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Counting numbers. Before 1200 AD Different cultures had it’s own system of symbols and rules to produce counting numbers. More advanced cultures also created “the fractions” and “decimals numbers” to solve problems involving divisions and the most advanced one could solve problems involving linear and quadratic equations. The following are a few more important systems: –Babylonian System. 2000 BC –Egyptian System. 2000 BC –Indo-Arabic System. 1400 BC –Chinese System. 1200 BC –Maya System. 1200 BC –Roman system. 600 BC –Greek system. 500 BC

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Babylonian used a sexagesimal system (base 60) of counting numbers, using two cuneiform (wedge) symbols. They used for 1 & for 10 ( to be reading in a vertical way ). 5(10) = 50 7x(1) = 7 57 2(60) = 120 3(10) = 30 4 (1) = 4 154 For numbers greater than 59 they use a space (acting like 0) For fractions (with numerator 1) the used a line over the denominator. Babylonian System. (2000+ BC) For example: ½ =, For examples: 57 was written 154 was written 1/5 =

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Egypt System 2000 BC Egyptians use a base 10 systems of hieroglyph for numerals, by this we mean that they have separate symbols for one unit, one ten, one hundred, one thousand and so on. They used 1) | for 1, || for 2, …,. 2) for 10, for 20, … 3) C for 100, CC for 200, … Example : 176 = They could solve arithmetic problems (including powers of numbers, decimals and fractions), linear equations and quadratic equations. CCCCCCC. . |||||||| C |||||| 718 =

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Indo-Arabic System 1400 BC They used a positional decimal system, using the cero to hold the place. Indi mathematics was very advanced. By 776 A.D. the Arab empire was beginning to take shape and one of the greatest advances was the introduction of "Arabic" numerals. The "Arabic" numerals were influenced by India's mathematics. It is a system based on positional values and a decimal system of tens. The following were the symbols used by that time and after. Western Arabic or Hindu-Arabic Numerals Modern Arabic Numerals developed from them. By 100 A.D. they used Brahmi numerals:

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Chinese System 1200 BC Around the14th century B.C. the first true evidence of mathematical activity was found in numeration symbols on tortoise shells and flat cattle bones. Most information about the early Chinese mathematics was lost or destroyed over the centuries; in particular, in 213 B.C the Emperor Shih Huang-ti ordered the burning of books that they used. 1)9 Chinese different symbols for 1,2,3,4,5,6,7,8,9 2)Special symbols represent the powers of 10. Chinese system is decimal and a positional one (very similar to our actual system) using the 0 from the beginning. Example: 348 as 3(100)4(10)8. ( 348 ? 1_2 =102 ?) The oldest existing texts, containing formal mathematical, goes to 300 B.C and contains an accurate process of division for finding out the square root of numbers and geometrical theorems including “Pythagorean Theorem”. By 130 B.C. the mathematician Shu used 3.1622 as value for .

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1200 B.C. Maya System They used a base 20 system, using the following symbols: 1)a dot for 1 a segment for 5, and combination of these symbols to represent number 1 tru19. For example = 7, = = 19 2)special symbol for 20 (like our cero), 3)for number greater than 20 they used a positional system, starting again with , , … for 20, 20 2 … 322 = 16(20) +2 = 2007 = 5(20 2 )+7 = Examples : They could solve arithmetic problems (including powers of numbers, decimals and fractions). Maya calendar was very good and more accurate than the Egyptians or Babylonians. This lead us think that his value for (and his mathematics) should be at least as good as the Egyptians.

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Greek system 500 B.C. They used the alphabet letters followed by an apostrophe, to represent number in the following way. 12345 678 9 Units digamma Tens koppa Hundreds sampi 1)The 9 first letters ( , , ,…, ) of the alphabet to represent 1, 2, …, 9. 2)The following 9 letters ( , ,…, koppa ) to represent 10, 20, …,90. 3)The last 8 letters ( , ,, … ) plus an extra symbol called “ sampi” (?) used for 100, 200,., 900. 4)Starting again with for1000 and so on. Later digamma and koppa were eliminated from the alphabet. Example: 334 = 300+30+4 = and 2334 =2000+300+30+4 = + =23+15 =38 =

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600 B.C. Roman system. Using the Greek’s idea, the Etruscans create a counting system using letters with different values: I, V, X, L, C, D and M representing respectively 1,5, 10, 50, 100, 500 y 1000. Any letter written on the right (left) of another of greater value add (subtracts) its value. Example XVII = 10+5+1+1=17, IX = 10 – 1 = 9, XXIV = 10+10 + (5 -1) = 24 and CDXLVIII is 448. A letter can’t be repeatedly used more than 3 times, for example 9 = IX and not VIIII. This system was used by roman’s up to 1200 AD and then they adopted the Hindu-Arabic system.

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Numbers After 1200 A.D. After Marco Polo trips to India the Indo-Arabic system was adopted and is the one we use actually. The set of counting number {1,2,3,4,5, n, n+1, …} is represented by N. When we have an expression involving numbers and operation it’ s necessary instructions in which order the operations should be performed. Hierarchy of the operation 1) Exponent and radicals 2) Multiplication and division 3) Addition and subtraction In each category operations have to performed from left to right

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Besides addition and multiplication are associative and commutative operations. Furthermore the multiplication is distributive with respect to addition. Examples: 6+3=3+6, 5·4=4·5 (3+2)+6 = 3+(2+6), 2· (3 · 4)= (2· 3)·4 5·(2+4) = 5·2 + 5·4

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For example from 0+0 = 0 multiplying both side for 4 4(0+0) = 4·0 +4·0 subtracting 4·0 both sides 4·0 - 4·0 = 4·0 +4·0 - 4·0 0 = 4·0 {…-3, -2, -1, 0, 1, 2 3, …} is the set of the integers, represented by Z. The hierarchy of operations on the operation is preserved on the integers. This required extra regulation for multiplication and division. For any counting number the same argument gives: n·0 =0 After 1300 A.D. the zero and the negative numbers were added to the number system. Defining -5 the number such that 5+(-5)=0 Again the hierarchy of operations provides information on working with integers. For example from 5+(-5) = 0 multiplying both sides by 4 4· [5+(-5)] = 4·0 using the distributive Law 20 + 4·(-5) = 0 implies that 4·(-5) = -20 Similarity from -4[5+(-5)] = 0 we get -20+(-4)·(-5) = 0, so (-4)·(-5) =20

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As we know every a rational number it can be written as a decimal number were the decimal part has a finite or periodic infinite numbers decimals. For example: Given de periodic decimal 12.3565656…find the fraction form for it. Let’s call x = 12.3565656…, so 1000x = 12356.565656… 10x = 123.565656… 990x = 1235-123 = 12233 x = 12233/990=203/90. The set of all positive or negative fractions (the whole numbers are fractions with 1 as denominator) it is was called the set of rational numbers, represented by Q. Reciprocally any number which decimal part has a period can be represented as ratio of two integers. In North of Netherlands Simon Stevin used decimals in the late 1500 ( Chinesse and Persian used many centuries before). Examples: 2 = 2.0, 3/2 = 1.5, 3/25 = 0.12, and 2/3 = 0. 666…, 14/15= 0.933333…, 125/84=1.488095238809523880952388…

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Since times of Pythagoras (400 AD) is known that sqrt(2) is not a fraction. Assume that exist two integers a, b such that sqrt(2) =a/b where a/b is in the simple form (cannot simplify ) Squaring both sides we get 2=a 2 /b 2 e.g 2a 2 = b 2 But a 2 & b 2 both have and even number of 2 in the prime decomposition, meaning that the left side has and odd number of 2 and the right and even number of 2, which impossible. So 2a 2 is not equal to b 2. In other words there is not such integers a&b. So it’s was necessary to add all decimals numbers that arises from radical or combination of them, which was called algebraic number ( numbers which arises as solution of polynomial equations with rational coefficients) Example: 2+sqrt(5) is an algebraic number, cause if we call x= 2+sqrt(5) (x-2) 2 = 5 x 2 -4x+4=5 x 2 -4x-1=0 Are this numbers sufficient to represent the solution of algebraic an geometric problems?

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By 1700 A.D. the notion of limits was introduced to develop Calculus ideas, demanding all decimals numbers. In the late 1700 L’Hermite proved that is irrational (not rational). In 1830 proved that is not solution of any polynomial equation. This type of numbers where called transcendental (meaning further than algebra) Finally, by 1820, Richard Dedekind (a German mathematician), gave an axiomatic for an extension of Q (denoted by R) accepting as numbers all possible decimal numbers (including those with non periodic decimal part). For example 2.101001000… and are in R but not in Q. In this presentation the addition, subtraction, multiplication, division, exponentiation, radicalization satisfies the same basic properties 1) thought 6) stated for Q.

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SUMMARY Counting Numbers 1,2,3,,... Negatives integers -… -3, -2, -1 0 INTEGERS RATIONAL NUMBERS ALGEBRAIC NUMBERS

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Real line Any line in which we have selected a point, called the origin, to represent the number 0 Starting from the origin, a pre-selected length as unit it’s applied consecutively to the left and right to represent the integer numbers (the positive to the right and the negative to the left). The rest of real numbers are located using the rule that a is to left if a < b (meaning that a - b is positive) is called a real line. | | | 0 a b

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So negative numbers are located to the left to the origin and positive to the right in such a way that its distance to the origin is the absolute value of the number. So there is a correspondence between real number and points in the real line and vice- versa. Exercise: In the real line give the relative position of and (86556004192)^(1/22). A real number is called irrational if it is not a rational number. The following figure represents different types of real numbers: In the figure irrational numbers are in R but out side Q. Real numbers that can be written using algebraic operations +,-,*,/, exponentiation and radicalization are called algebraic numbers (cause they are solution of a polynomial equation with integer coefficients), any other real number is called transcendental (cause they show up from branches of mathematics different that Algebra). For examples 4, -1/2, 3/5, -sqrt(2), 1+ sqrt(5) are algebraic numbers, but , e, sin20, ln3, are transcendental. If we pick randomly a real number it’s a very difficult to prove if it’s a transcendental or not.

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Section 1.1 Numbers and Their Properties.

Section 1.1 Numbers and Their Properties.

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