Chapter 1.1 Real numbers

# However this formula doesnt work well for examples 5

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However, this formula doesn’t work well for examples like 33 1000 + 5 2000 (try it). A better method is to rewrite the first fraction 33 1000 as follows: multiply numerator and denominator by 2 to get 33 1000 = 33 1000 · 2 2 = 33 · 2 1000 · 2 = 66 2000 . Then the fractions can be added easily because their denominators are the same. 33 1000 + 5 2000 = 66 2000 + 5 2000 = 71 2000 The best way to add fractions is to rewrite them with a common denominator that is as small as possible. This will be the denominators’ least common multiple (LCM). Procedure To find the least common multiple (LCM) of two or more natural numbers: Write down the prime power factorization of each number. For each prime that appears, determine the highest power of that prime in those factorizations. The LCM is the product of those highest powers. Stanley Ocken M19500 Precalculus Chapter 1.1: Real numbers

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Welcome Classifying real numbers Factoring Fraction rules Adding fractions; LCD Intervals Infinite intervals Absolute value Exercises Quiz review Example 8: Find the LCM of 36 and 120 Solution: 36 = 4 · 9 = 2 2 · 3 2 and 120 = 8 · 15 = 2 3 · 3 1 · 5 1 . The primes that appear are 2 , 3 , and 5 . The highest power of 2 is 2 3 , which appears in the factorization of 120 . The highest power of 3 is 3 2 , which appears in the factorization of 36 . The highest power of 5 is 5 1 , which appears in the factorization of 120 . The product of those highest powers is 2 3 · 3 2 · 5 1 . Answer: The LCM of 36 and 120 is 2 3 · 3 2 · 5 = 8 · 9 · 5 = 360 . Finding the LCD of fractions The LCD of fractions is the LCM of their denominators. Procedure To add fractions by using their LCD: Build (rewrite) each fraction with that LCD as its denominator. Add the rewritten fractions by adding their numerators. Reduce the resulting fraction. Stanley Ocken M19500 Precalculus Chapter 1.1: Real numbers
Welcome Classifying real numbers Factoring Fraction rules Adding fractions; LCD Intervals Infinite intervals Absolute value Exercises Quiz review Example 9: Find 5 36 + 7 120 by using their LCD. Solution: Find the LCD quickly, just as above: 36 = 4 · 9 = 2 2 · 3 2 and 120 = 8 · 15 = 2 3 · 3 1 · 5 1 . and so the LCD is 2 3 · 3 2 · 5 = 360 Since 360 = 36 · 10 , we write 5 36 = 5 · 10 36 · 10 = 50 360 Since 360 = 120 · 3 , we write 7 120 = 7 · 3 120 · 3 = 21 360 Add the rewritten fractions: 50 360 + 21 360 = 71 360 . This can’t reduce: 71 is prime. You may have seen the following procedure. It always works, but it can take a long time. Furthermore, it doesn’t work when the fractions involve variables. Procedure To find the LCD of numerical fractions, list multiples of the smallest denominator until you get a number that is a multiple of the other denominators. Example 10: Find the LCD of 36 and 120 . Solution: List the multiples of 36 until you get a multiple of 120 , as follows: 36 , 72 , 108 , 144 , 180 , 216 , 252 , 288 , 324 , 360 , which equals 3 · 120 . Thus the LCD is 360 . Stanley Ocken M19500 Precalculus Chapter 1.1: Real numbers

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Welcome Classifying real numbers Factoring Fraction rules Adding fractions; LCD Intervals Infinite intervals Absolute value Exercises Quiz review Intervals on the real number line As we have discussed earlier, the real number line includes:

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