Properties of operations

Definitions:

An operation, \(\star\), has an identity, \(e\), if \(a \star e = a\) and \(e \star a = a\). For example, the identity for addition is 0 because \( a + 0 = a\) and \(0 + a = a\).

  1. Does multiplication have an identity element? If so, what is it?
  2. Does subtraction have an identity element? If so, what is it?
  3. Does division have an identity element? If so, what is it?

An operation, \(\star\), has an inverse, \(x\), if \(a \star x = e\) and \(x \star a = e\). For example, in addition \(-a\) is the inverse of \(a\) because \( a + (-a) = 0\) and \((-a) + a = 0\).

  1. Does multiplication have an inverse? Why or why not?
  2. Does subtraction have an inverse? Why or why not?
  3. Does division have an inverse? Why or why not?

An operation, \(\star\), is commutative if \(a \star b = b \star a\). For example, addition is commutative because \(2 + 3 = 3 + 2\), which works for any two numbers.

  1. Is multiplication commutative? If so, give an example. If not, give a counter-example.
  2. Is subtraction commutative? If so, give an example. If not, give a counter-example.
  3. Is division commutative? If so, give an example. If not, give a counter-example.

An operation, \(\star\), is associative if \(a \star (b \star c) = (a \star b) \star c\). For example, addition is associative because \(1+ (2 + 3) = (1 + 2) + 3\), which works for any three numbers.

  1. Is multiplication associative? If so, give an example. If not, give a counter-example.
  2. Is subtraction associative? If so, give an example. If not, give a counter-example.
  3. Is division associative? If so, give an example. If not, give a counter-example.

What properties were used in the following equations? Use the vocabulary from CCSSM (page 90, table 3).

  1. \( a + (n + 49) = (a + n) + 49 \)
  2. \( 6\frac{4}{5} \times \frac{1}{2} = \frac{1}{2} \times 6\frac{4}{5} \)
  3. \( (2b)c = 2(bc) \)
  4. \( z + -z = 0 \)
  5. \( 29 \times \frac{7}{9} \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)
  6. \( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)
  7. \( xy^2 + 0 = xy^2 \)
  8. \( 1 \cdot x = x \)

Use properties to make the following computations easier to do mentally.

  1. \( (7 + 40) + 3 \)
  2. \( 25 \times (4 \times 27.2 ) \)
  3. \( 24 \times 38 + 24 \times 12 \)
  4. \( 57^2 + (43 \times 57) \)

Source: Reconceptualizing Mathematics, Chapter 12.1, by Sowder, Sowder and Nicholson


Challenge

Suppose Δ is a new operation that only works for numbers 1 through 5. The table below gives the results of the operation. For example, 2 Δ 5 = 4.

Δ 1 2 3 4 5
1 4 5 1 2 3
2 5 1 2 3 4
3 1 2 3 4 5
4 2 3 4 5 1
5 3 4 5 1 2

 

  1. Does Δ have an identity? If so, what?
  2. Does Δ have inverses? If so, what are they?
  3. Is Δ commutative? Why or why not?
  4. Is Δ associative? If so, give an example. If not, give a counter-example.