02Bsolna

= Number Laws = toc

Solutions to Activities

Activity 1
Use the values 5 and 10 to demonstrate the commutative law for addition, subtraction, multiplication and division.

__**Solution:**__

math \\ \textbf{(1)} \qquad 5 + 10 = 10 + 5 \qquad 15 = 15 \\ \\ \textbf{(2)} \qquad 5 - 10 \neq 10 - 5 \qquad -5 \neq 5 \\ \\ \textbf{(3)} \qquad 5 \times 10 = 10 \times 5 \qquad 50 = 50 \\ \\ \textbf{(4)} \qquad 5 \div 10 \neq 10 \div 5 \qquad \frac{1}{2} \neq 2 math

Activity 2
Find a pair of numbers for which the commutative law is true for subtraction and division.

__**Solution:**__

Answers will vary.

5 and 5 is one possible answer math \\ \textbf{(1)} \qquad 5 - 5 = 5 - 5 \qquad 0 = 0 \\ \\ \textbf{(2)} \qquad 5 \div 5 = 5 \div 5 \qquad 1 = 1 math

Activity
Use the values 6, 2 and 4 (in that order) to demonstrate the associative law for addition, subtraction, multiplication and division

__**Solution:**__

math \\ \textbf{(1)} \qquad (6+2)+4 = 6+(2+4) \qquad 12 = 12 \\ \\ \textbf{(2)} \qquad (6-2)-4 \neq 6 - (2-4) \qquad 0 \neq 8 \\ \\ \textbf{(3)} \qquad (6 \times 2) \times 4 = 6 \times (2 \times 4) \qquad 48 = 48 \\ \\ \textbf{(4)} \qquad (6 \div 2) \div 4 \neq 6 \div (2 \div 4) \qquad \frac{3}{4} \neq 12 math

Activity
Find the reciprocal of 2¼ and demonstrate that the multiplicative inverse law works for these values (ie their product is 1)

__**Solution:**__

math \\ \text{As an improper fraction : } 2\dfrac{1}{4} = \dfrac{9}{4} \\ \\ \text{The reciprocal of } \; \dfrac{9}{4} \; \text{ is } \; \dfrac{4}{9} \\ \\ \text{hence} \\ \\ \dfrac{9}{4} \times \dfrac{4}{9} = 1 math so the multiplicative inverse law works in this example.

Activity 1
Make up examples that demonstrate why the set of positive integers is not closed for subtraction and division

__**Solution:**__

Answers will vary

2 and 4 is one possible answer: math \\ \textbf{(1)} \quad 2 - 4 = -2 \qquad -2 \textit{ is not a positive integer} \\ \\ \textbf{(2)} \quad 2 \; \div \; 4 = \frac{1}{2} \qquad \frac{1}{2} \textit{ is not a positive integer} math

Activity 2
Make up examples that demonstrate why the set of irrational numbers is not closed for multiplication and division

__**Solution:**__

math \sqrt{8} \; \text{ and } \; \sqrt{2} \; \textit{ is one possible answer} math

math \\ \textbf{(1)} \quad \sqrt{8} \times \sqrt{2} = \sqrt{16} = 4 \qquad 4 \textit{ is not an irrational number} \\ \\ \textbf{(2)} \quad \sqrt{8} \div \sqrt{2} = \sqrt{4} = 2 \qquad 2 \textit{ is not an irrational number} math

Activity 3
State whether the set of rational numbers is closed or not closed for addition, subtraction, multiplication and division. Give examples to support your statement.

__**Solution:**__

For the set of rational numbers
 * Addition is closed
 * Subtraction is closed
 * Multiplication is closed
 * Division is closed

Examples will vary. Remember that rational numbers includes all fractions and all whole numbers(integers).

2 and 5 is one possible answer.

math \\ \textbf{(1)} \quad 2 + 5 = 7 \qquad 7 \textit{ is rational} \\ \\ \textbf{(2)} \quad 2 - 5 = -3 \qquad -3 \textit{ is rational} \\ \\ \textbf{(3)} \quad 2 \times 5 = 10 \qquad 10 \textit{ is rational} \\ \\ \textbf{(4)} \quad 2 \div 5 = \frac{2}{5} \qquad \frac{2}{5} \textit{ is rational} math

Extension 1
To __prove__ that a set is __not__ closed, you only have to give one example. For a set to be closed, it has to be true for __all__ possible combinations. Giving one example does not __prove__ that it is true.

How could you __prove__ that the set of rational numbers is closed for a particular operation?

Solution to Extensions

Activity 4
Let A = the set of integer powers of 2 {ie 2 n where n is any integer (see below)}. State whether A is closed or not closed for addition, subtraction, multiplication and division. Give examples to support your statement. math A = \big\{ ... \; \frac{1}{8}, \; \frac{1}{4}, \; \frac{1}{2}, \; 1, \; 2, \; 4, \; 8, \; 16, \; ... \big\} math

__**Solution:**__

For the set of integer powers of 2 (see A)
 * Addition is not closed
 * Subtraction is not closed
 * Multiplication is closed
 * Division is closed

Examples will vary.

8 and 2 is one possible answer

math \\ \textbf{(1)} \quad 8 + 2 = 10 \qquad 10 \textit{ is not in A} \\ \\ \textbf{(2)} \quad 8 - 2 = 6 \qquad 6 \textit{ is not in A} \\ \\ \textbf{(3)} \quad 8 \times 2 = 16 \qquad 16 \textit{ is in A} \\ \\ \textbf{(4)} \quad 8 \div 2 = 4 \qquad 4 \textit{ is in A} math

Extension 2
How could you __prove__ that the set of integer powers of 2 (2 n where n is any integer) is closed for multiplication and division?

Solution to Extensions Go to top of page flat

.