Any real number can marry any other real number. Any real number can kiss any other real number too, including itself!
Any real number can break up with any real number EXCEPT a particular number which shall, for the time being, remain nameless.
In Section 1.3 we talked about positive reals only, and in Section 1.4 the negatives showed up and we had to make some rules which govern how negatives and positives interact with regard to addtion and subtraction.
Now we must set the rules for how positives and negatives interact with respect to multiplication and division.
Rules! Rules! Rules! More Ground Rules:
Rule #1: Kissing. $$ \begin{array}{c} (+)(+)=(+)\\ (+)(-)=(-)\\ (-)(+)=(-)\\ (-)(-)=(+)\\ \end{array} $$
Example: $3.1 \cdot (-5.8)$.
Rule #2: Breaking up is not hard to do. $$ \begin{array}{c} (+) \div (+)=(+)\\ (+) \div (-)=(-)\\ (-) \div (+)=(-)\\ (-) \div (-)=(+)\\ \end{array} $$
Example: $\frac{3}{2} \div \left( -\frac{5}{6} \right)$.
Rule #3: Multiplication is associative. That is, kissing can be performed in any order. $$abc=(ab)c=a(bc)$$ Example: $3.1 \cdot(-5.8) \cdot \frac{1}{2}$.
Rule #4: Multiplication is commutative. That is, if $a$ kisses $b$, or $b$ kisses $a$, the result is the same. $$ab=ba$$
Example: $3.1 \cdot (-5.8)$ is the same as $-5.8 \cdot 3.1$.
Rule #5: You can kiss zero, but you can't break up with zero.
Example: $6 \cdot 0$
Example: $0 \cdot 6$
Example: $0 \div 6$
Example: $6 \div 0$
Application: A knitter can knit 100 rows of a design over 7 hours. What is the knitter's average hourly knitting rate? Round your answer to the nearest row.
Application: In the state of Jefferson, sales tax is 4$\%$. If subtotal is $\$$8.38, what will they pay in sales tax? Round to the nearest penny.
Application: has decided to build a rectangular fish pond by the patio. It needs to be 13 feet wide, 18 feet long, and 2.5 feet deep. What is the volume of water it will hold?