1. Chapter 1
Section 6

2. Multiplying and Dividing Real Numbers
1.6
Multiplying and Dividing Real Numbers
Find the product of a positive number and a negative number.
Find the product of two negative numbers.
Identify factors of integers.
Use the reciprocal of a number to apply the definition of division.
Use the rules for order of operations when multiplying and dividing signed numbers.
Evaluate expressions involving variables.
Translate words and phrases involving multiplication and division.
Translate simple sentences into equations. 2 3 4 5 6 7 8

3. Multiplication by Zero
Definition
The result of multiplication is called the product. We already know how to multiply positive numbers, and we know that the product of two positive numbers is positive.
We also know that the product of 0 and any positive number is 0, so we extend that property to all real numbers.
Multiplication by Zero
For any real number x,
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4. Find the product of a positive and a negative number.
Objective 1
Find the product of a positive and a negative number.
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5. Multiplying Numbers with Different Signs
Find the product of a positive number and a negative number.
The product of a 3(−1) represents the sum
Also,
and
These results maintain the pattern which suggests a new rule.
Multiplying Numbers with Different Signs
For any positive real numbers x and y,
and
That is, the product of two numbers with opposite signs is negative.
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6. Multiplying a Positive Number and a Negative Number
EXAMPLE 1
Multiplying a Positive Number and a Negative Number
Find the product.
Solution:
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7. Find the product of two negative numbers.
Objective 2
Find the product of two negative numbers.
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8. Multiplying Two Negative Numbers
Find the product of two negative numbers.
Multiplying Two Negative Numbers
For any positive real numbers x and y,
That is, the product of two negative numbers is positive.
Example:
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9. Multiplying Two Negative Numbers
EXAMPLE 2
Multiplying Two Negative Numbers
Find the product.
Solution:
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10. Identify factors of integers.
Objective 3
Identify factors of integers.
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11. Identify factors of integers.
In Section 1.1, the definition of a factor was given for whole numbers. The definition can be extended to integers. If the product of two integers is a third integer, then each of the two integers is a factor of the third.
The table below show several examples of integers and factors of those integers.
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12. Use the reciprocal of a number to apply the definition of division.
Objective 4
Use the reciprocal of a number to apply the definition of division.
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13. Reciprocal or Multiplicative Inverse
Use the reciprocal of a number to apply the definition of division.
The quotient, of two numbers is found by multiplying by the reciprocal, or multiplicative inverse, of the second number. The following table shows several numbers and their multiplicative inverses.
Reciprocal or Multiplicative Inverse
Pairs of numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other.
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14. Definition of Division
Use the reciprocal of a number to apply the definition of division. (cont’d)
Definition of Division
For any real numbers x and y, with y ≠ 0,
That is, to divide two numbers, multiply the first by the reciprocal, or multiplicative inverse, of the second.
Recall that an equivalent form of is where x is called the
dividend and y is called the divisor.
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15. Using the Definition of Division
EXAMPLE 3
Using the Definition of Division
Find each quotient, using the definition of division.
Solution:
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16. Dividing Signed Numbers
Use the reciprocal of a number to apply the definition of division. (cont’d)
Division Involving 0
For any real numbers x, with x ≠ 0,
When dividing fractions, multiplying by the reciprocal works well. However, using the definition of division directly with integers is awkward. It is easier to divide in the usual way and then determine the sign of the answer.
Dividing Signed Numbers
The quotient of two numbers having the same sign is positive.
The quotient of two numbers having different signs is negative.
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17. Dividing Signed Numbers
EXAMPLE 4
Dividing Signed Numbers
Find each quotient.
Solution:
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18. Equivalent Forms Equivalent Forms
Use the reciprocal of a number to apply the definition of division. (cont’d)
Equivalent Forms
For any positive numbers x and y,
Equivalent Forms
For any positive real numbers x and y,
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19. Objective 5
Use the rules for order of operations when multiplying and dividing signed numbers.
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20. Using the Rules for Order of Operations
EXAMPLE 5
Using the Rules for Order of Operations
Perform each indicated operation.
Solution:
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21. Evaluate expressions involving variables.
Objective 6
Evaluate expressions involving variables.
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22. Evaluating Expressions for Numerical Values
EXAMPLE 6
Evaluating Expressions for Numerical Values
Evaluate if and
Solution:
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23. Translate words and phrases involving multiplication and division.
Objective 7
Translate words and phrases involving multiplication and division.
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24. Translate words and phrases involving multiplication and division.
The word product refers to multiplication. The table gives other key words and phrases that indicate multiplication in problem solving.
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25. Translating Words and Phrases (Multiplication)
EXAMPLE 7
Translating Words and Phrases (Multiplication)
Write a numerical expression for the phrase, and simplify the expression.
Three times the difference between 4 and −11.
Three-fifths of the sum of 2 and −7.
Solution:
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26. Translate words and phrases involving multiplication and division.
The word quotient refers to division. In algebra, quotients are usually represented with a fraction bar; the symbol ÷ is seldom used. The table gives some key phrases associated with division.
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27. Interpreting Words and Phrases Involving Division
EXAMPLE 8
Interpreting Words and Phrases Involving Division
Write a numerical expression for the phrase, and simplify the expression.
The product of −9 and 2, divided by the difference between 5 and −1.
Solution:
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28. Translate simple sentences into equations.
Objective 8
Translate simple sentences into equations.
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29. Translating Sentences into Equations
EXAMPLE 9
Translating Sentences into Equations
Write each sentence as an equation, using x as the variable. Then find the solution from the list of integers between −12 and 12, inclusive.
The quotient of a number and −2 is 6.
Twice a number is −6.
Solution:
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