real_number_arithmetic

# real_number_arithmetic - Arithmetic Operations on Real...

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Arithmetic Operations on Real Numbers Arithmetic operations performed on rational numbers . Two fractions (rational numbers) a b and c d can be added by the means of the formula: = + a b c d + a d b c b d . For example, = + 7 6 4 15 + ( ) 7 . ( ) 15 ( ) 4 . ( ) 6 ( ) 6 . ( ) 15 = + 105 24 90 = 129 90 . The numerator 129 and denominator 90 have the greatest common divisor of 3, so we can divide the numerator and denominator by 3 to obtain 43 30 . Hence = + 7 6 4 15 43 30 , where the result is given in its " lowest terms ", that is, with the numerator and denominator having no common factor. Alternatively, one can look for the least common multiple of the two denominators 6 and 15 in the original two fractions. This is the lowest number that both 6 and 15 divide into exactly. Since 6 = 2 x 3 and 15 = 3 x 5, the least common multiple of 6 and 15 is 2 x 3 x 5 = 30. The two fractions can now both be written with the common denominator of 30. This is achieved by multiplying both numerator and denominator of 7 6 by 5 to give = 7 6 35 30 , and by multiplying both numerator and denominator of 4 15 by 2 to give = 4 15 8 30 . Hence = + 7 6 4 15 + 35 30 8 30 = 43 30 , as before. The calculation is often written as follows. = + 7 6 4 15 + 35 8 30 = 43 30 . The first number 35 in the numerator of the middle fraction is obtained by dividing the denominator 6 of the first fraction into the common denominator 30 to give 5, and then multiplying this number by the numerator 7. The second number 8 is obtained by dividing the denominator 15 of the second fraction into the common denominator 30 to give 2, and then multiplying this number by the numerator 4. This calculation can be checked, at least partially, by performing an approximate decimal calculation with a calculator. 7 6 ~ 1.166666667 and 4 15 ~ 0.2666666667, so that + 7 6 4 15 ~ 1.433333334. A decimal approximation for 43 30 can also be obtained. 43 30 ~ 1.433333333. Two fractions (rational numbers) a b and c d can be multiplied by the means of the formula: a b x = c d a c b d . For example, 6 35 x = 7 9 42 315 .

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The fraction 42 315 can be reduced to its lowest terms as follows. = 42 315 6 45 = 2 15 . Hence 6 35 x = 7 9 2 15 . Another way to perform the computation is to write the product in the intermediate form 6 35 x = 7 9 6 . 7 35 . 9 and look for common factors of the top and bottom in this form. 7 is a common factor of the top and bottom so the 7 at the top can be divided into the 35 at the bottom to give = 6 . 7 35 . 9 6 5 . 9 . The top and bottom can also be divided by 3 (6 and 9 have the common factor 3). Hence = 6 5 . 9 2 5 . 3 = 2 15 . The complete calculation may written in the form 6 35 x = 7 9 6 . 7 35 . 9 = 6 5 . 9 = 2 15 . It is common practice to use "
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## This note was uploaded on 11/12/2011 for the course MATH 111 taught by Professor Uri during the Spring '08 term at Rutgers.

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real_number_arithmetic - Arithmetic Operations on Real...

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