My High School Math Notebook - Smarandache Notions

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Student Florentin Smarandache(1973 – 1974)Râmnicu Vâlcea (Romania)My High School Math NotebookVol. 1[Arithmetic, Plane Geometry, and Space Geometry]Educational Publishing20141

Copyright 2013Education Publishing1313 Chesapeake AvenueColumbus, Ohio 43212USAPeer-Reviewers:Prof. Angelo de Oliveira, Unir – Departamento deMatemática e Estatística, Ji-Parana, RO, Brazil.Said Broumi, Univ. of Hassan II Mohammedia,Casablanca, Morocco.Prof. Xingsen Li, Ningbo Institute of Technology,Zhejiang University, Ningbo 315100P. R. China.EAN: 9781599732602ISBN: 978-1-59973-260-22

PrefaceSince childhood I got accustomed to study with a pen in my hand.I extracted theorems and formulas, together with the definitions, from my text books.It was easier, later, for me, to prepare for the tests, especially for the final exams at theend of the semester.I kept (and still do today) small notebooks where I collected not only mathematical butany idea I read in various domains.These two volumes reflect my 1973-1974 high school studies in mathematics.Besides the textbooks I added information I collected from various mathematical booksof solved problems I was studying at that time.In Romania in the 1970s and 1980s the university admission exams were verychallenging. Only the best students were admitted to superior studies. For science and technicaluniversities, in average, one out of three candidates could succeed, since the number of placeswas limited. For medicine it was the worst: only one out of ten!The first volume contains: Arithmetic, Plane Geometry, and Space Geometry.The second volume contains: Algebra (9th to 12th grades), and Trigonometry.Florentin Smarandache3

Table of ContentsPreface . 3Table of Contents . 4ARITHMETIC . 11Systems of numeracy . 12Numeration systems in the decimal base . 13Positional system . 13Transformation of a number from a given base to another base . 13Transformation of a number from a given base to base 10. . 13Transformation of a number from base 10 to another base . 13Transformation of a number from a random base to another base different of 10 . 13Addition . 14Properties of Additions . 14Subtraction . 14Properties of subtraction . 14The subtraction in a given positional number system . 15Multiplication. 15The properties of multiplication . 15The multiplication of the numbers written in a given positional number system . 15The number of the digits of a product . 15Division . 16The properties of division . 16Division with remainder . 16Partial remainders. 16Algorithm . 17Measuring quantities . 17Arithmetic mean. 17Another mode to compute the arithmetic mean . 17Properties of arithmetic’s mean . 18The square of deviations . 18Pondered average . 18Problems of mixtures and alloy . 18Geometric mean . 194

Harmonic mean . 19Methods of solving arithmetic problems . 19The figurative method . 19The method of reduction to the same comparative term . 20The method of the false hypothesis . 20The method of resolving by starting from the end to the beginning of a problem. 21Combined problems. 21Divisibility of the natural numbers . 21The rules of divisibility . 21Divisibility criteria. 22Theorems referring to Euclid’s algorithm . 23Euclid’s algorithm (Euclidean Algorithm) . 23Consequence of Euclid’s algorithm . 23Properties. 24Common multiples . 24Diophantine equations of first degree with two unknowns . 24Prime Numbers . 25Properties of the divisors of a composed number . 26The sieve of Eratosthenes (III century BC) . 26Ordinary fractions . 27Comparison of fractions . 27Amplification of fractions. 27Simplification of a fraction . 28Solved problems. 28Decimal fractions Decimal numbers) . 29How to transform a decimal number to a number of another numeration bases. 29The division of decimal numbers . 29Transformation of ordinary fractions in decimal fractions . 30Simple periodical fractions. 30Mixed periodical functions . 30The transformation of the periodical decimal fractions in ordinary fractions . 30Resolved problems. 30Fractional units or principal fractions . 32Approximate calculations . 335

Absolute error . 33Approximate values resulted from computations . 33Operations with approximated numbers . 33Addition. 33Subtraction. 34Multiplication . 34Division . 35The quotient’s error . 35Relative errors . 36The relative error of a product . 36The relative error of a ratio. 36Ratio and proportions. 37The fundamental property of a ratio . 37Proportions . 37The fundamental property of a proportion . 37Derived proportions . 37Average proportions . 39Multiple equal proportions . 39The division of a number in proportional parts with given numbers . 39The division of a number in invers proportional parts with given numbers . 39Direct proportional measures . 40Invers proportional measures . 40The fundamental rule of proportions . 40The compound proportions’ fundamental rule . 40Percentages . 41Ratio to percentage . 41Successive percentages. 42Periodical simple fractions . 42PLANE GEOMETRY . 44Lines. 45Angles . 46Triangles . 47Important lines in a triangle . 47The properties of an equilateral triangle . 48The properties of the isosceles triangle . 486

The cases of equality of two random triangles . 49The cases of equality of two rectangle triangles . 49Relations between the sides and the angles of a triangle . 49Inequalities between the sides of a triangle . 49Angles formed by two parallel lines intersected by a secant. . 50Geometrical loci . 51Quadrilaterals . 51Parallelogram . 51Rhombus . 52Rectangle. 52Square . 52Trapezoid . 52Geometrical loci . 54Circle . 55The position of point with respect to a circle . 55The position of line with respect to a circle . 55The position of two circles . 55Arc subtended by a given angle . 58Inscribable quadrilaterals . 58First Theorem of Ptolemy . 59Second Theorem of Ptolemy. 59The power of a point in relation to circle . 59Excircle or escribed circle to triangle . 60Regular polygons . 62The common tangent exterior of two circles. 62The common tangent interior of two circles . 63Quadrilateral complete . 63Gauss’ Theorem .

1 . Student Florentin Smarandache (1973 – 1974) Râmnicu Vâlcea (Romania) My High School Math Notebook . Vol. 1 [Arithmetic, Plane Geometry, and Space Geometry

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