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**Unformatted text preview: **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 . 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 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 6 = 0 and d 6 = 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!!!...

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