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Real Numbers
and Their Basic Properties
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2. Copyright © Cengage Learning. All rights reserved.
Section
Fractions
1.2
Copyright © Cengage Learning. All rights reserved.

3. Simplify a fraction
In the fractions the number above the bar is called the numerator, and the number below the bar is called the denominator.

4. Fraction to Indicate Parts of Whole
We often use fractions to indicate parts of a whole. The fraction indicates how much of the figure is shaded.
(a)
(b)
Figure 1-12

5. Fraction to Indicate Division
We can also use fractions to indicate division. For example, the fraction indicates that 8 is to be divided by 2:

6. Interpreting a Fraction
What kind of a number is ? Lets call it x, i.e., x is such that: x ∙ 2 = 8, or Can you see why is undefined? There is no such number.
8 x =
∙ 2 = 8 2
∙ 0 = 8 ?! 0

7. Fraction in Its Lowest Terms
A fraction is said to be in lowest terms (or simplest form) when no integer other than 1 will divide both its numerator and its denominator exactly. is in its lowest terms is not in lowest terms

8. Simplify a fraction
We can simplify a fraction that is not in lowest terms by dividing its numerator and its denominator by the same number.
Figure 1-13

9. Factoring a Number
When a composite number has been written as the product of other natural numbers, we say that it has been factored. For example, 15 can be written as the product of 5 and = 5  3 The numbers 5 and 3 are called factors of 15. When a composite number is written as the product of prime numbers, we say that it is written in prime-factored form.

10. Example 1
Write 210 in prime-factored form. Solution: We can write 210 as the product of 21 and 10 and proceed as follows: 210 = 21  = 3  7  2  5

11. Simplify a fraction
To simplify a fraction, we factor its numerator and denominator and divide out all factors that are common to the numerator and denominator. For example,

12. Simplify a fraction
The Fundamental Property of Fractions If a, b, and x are real numbers,

13. Multiply and divide two fractions2.

14. Multiply and divide two fractions
To multiply fractions, we use the following rule. Multiplying Fractions To multiply fractions, we multiply their numerators and multiply their denominators. In symbols, if a, b, c, and d are real numbers, For example,

15. Multiply and divide two fractions
To justify the rule for multiplying fractions, we consider the square in Figure Because the length of each side of the square is 1 unit and the area is the product of the lengths of two sides, the area is 1 square unit. If this square is divided into 3 equal parts vertically and 7 equal parts horizontally, it is divided into 21 equal parts, and each represents of the total area.
Figure 1-14

16. Multiply and divide two fractions
The area of the shaded rectangle in the square is , because it contains 8 of the 21 parts. The width, w, of the shaded rectangle is its length, l, is and its area, A, is the product of l and w: A = l  w

17. Multiply and divide two fractions
This suggests that we can find the product of by multiplying their numerators and multiplying their denominators. Fractions whose numerators are less than their denominators, such as , are called proper fractions. Fractions whose numerators are greater than or equal to their denominators, such as , are called improper fractions.

18. Example 3 Perform each multiplication. a. b.
Multiply the numerators and multiply the denominators. There are no common factors.
Multiply in the numerator and multiply in the denominator.
Write 5 as the improper fraction
Multiply the numerators and multiply the denominators.

19. Example 3 cont’d To simplify the fraction, factor the denominator.
Divide out the common factors of 3 and 5.

20. Multiply and divide two fractions
One number is called the reciprocal of another if their product is 1. For example, is the reciprocal of , because Dividing Fractions To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. In symbols, if a, b, c, and d are real numbers,

21. Add and subtract two or more fractions3.

22. Add and subtract two or more fractions
Adding Fractions with the Same Denominator To add fractions with the same denominator, we add the numerators and keep the common denominator. In symbols, if a, b, and d are real numbers,

23. Add and subtract two or more fractions
For example, Figure 1-15 illustrates why
Add the numerators and keep the common denominator.
Figure 1-15

24. Add and subtract two or more fractions
To add fractions with unlike denominators, we write the fractions so that they have the same denominator. For example, we can multiply both the numerator and denominator of by 5 to obtain an equivalent fraction with a denominator of 15:

25. Add and subtract two or more fractions
To write as an equivalent fraction with a denominator of 15, we multiply the numerator and the denominator by 3: Since 15 is the smallest number that can be used as a common denominator for and , it is called the least (or lowest) common denominator (the LCD).

26. Add and subtract two or more fractions
To add the fractions and , we write each fraction as an equivalent fraction having a denominator of 15, and then we add the results:

27. Example 6
Add: Solution: To find the LCD, we find the prime factorization of each denominator and use each prime factor the greatest number of times it appears in either factorization: 10 = 2  5 28 = 2  2  7
LCD = 2  2  5  7 = 140

28. Example 6 – Solution
cont’d
Since 140 is the smallest number that 10 and 28 divide exactly, we write both fractions as fractions with denominators of 140.
Write each fraction as a fraction with a denominator of 140.
Do the multiplications.
Add the numerators and keep the denominator.

29. Example 6 – Solution
cont’d
Since 67 is a prime number, it has no common factor with 140. Thus, is in lowest terms.

30. Add and subtract two or more fractions
Subtracting Fractions with the Same Denominator To subtract fractions with the same denominator, we subtract their numerators and keep their common denominator. In symbols, if a, b, and d are real numbers, For example,

31. Add and subtract two or more fractions
To subtract fractions with unlike denominators, we write them as equivalent fractions with a common denominator. For example, to subtract from , we write , find the LCD of 4 and 5, which is 20, and proceed as follows:
Write each fraction as a fraction
with a denominator of 20.
Do the multiplications.
Add the numerators and
keep the denominator.

32. Add and subtract two or more mixed numbers4.

33. Add and subtract two or more mixed numbers
The mixed number represents the sum of 3 and . We can write as an improper fraction as follows:
Add the numerators and keep the denominator.

34. Add and subtract two or more mixed numbers
To write the fraction as a mixed number, we divide 19 by 5 to get 3, with a remainder of 4.

35. Example 8
Add: Solution: We first change each mixed number to an improper fraction.

36. Example 8 – Solution
cont’d
Then we add the fractions. Finally, we change to a mixed number.
Write each fraction with the LCD of 12.

37. Add, subtract, multiply, and divide two or more decimals5.

38. Add, subtract, multiply, and divide two or more decimals
Rational numbers can always be changed to decimal form. For example, to write and as decimals, we use long division: The decimal 0.25 is called a terminating decimal.

39. Add, subtract, multiply, and divide two or more decimals
The decimal (often written as ) is called a repeating decimal, because it repeats the block of digits 27. Every rational number can be changed into either a terminating or a repeating decimal. Terminating decimals Repeating decimals

40. Add, subtract, multiply, and divide two or more decimals
The decimal 0.5 has one decimal place, because it has one digit to the right of the decimal point. The decimal 0.75 has two decimal places, and has three. To add or subtract decimals, we align their decimal points and then add or subtract.

41. Example 10
Add and 2.74 using a vertical format. Solution: We align the decimal points and add the numbers, column by column,

42. Add, subtract, multiply, and divide two or more decimals
To multiply decimals, we multiply the numbers and place the decimal point so that the number of decimal places in the answer is equal to the sum of the decimal places in the factors. To divide decimals, we move the decimal point in the divisor to the right to make the divisor a whole number. We then move the decimal point in the dividend the same number of places to the right.

43. Round a decimal to a specified number of places6.

44. Round a decimal to a specified number of places
We often round long decimals to a specific number of decimal places. For example, the decimal rounded to one place (or to the nearest tenth) is Rounded to two places (or to the nearest one-hundredth), the decimal is

45. Round a decimal to a specified number of places
Rounding Decimals 1. Determine to how many decimal places you want to round. 2. Look at the first digit to the right of that decimal place. 3. If that digit is 4 or less, drop it and all digits that follow. If it is 5 or greater, add 1 to the digit in the position to which you want to round, and drop all digits that follow.

46. Example 13
Round to two decimal places. Solution: Since we are to round to two digits, we look at the digit to the right of the 8, which is 6. Since 6 is greater than 5, we add 1 to the 8 and drop all of the digits that follow. The rounded number is 2.49.

47. Use the appropriate operation for an application7.

48. Use the appropriate operation for an application
A percent is the numerator of a fraction with a denominator of 100. For example, percent, written , is the fraction , or the decimal In problems involving percent, the word of usually indicates multiplication. For example, of 8,500 is the product (8,500).

49. Example 14 – Auto Loans
Juan signs a one-year note to borrow $8,500 to buy a car. If the rate of interest is , how much interest will he pay? Solution: For the privilege of using the bank’s money for one year, Juan must pay of $8,500. We calculate the interest, i, as follows: i = of 8,500 =  8,500 = Juan will pay $ interest.
In this case, the word of means times.