Class work 8. Algebra..

Algebra.

Positive is a number which can be represented as a of two natural numbers: 푝 푎 = ; 푝, 푞 ∈ 푁 푞 As we know such number is also called a , 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 multiplied 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 , 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. This fact can be verified with the help of the 7 long , for example is a proper fraction, using the long division this 8 7 fraction can be written as a decimal = 0.875. Indeed, 8 7 7 7 ∙ 5 ∙ 5 ∙ 5 7 ∙ 53 7 ∙ 125 875 875 ======0.875 8 2 ∙ 2 ∙ 2 2 ∙ 2 ∙ 2 ∙ 5 ∙ 5 ∙ 5 2353 (2 ∙ 5)3 103 1000

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 can be represented as an irreducible 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. (Irreducible fraction can be represented as a finite decimal if and only if it has denominator containing 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 periodic decimal. At each step during this division we will have a remainder. At some point during the process we will see the remainder which occurred before. Process will start to repeat 5 itself. On the example on the left, , after 7, 1, 4, 2, 8, 5, remainder 7 appeared again, 7 5 the fraction can be represented only as an infinite periodic decimal and should be 7 5 written as = 0. ̅714285̅̅̅̅̅̅̅̅̅. (Sometimes you cen find the periodic infinite decimal 7 written as 0. 714285̅̅̅̅̅̅̅̅̅̅ = 0. (714285) ). How we can represent the periodic decimal as a fraction? 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 = 8 1000푥 = 2357. 7̅ 10000푥 − 푥 = 108 9푥 = 8 1000푥 − 100푥 = 2357. 7̅ − 235. 7̅ 108 12 푥 = = 8 = 2122 9999 1111 푥 = 9 2122 1061 푥 = = 900 450

Now consider 2.40̅ and 2.39̅ 푥 = 2.40̅ 100푥 − 10푥 = 240 − 24 10푥 = 24. 0̅ 240 − 24 216 푥 = = = 2.4 100푥 = 240. 0̅ 90 90

푥 = 2.39̅ 100푥 − 10푥 = 239 − 23 10푥 = 23. 9̅ 239 − 23 216 푥 = = = 2.4 100푥 = 239. 9̅ 90 90

Exercises.

1. Represent the following as :

3 7 123 푎. , 푑. ; 푔. ; 2000 4 20 17 3 783 푏. ; 푒. ; ℎ. ; 40 2 540 28 9 324 푐. ; 푓. ; 푖. ; 140 5 25

2. Write as a fraction:

푎. 0. 3̅, 푒. 0.12̅, 푖. 7.54̅,

푏. 0.3, 푓. 0. 12̅̅̅̅, 푗. 1.012̅̅̅̅.

푐. 0. 7̅, 푔. 0.12,

푑. 0.7, ℎ. 1.123̅,

3. Evaluate the following using decimals: 1 3 2 1 푎. 0.36 + ; 푏. 5.8 − ; 푐. : 0.001; 푑. 7.2 ∙ 2 4 5 1000 4. Evaluate the following using fractions: 2 1 5 8 푎. + 0.6; 푏. 1 − 0.5; 푐. 0.3 ∙ ; 푑. : 0.4; 3 6 9 11 5. Evaluate: 1 1 3 1 1 2 2 5 1 ∙ 2 ∙ 3 1 ∙ 2 ∙ 0.36 0.38 ∙ 0.17 ∙ 2 ∙ 2.7 푎. 7 ; 푏. 3 11 2 ; 푠. 2 3 ; 푑. 15 3 1 1 9 1 1 4 3 ∙ 4 ∙ 3 0.6 ∙ 2 ∙ 1 5.1 ∙ 3 ∙ 0.064 14 2 6 11 4 3 5

Geometry.

Special segments of a triangle.

From each vertices of a tringle to the opposite side 3 special segment can be constructed. An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude.

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.