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# Fraccion Generatriz

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### Fraccion Generatriz

1. 1. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ FRACTIONS AND DECIMALS All fractions can be written as decimals .Example: If the denominator of a fraction only has the prime factors 2 or 5, its decimal terminates ( stops after a certain number of digits). 1  1 : 5  0.2 5 Other denominators produce recurring decimals Recurring decimals contain a group of repeating digits that “go for ever”. They are shown by:  A single dot above a single recurring digit  A dot above the first and last digit of a set of recurring digits. For example: 1   0.3333333333..........  0. 3 3 123    0.123123123.....  0.1 2 3 999 Changing terminating decimal as a fraction: You can write a terminating decimal as a fraction. Remember: a terminating decimal ends after a definite number of digits Divide the number without the decimal point by the power of 10 that corresponds to the number of digits in the decimal part. Examples: 342 171 3.42   100 50 325 65 13 0.0325    1000 200 40 Changing recurring decimals to fractions Recurring decimals can be converted into fractions Let x= the recurring decimal. Multiply both sides by the power of 10 that corresponds to the number of digits in the recurring pattern. E.g. by 101 =10 if only 1 digit recurs, by 102  100 if 2 digits recur, by 103  1000 if 3 digits recur, IES OSTIPPO SECCIÓN BILINGÜE 1
2. 2. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ  Subtract the original equation from the new equation.  Solve the resulting equation for x.  Make sure that the answer is a fraction in its simplest form. Example: Write down the fraction, in its simplest form, which is equal to these recurring decimals.  a) 0. 4 x  0.444444444....... Only 1 digit recurs, so, multiply both sides by 10 10 x  4.44444444444....... Subtract the original equation from the new equation. 10 x  4.44444444444.......  x  0.444444444444... 9x  4 4 x 9  4 0.4  9 b) x  0.575757....... 2 digits recur, so, multiply both sides by 100 100 x  57.575757....... Subtract the original equation from the new equation. 100 x  57.575757....... x  0.575757....... 99 x  57 Divide both sides by 99. 57 19 x  (in its simplest form) 99 33   19 0.57  33 IES OSTIPPO SECCIÓN BILINGÜE 2
3. 3. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ REAL NUMBERS Real numbers are either rational or irrational. All terminating and recurring decimals are rational numbers. An irrational number cannot be written as a fraction, Irrational numbers include:  Square roots of non-square numbers  Cube roots of non-cube numbers. Examples of irrational numbers are: 3 2 7  13 Examples:  1.- Show that 0.425 and 0. 2 are rational numbers. 0.425 is a terminating decimal. 425 17 0.425   1000 40  0. 2 is a recurring decimal. x  0.2222222..... 10 x  2.2222...  x  0.22222....... 9x  2 2 x 9 a All terminating and recurring decimals can be written in the form so b they are all rational numbers. 2.- State whether each of the following are rational or irrational numbers. Where a number is rational write it as a fraction in its simplest form. 3 0.6  7 36 6 3 a) 0.6   ……………………..rational 10 5 a A rational number expressed in the form is in its simplest form if a and b b have no common factor. IES OSTIPPO SECCIÓN BILINGÜE 3
4. 4. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ b)   3.141592654... irrational  is a non-recurring decimal and has no exact value. 3 c) 7  1.91293118..... irrational 3 7 is a non-recurring decimal and has no exact value. d) 36  6 rational IES OSTIPPO SECCIÓN BILINGÜE 4