corollaries are indicated by the symbol ♦. Abdulqader Othman Department of Mathematics, Faculty of Science & Health Koya University 2025
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| Additional Information: | پوختە ئەم کتيبە لەسەر بنەمای ئەم وانانە نوسراوە کە نوسەر لە خوێندنی بەرايی و ماستەر و دکتۆرا ووتويەتيەوە لە بەشی ماتماتيک لە زانکۆی کۆيە. هەروەها لەسەر بنەمای خويندنی باڵای ماستەر و دکتۆرا لە زانکۆکانی صلاح الدين-هەولێر و زانکۆی سلێمانی کە نوسەر ئامادەيی هەبوو. لە داڕشتنی بيرکاری و فێربونيدا، بيرکاری سنوردار کۆڵەگەيەکە لە کۆڵەگە بنچينيەکانی کە جيايە لە داتاشرار و تەواوکاری. لەو کاتەی کتێبی بنچينيەکانی بيرکاريمان)حمدامين ٢٠٢٠ ( دەنوسی بيرمان لەوە کردەوە کە پێويستە کتێبی ماتماتيکی سنورداريشمان هەبێت، تاکو زانستی بيرکاری گشتگيريانە پێکەوە بنرێت. بە واتايەکی تر نوسينی ئەم بەرهەمە مەرج و ويست بوو بۆ بواری فێربونی خوێندکاران لە بوارە ژيری و تيۆری و توانای جێبەجێکردن و پراکتيکی لە: )ماتماتيکی پراکتيک، زانستە کۆمەڵايەتيەکان، کۆمپيوتەر، زانستە کارگوزاريەکانی ئامار و ئەگەرکان، پزيشکی، فيزيا، کيميا، زيندەوەرزانی، زۆربەی زانستە ئەندازەييەکان، وە هەروەها لە بواری تەکتيکە زانستيەکان(. ناوەڕۆکی ئەم کتێبە ڕەچاوی پێداويستی خوێندکار دەکات لە گشت ئەو بوارانەی کە لە سەرەوە باسکران، هەروەها ئامانجە ئەکاديميەکانيش بريتين لە:- ( i ( پێناساندنی دەرئەنجامی بيرکاری و پرەنسيپەکانی بە خوێندکاران و چۆنيەتی بەکارهێنانی لە ژيانی ڕۆژانە و بوارە پراکتيکيەکان. ( ii ( هاندانی خوێندکار بۆ ئاشنابوون بە ژمارە ئاوێتەکان و دەستنيشانکردنيان لە ڕووتەختدا و کرداری جەبری بە سەرياندا. دەربڕينی ژمارە ئاوێتەکان بە جەمسەرە پێوەر، بەکارهێنانی سەلمێندراوی دی مواڤر لە بواری ژمارە ئاوێتەکان. دۆزينەوەی ئاوەڵ ژمارەی ئاوێتە و جگە لە لاسەنگەی ژمارەی ئاوێتەيی و ڕەگەکانيان. ( iii ( ناساندنی زۆر ڕادەدارەکان و تايبەتمەنديان، خەوارزمی دابەشکردنيان بەسەر يەکدا. دۆزينەوەی گەورەترين کۆلکەی هاوبەش و بچوکترين چەند جارەی هاوبەش. چۆنيەتی شيکارکردنی هاوکيشەی سێجا و پلە چوار. ( iv ( يارمەتيدانی خوێندکار بۆ دۆزينەوەی شيکاری ژمارەيی بۆ هاوکێشە ناهێڵيەکان. هەروەها دۆزينەوەی نرخی نزيکی ڕەگی هاوکێشە ناهێڵيەکان بە ڕێگای ديکارت و هۆنەر لەگەڵ چەند ڕێگايەکی تری پراکتيکی وەکو: نيوتن-رافسۆن و بێرج-ڤيتا و گريف کە توانای پراکتيکيان هەيە لە زۆر بواری ژيان. ( v ( ناساندنی ڕيزکراوەکان و کردار لە سەريان و لەگەڵ جۆری ڕيزکراوەکان و تايبەتمەنديەکانيان. ناساندنی بڕ و ئاڕاستەکان و دەستنيشانکردنی جۆرەکانيان لە ڕووی وابەستە و سەربەخۆ. گۆڕينی ڕيزکراوەکان بە کردار لەسەر ڕيز و ستونەکانيان. ( vi ( ناساندنی دياريکەری ڕيزکراوەکان و ڕۆڵيان لە جێبەجێکردنی ڕيزکراوەکان لە بوارەکانی ژيانی پراکتيکی. هەروەها پشت بەستن بە دياريکەر بۆ دۆزينەوەی هەڵگەڕاوەی ڕيزکراوەکان. ( vii ( هاندانی خوێندکار بۆ بەکارهێنانی ڕێگاکانی کرامەر، گاوس و چەند ڕێگايەکی تر بۆ دۆزينەوەی شيکاری چەند کێشەی ڕاستەقينەی ژيان و ڕاپەڕاندنيان بە کۆمپيوتەر. ( viii ( بەکارهێنانی چەمکەکانی بەها تايبەتيەکان و ئاڕاستە تايبەتيەکان و ڕوونکردنەوەی ڕۆڵيان لە گۆڕينی شوێن و ئاڕاستە و گەورەکردن و بچوککردنەوەی تەنەکان لە ڕووتەخت و بۆشاييدا و گرێدانيان بە چەمکی ڕيزکراوەکان. ( ix ( بەکارهێنانی چەمکی گۆڕين و گونجێن و ئەگەرەکانيان لە پرەنسيپی بنچينەيی ژماردن و تايبەتمەنديەکان. دەرخستنی جياوازی نێوانيان و جێبەجێکردنی سەلمێندراوی دوو ڕادەيی و سەلمێندراوی زۆر ڕادەداری. ئەم کتێبە پشتی بە زۆر سەرچاوەی زانستی و ئەکاديمی بەستوە بۆ چەسپاندن و ناساندن و شيکاری چەمکە وەرگيراوەکان و کێشە چارەسەرکراوەکانی ناواخنی بەندەکان و ڕاهێنانەکانی. ئەم کتێبە لە ١٠ بەش پێک دێت کە بەم جۆرە ڕێکخراوە: -بەشی يەکەم: دەرئەنجامی بيرکاری. -بەشی دووەم: ژمارە ئاوێتەکان. -بەشی سێيەم: زۆر ڕادەدارەکان. -بەشی چوارەم: شيکاری ژمارەی ناهێڵيەکان. -بەشی پێنجەم: ڕيزکراوەکان. - بەشی شەشەم: دياريکەرەکان. - بەشی حەوتەم: دژە ڕيزکراوەکان. - بەشی هەشتەم: شيکاری ژمارەيی سيستەمی هاوکێشە هێڵيەکان. - بەشی نۆيەم: بەها تايبەتەکان و ئاراستەبڕە تايبەتەکان. - بەشی دەيەم: گۆڕين و گونجێنەکان. لە کۆتايشدا هێما بەسەرچاوە بەکارهاتوەکان کراوە و کتێبەکەش ناواخن کراوە. |
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science and Health > Department of Mathematics |
| Depositing User: | Mr. Rebwar Mohammed Jarjis |
| Date Deposited: | 13 Oct 2025 07:33 |
| Last Modified: | 13 Oct 2025 07:33 |
| URI: | http://eprints.koyauniversity.org/id/eprint/506 |
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