Since changing the order of the division did not give the same result,
division is not commutative . The commutative properties only apply to addition and multiplication!
Addition and multiplication
are commutative.
Subtraction and Division
are not commutative.
If you were asked to simplify this expression, how would you do it and what would your answer be?
$7+8+2$
Some people would think
$7+8\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}15$ and then
$15+2\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}17.$ Others might start with
$8+2\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}10$ and then
$7+10\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}17.$
Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.
When multiplying three numbers, changing the grouping of the numbers gives the same result.
You probably know this, but the terminology may be new to you. These examples illustrate the
associative property .
Associative property
$\begin{array}{cccc}\mathbf{\text{of Addition}}\hfill & & & \text{If}\phantom{\rule{0.2em}{0ex}}a,b,c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\phantom{\rule{0.2em}{0ex}}(a+b)+c=a+(b+c)\hfill \\ \mathbf{\text{of Multiplication}}\hfill & & & \text{If}\phantom{\rule{0.2em}{0ex}}a,b,c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\phantom{\rule{0.2em}{0ex}}(a\xb7b)\xb7c=a\xb7(b\xb7c)\hfill \end{array}$
When adding or multiplying, changing the
grouping gives the same result.
Let’s think again about multiplying
$5\xb7\frac{1}{3}\xb73.$ We got the same result both ways, but which way was easier? Multiplying
$\frac{1}{3}$ and
$3$ first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?