For the table below, answer the following questions considering the operations +, -, ×, ÷, ^:
Which operations are commutative? Give a counter example for each non-commutative operation.
commutative: \(+, \times \);
\(1 - 2 \neq 2 - 1\)
\(1 \div 2 \neq 2 \div 1 \)
\( 1^2 \neq 2^1 \)
Which operations have an identity element? Please state them.
identity element for addition: 0
identity element for multiplication: 1
subtraction, division, and exponents do not have identity elements because the operations are not commutative
Which operations have inverse elements? Please state them.
inverse element for addition: \(-a\)
inverse element for multiplication: \( \dfrac{1}{a} \)
Which operations are associative? Give a counter example for each non-associative operation.
associative: \(+, \times\);
\( 10 - (9 - 8) \neq (10 - 9) - 8 \)
\( 10 \div (5 \div 1) \neq (10 \div 5) \div 1 \)
\( 1^{(2^3)} \neq (1^2)^3 \)
What is the full name of the distributive property? Please give an example for how the property works.
distributive property of multiplication and addition
\( a (b + c) = a \cdot b + a \cdot c \)
be careful as students will often "distribute" any time there is a coefficient in front of parentheses, e.g. \( a(b \cdot c) \rightarrow ab \cdot bc \)
Identify the properties
What properties were used in the following equations?
\( a + (n + 49) = (a + n) + 49 \)
associative property of addition