Accelerated Math. Class work 18. Algebra. Positive rational number is a number which can be represented as a ratio of two natural numbers: 푝 푎 = ; 푝, 푞 ∈ 푁 푞 As we know such number is also called a fraction, p in this fraction is a nominator and q is a denominator. Any natural number can be represented as a fraction with denominator 1: 푏 푏 = ; 푏 ∈ 푁 1 Basic property of fraction: nominator and denominator of the fraction can be multiply by any non-zero number n, resulting the same fraction: 푝 푝 ∙ 푛 푎 = = 푞 푞 ∙ 푛 푝 In the case that numbers p and q do not have common prime factors, the fraction is irreducible 푞 fraction. If 푝 < 푞, the fraction is called “proper fraction”, if 푝 > 푞, the fraction is called “improper fraction”. If the denominator of fraction is a power of 10, this fraction can be represented as a finite decimal, for example, 37 37 3 3 12437 12437 = = 0.37, = = 0.3, = = 12,437 100 102 10 101 1000 103 10푛 = (2 ∙ 5)푛 = 2푛 ∙ 5푛 2 2 2 ∙ 21 4 = = = = 0.4 5 51 51 ∙ 21 10 Therefore, any fraction, which denominator is represented by 2푛 ∙ 5푚 can be written in a form of finite decimal. Also, any finite decimal can be represented as a fraction with denominator 10푛. 375 3 3 65 13 ∙ 5 13 0.375 = = = ; 0.065 = = = ; 1000 8 23 1000 5323 5223 672 168 168 34 17 ∙ 2 17 6.72 = = = ; 0.034 = = = ; 100 25 52 1000 5323 5322 In other words, if the finite decimal is represented as an irreduceble fraction, the denominator of this fraction will not have other factors besides 5푚 and 2푛. Converse statement is also true: if the irreducible fraction has denominator which only contains 5푚 and 2푛 than the fraction can be written as a finite decimal. (Irrdeuceble fraction can be represented as a finite decimal if and only if it has determinator conteining only 5푚 and 2푛 as factors.) If the denominator of the irreducible fraction has a factor different from 2 and 5, the fraction cannot be represented as a finite decimal. If we try to use the long division process we will get an infinite decimal. At each step during this division we have a 0.0675 remainder. They are the numbers from 1 to 73. At 74 ) 5.00000 some point during this division we will see the 4.44 remainder which occurred before. Process will start 560 to repeat itself. 518 420 How we can represent the periodic decimal as a 370 fraction? 500 Let’s take a look on a few examples: 0. 8̅, 2.357̅, 0. 0108̅̅̅̅̅̅̅. 1. 0. 8̅. 2. 2.357̅ 3. 0. 0108̅̅̅̅̅̅̅ 푥 = 0. 8̅ 푥 = 2.357̅ 푥 = 0. 0108̅̅̅̅̅̅̅ 10푥 = 8. 8̅ 100푥 = 235. 7̅ 10000푥 = 108. 0108̅̅̅̅̅̅̅ 10푥 − 푥 = 8. 8̅ − 0. 8̅ 1000푥 = 2357. 7̅ 10000푥 − 푥 = 108 = 8 1000푥 − 100푥 108 푥 = 9푥 = 8 = 2357. 7̅ − 235. 7̅ 9999 8 = 2122 푥 = 9 2122 푥 = 900 Now consider 2.40̅ and 2.39̅ 푥 = 2.40̅ 100푥 − 10푥 = 240 − 24 10푥 = 24. 0̅ 100푥 = 240. 0̅ 240 − 24 216 푥 = = = 2.4 90 90 푥 = 2.39̅ 100푥 − 10푥 = 239 − 23 10푥 = 23. 9̅ 239 − 23 216 푥 = = = 2.4 90 90 100푥 = 239. 9̅ Exercises. Write as a fraction: 0. 3̅, 0. 7̅, 0.12̅, 1.123̅, 7.54̅, 0. 12̅̅̅̅, 1.012̅̅̅̅. Geometry. Pythagorean theorem. In a right triangle the side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Pythagorean theorem is states that the square of the hypotenuse is equal to the sum of square of legs. 푐2 = 푎2 + 푏2 There are numneros proofs if this theorem, but let’s see one. 4 identical right triangles are arranged as shown on the picture. He area of the big square is 푆 = (푎 + 푏) ∙ (푎 + 푏) = (푎 + 푏)2, the are of the small square is 푠 = 푐2. 1 The area of 4 triangles is 4 ∙ 푎푏 = 2푎푏. But also cab be 2 represented as 푆 − 푠 = 2푎푏 2푎푏 = (푎 + 푏) ∙ (푎 + 푏) − 푐2 = 푎2 + 2푎푏 + 푏2 − 푐2 푎2 + 푏2 = 푐2 .