02Bnumberlaws

= Number Laws = toc Here we are putting into formal language some things you already know.

You do not __need__ to remember the __names__ of these laws.

Commutative Law
The ** commutative law ** says that for __some__ operations, it doesn't matter what order you write the values in.

math \\ \textbf{(1)} \qquad x + y = y + x \\ \\ . \qquad \qquad \text{Eg : } 3 + 2 = 5 \;\; \textit{ and } \;\; 2 + 3 = 5 math

math \\ \textbf{(2)} \qquad x - y \neq y - x \\ \\ . \qquad \qquad \text{Eg : } 3 - 2 = 1 \;\; \textit{ but } \;\; 2 - 3 = -1 math

math \\ \textbf{(3)} \qquad x \times y = y \times x \\ \\ . \qquad \qquad \text{Eg : } 3 \times 2 = 6 \;\; \textit{ and } \;\; 2 \times 3 = 6 math

math \\ \textbf{(4)} \qquad x \div y \neq y \div x \\ \\ . \qquad \qquad \text{Eg : } 3 \div 2 = \frac{3}{2} = 1.5 \;\; \textit{ but } \;\; 2 \div 3 = \frac{2}{3} \approx 0.667 math

So : The ** commutative law ** is __true__ for addition and multiplication, but __not true__ for subtraction and division.

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

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

Solutions

Associative Law
The ** associative law ** says that for __some__ operations, it doesn't matter the way you group the calculations.

math \\ \textbf{(1)} \qquad (x + y) + z = x + (y + z) \\ \\ . \qquad \qquad \text{Eg : } (1 + 2) + 3 = 6 \;\; \textit{ and } \;\; 1 + (2 + 3) = 6 math

math \\ \textbf{(2)} \qquad (x - y) - z \neq x - (y - z) \\ \\ . \qquad \qquad \text{Eg : } (1 - 2) - 3 = -4 \;\; \textit{ but } \;\; 1 - (2 - 3) = 2 math

math \\ \textbf{(3)} \qquad (x \times y) \times z = x \times (y \times z) \\ \\ . \qquad \qquad \text{Eg : } (1 \times 2) \times 3 = 6 \;\; \textit{ and } \;\; 1 \times (2 \times 3) = 6 math

math \\ \textbf{(4)} \qquad (x \div y) \div z \neq x \div (y \div z) \\ \\ . \qquad \qquad \text{Eg : } (1 \div 2) \div 3 = \frac{1}{6} \;\; \textit{ but } \;\; 1 \div (2 \div 3) = \frac{3}{2} math

So : The ** associative law ** is __true__ for addition and multiplication, but __not true__ for subtraction and division.

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

Solutions

Identity Laws
The **identity laws** deal with operations that leave the pronumeral unchanged (ie they keep their __identity__)

math \\ \textbf{(1)} \qquad x + 0 = x \\ \\ \textbf{(2)} \qquad x - 0 = x \\ \\ \textbf{(3)} \qquad x \times 1 = x \\ \\ \textbf{(4)} \qquad x \div 1 = x math

Inverse Laws
The ** additive inverse law ** says that if you __add__ a value and its negative together, you get zero

math \textbf{(1)} \qquad x + (-x) = 0 math

The ** multiplicative inverse law ** says that if you __multiply__ a value by its __reciprocal__, you get one.

math . \qquad \qquad \text{The } \textbf{reciprocal} \text{ of a number } (x) \text{ is the result of calculating } 1 \div x math

math \\ \textbf{(2)} \qquad x \times \dfrac{1}{x} = 1 \\ \\ . \qquad \qquad \text{Eg : } 3 \times \dfrac{1}{3} = 1 \\ \\ . \qquad \qquad \text{Eg : } \dfrac{3}{4} \times \dfrac{4}{3} = 1 math

Note : You are used to finding the reciprocal from learning how to divide fractions.

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

Solutions

Closure Law
The ** closure law ** says that for __some__ operations, if you perform that operation on values from a set of numbers, the answer will be in the same set of numbers.

For example, using the set of positive integers (or the set of positive whole numbers -- called the **natural numbers** or "counting numbers")
 * Addition is closed. (the sum of any two positive integers will always be a positive integer)
 * Subtraction is __not__ closed. (subtracting two positive integers doesn't always give a positive integer as an answer)
 * Multiplication is closed. (the product of any two positive integers will always be a positive integer)
 * Division is __not__ closed. (dividing two positive integers doesn't always give a positive integer as an answer)

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

For example, using the set of irrational numbers (including surds)
 * Addition is closed
 * Subtraction is closed
 * Multiplication is not closed
 * Division is not closed

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

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.

Extension: 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?

Activity 4 : Let A = the set of integer powers of 2 {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

Extension: How could you __prove__ that A is closed for a particular operation?

Solutions Go to top of page flat