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**Thinking Mathematically**

The Rational Numbers

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The Rational Numbers The set of rational numbers is the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not equal to 0. The integer a is called the numerator and the integer b is called the denominator. Note that every integer is a rational number. For example 17 can be written 17/1.

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**The Fundamental Principle of Rational Numbers**

If is a rational number and c is any number other than 0, The rational numbers and are called equivalent fractions

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Lowest Terms A rational number can be reduced to its lowest terms by dividing the numerator and denominator by the greatest common divisor. Example: Reduce 24/40 to lowest terms. The greatest common divisor of 24 and 40 is 2 x 2 x 2 = 8. (24/8) = (40/8) 40 24 5 = 3

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**Converting a Positive Mixed Number to an Improper Fraction**

Multiply the denominator of the rational number by the integer and add the numerator to this product. Place the sum in step 1 over the denominator in the mixed number.

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**Converting a Positive Improper Fraction to a Mixed Number**

Divide the denominator into the numerator. Record the quotient and the remainder. Write the mixed number using the following form:

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**Rational Numbers and Decimals**

Any rational number can be expressed as a decimal. The resulting decimal will either terminate, or it will have a digit that repeats or a block of digits that repeat. Examples: 5 3 = = .60 The “line” over the zero indicates that it is to be repeated infinitely often. 3 1 = = .3

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**Expressing a Repeating Decimal as a Quotient of Integers**

Step 1 Let n equal the repeating decimal. Step 2 Multiply both sides of the equation in step 1 by 10 if one digit repeats, by 100 if two digits repeat, by 1000 if three digits repeat, and so on. Step 3 Subtract the equation in step 1 from the equation in step 2. Step 4 Divide both sides of the equation in step 3 by an appropriate number and solve for n.

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**Multiplying Rational Numbers**

The product of two rational numbers is the product of their numerators divided by the product of their denominators. Example: 5 3 x 7 4 = (5 x 7) (3 x 4) 35 12

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**Dividing Rational Numbers**

The quotient of two rational numbers is the product of the first number and the reciprocal of the second number. If a/b and c/d are rational numbers, and c/d is not 0, then

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**Adding and Subtracting Rational Numbers with Identical Denominators**

The sum or difference of two rational numbers with identical denominators is the sum or difference of their numerators over the common denominator. If a/b and c/d are rational numbers, then

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**Density of Rational Numbers**

If r and t represent rational numbers, with r<t, then there is a rational number s such that s is between r and t. r < s < t.

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**Thinking Mathematically**

The Rational Numbers

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