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4 MULTIPLYING AND DIVIDING We can only do arithmetic in the usual way. To calculate 5(−2), we have to do 5· 2 = 10 -- and then decide on the sign. Is it +10 or −10? To decide, we have the followng Rule of Signs. 1. What is the Rule of Signs for multiplying, dividing, and fractions? To see the answer, pass your mouse over the colored area.
Like signs produce a positive number;
For an explaination of these rules, see below. 2. Write the formal Rule of Signs as it applies to fractions.
Problem 1. Calculate the following.
Problem 2. Evaluate the following. (Be careful to distinguish the operations.)
Example 2. The form a − b(−c). Consider a problem in this form: 3 − 5(−2). We are to subtract 5 times −2:
And so even though the problem means to subtract (5 times −2), we may interpret it to mean: −5 times −2 = +10. We may simply write
In other words, any problem that looks like this -- a − b(−c) -- we may evaluate like this: a + bc. Problem 3. Evaluate the following.
Problem 4. Two variables. Let the value of y depend on the value y = 3x − 6. Calculate the value of y that corresponds to each value of x: When x = 0, y = 3· 0 − 6 = 0 − 6 = −6. When x = 1, y = 3· 1 − 6 = 3 − 6 = −3 . When x = −1, y = 3· −1 − 6 = −3 − 6 = −9. When x = 2, y = 3· 2 − 6 = 6 − 6 = 0. When x = −2, y = 3· −2 − 6 = −6 − 6 = −12. When x = 3, y = 3· 3 − 6 = 9 − 6 = 3. When x = −3, y = 3· −3 − 6 = −9 − 6 = −15. Problem 5. Negative factors.
Problem 6. According to the previous problem: An even number of negative factors produces a positive number. While an odd number of negative factors produces a negative number. Problem 7. Calculate.
Problem 8. Evaluate each of the following as a positive or negative fraction in lowest terms, or as an integer.
To multiply fractions, multiply the numerators, and Problem 9. Multiply.
An explanation of the Rule of Signs The inclusion of a negative factor changes the sign of a product. That is, if ab is positive, then (−a)b cannot also be positive. It must be negative. For, suppose (−a)b were also positive. Then ab + (−a)b would be positive. But that sum is actually 0 ab + (−a)b = [a + (−a)]b = 0· b = 0 Therefore, (−a)b cannot be positive. It must be negative. In fact, since ab + (−a)b = 0, (−a)b must be the negative of ab. (−a)b = −ab. Similarly, the inclusion of two negative factors will change the sign twice -- the sign will return to positive (−a)(−b) = ab. Next Lesson: Some rules of algebra Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |