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Ch3_Fractions

# Ch3_Fractions - MA100 Fall 2010 PART 1 Precalculus[3...

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MA100, Fall 2010, PART 1: Precalculus [3] Rational numbers 1. Fractional and decimal notations As you have learned in high school, rational numbers are numbers obtained as ratios of integers. Rational numbers can be written in two forms: - as ratios (or fractions); for example 1 2 , - in decimal notation; for example, 0 . 5 . Any (terminating or repeating) decimal number can be written as a fraction and vice versa. For instance, 3 . 7493 = 37493 10000 or 0 . 00103 = 103 100000 . Your calculators always give you the answer in decimal notation. However, in this course we will prefer the fractional notation. This is because, unless a ratio has a finite number of decimals, any calculator displays an approximation of that ratio (which might not be good enough!). Definition Two ratios a b and c d are equal if and only if ad = bd. As a relation, we write: a b = c d ⇐⇒ ad = bc . If we have a negative ratio, the “minus” sign can be assigned to either numerator (the “top”), or the denominator (the “bottom”), but not both! For example - 5 7 = - 5 7 = 5 - 7 , Conventionally, we prefer to think of the “minus” sign as being in front of the numerator, so - 5 7 = - 5 7 . Example: Determine x so that x - 2 3 = - 8 5 Solution: x - 2 3 = - 8 5 ⇐⇒ x - 2 3 = ( - 8) 5 ⇐⇒ 5( x - 2) = 3 × ( - 8) Now we solve: 5( x - 2) = 3 × ( - 8) ⇐⇒ 5 x - 10 = - 24 ⇐⇒ 5 x = - 24 - 10 ⇐⇒ 5 x = - 34 . 1

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Finally, 5 x = - 34 = x = - 34 5 or x = - 34 5 . Observation Any integer can be written as a ratio with denominator 1. For example, 11 = 11 1 or, as another example, - 23 = - 23 1 = - 23 1 . Simplification A fraction is a ratio of two integers, the numerator (the “top”) and the denomina- tor (the “bottom”). Say we are able to factor the “top” (numerator) and “bottom” (denominator). If some of these factors are identical, then they can be simplified. For example 88 12 = 2 × 44 2 × 6 = 44 6 = 2 × 22 2 × 3 = 22 3 or, directly, 88 12 = 4 × 22 4 × 3 = 22 3 2. Operations with rational numbers Given two rational numbers, we can add (subtract) and multiply (divide) them. Addition : Given a, b, c, d Z , b = 0 and d = 0, the simplest way to deal with the addition of a b and c d is defining that: a b + c d = ad + bc bd If we deal with more fractions, then we do pretty much the same thing: a b + c d + e f = adf + cbf + bde bdf Example: 2 3 + 5 8 = 2 × 8 + 3 × 5 24 = 16 + 15 24 = 31 24 Example: 4 + 5 8 = 4 1 + 5 8 = 4 × 8 + 1 × 5 8 = 32 + 5 8 = 37 8 Subtraction is similar: a b - c d = ac - bd bd Observation A common mistake is to “simplify” when adding terms on the “top.” DO NOT confound addition with multiplication! (This might cost you failing this course.) For example, 5 + 4 5 = 1 + 4 1 IS WRONG!!! 2
The correct way is: 5 + 4 5 = 9 5 The denominator of the summed fraction is called the common denominator (CD). There are more involved ways to find the common denominator for two or more fractions. A special case is when one (or more) denominators is a multiple of another. Then, as rules, we have: a b + c bd = ad + c bd (when adding two fractions; here the CD = bd ) a b + c d + e bdf = ad + cb + e bdf (when adding two fractions; here the CD = bdf ) Example: Calculate: 3 4 + 5 3 - 11 12 The CD is 12 . We have: 3 4 + 5

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Ch3_Fractions - MA100 Fall 2010 PART 1 Precalculus[3...

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