Principle 1.12. Properties of Real Numbers.
- Commutativity: \(a+b=b+a\) and \(ab = ba\)
- Associativity: \(a+(b+c)=(a+b)+c\) and \(a(bc) = (ab)c\)
- Distributivity: \(a(b+c) = ab+ac\) and \((a+b)c = ac+bc\)
Section 1.4 Properties of Real Numbers
The real numbers and the arithmetic operations we perform with them fundamental properties that you should be aware of.
Example 1.13. Using Commutativity.
Commutativity says it doesn’t matter the order you add or the order you multiply real numbers, as the results will be the same. For addition:
\begin{equation*}
-2 + 7 = 7+(-2) = 7-2 =5.
\end{equation*}
For multiplication:
\begin{equation*}
-2\cdot 7 = 7(-2) = -14.
\end{equation*}
Example 1.14. Using Associativity.
Associativity says the way you group multiple additions or products together doesn’t matter. This ensures that it makes perfect sense to drop parenthesis altogether as in the following example involving addition:
\begin{equation*}
-2+7+3 = -2+(7+3) = -2 + 10 = 8.
\end{equation*}
This is the same if computed as
\begin{equation*}
-2+7+3 = (-2+7)+3 = 5+3 = 8.
\end{equation*}
Similarly for multiplication,
\begin{equation*}
-2\cdot 7\cdot 3 = (-2\cdot 7)\cdot 3 = -14\cdot 3 =-42.
\end{equation*}
Or, equivalently
\begin{equation*}
-2\cdot 7\cdot 3 = -2\cdot (7\cdot 3) = -2\cdot 21 =-42.
\end{equation*}
Example 1.15. FOIL’ing.
The FOIL (First-Outer-Inner-Last) rule for expanding expressions of the form \((a+b)(c+d)\) is a consequence of the distributive property.
\begin{align}
(a+b)(c+d) \amp=\quad (a+b)c + (a+b)d \tag{1.1}\\
\amp=\quad ac+bc + ad+bd \tag{1.2}\\
(a+b)(c+d) \amp=\quad \underbrace{ac}_{\text{first}} + \underbrace{ad}_{\text{outer}} + \underbrace{bc}_{\text{inner}} + \underbrace{bd}_{\text{last}}\tag{1.3}
\end{align}
Example 1.16.
Let’s evaluate \(31\times 49\) using properties of real numbers to avoid needing a calculator. I can write \(31 = 30+1\) and \(49 = 50-1\) which relates them to a nearby multiple of \(10\text{.}\) Then
\begin{align*}
31\times 49 \amp=\quad (30+1)\times (50-1)\\
\amp=\quad 30\times 50-30+50-1\\
\amp=\quad 30\times 50+20-1\\
\amp=\quad 30\times 50+19\\
\amp=\quad 3\times 10\times 5\times 10+19\\
\amp=\quad 15\times 100+19\\
\amp=\quad 1500+19\\
\amp=\quad 1519.
\end{align*}
Checkpoint 1.17.
Multiply \(61\times 18\) in a creative way to illustrate the use of commutativity, associativity, and distributivity. Think about which property you are using in each step.
