In number systems, it is sometimes useful to find identity and inverse elements. Such elements can tell us more about the behavior of numbers in a particular system. Let's explore what this means.
The identity element for addition (i.e., the additive identity element) is a number that, when added to any other number in the table, doesn't change its value.
The inverse element for addition (i.e., the additive inverse element) is a number that, when added to any other number in the table, gives back the identity element.
What do these elements tell us about the finite number system?
Problem A3
Find the additive identity element in the addition table you created. Is there one number that works for every other number in your table?
Problem A4
a.
Find the additive inverse element for the number 4 in the table you created.
Remember, the additive identity element for this number system is 0. Close Tip
b.
Find the additive inverse for as many of the elements in your set as you can. Does every number in your table have an additive inverse?
Problem A5
a.
The commutative law for addition states that a + b = b + a. Does this law hold for the finite number system in your table? Why or why not? Note 1
b.
The associative law for addition states that (a + b) + c = a + (b + c). Does this law hold for the finite number system in your table? Why or why not?
Problem A6
a.
How could you use the addition table to subtract?
Consider subtraction as a way of undoing addition. Think of what number you'd need to add to the second number in the subtraction problem in order to obtain the first number (a - b = x; or b + x = a). Close Tip
b.
Is it possible to subtract any number from any other number using this number system?
A set is said to be closed under a given operation if the result of the operation is always in the set. For example, the integers as we know them are closed under addition, because whenever you add two integers, you get an integer. They are not closed under division because 5 divided by 3 is not in the set -- it is not an integer.
Problem A7
Is this finite set closed under addition?
Note that you can further explore units digit arithmetic in Learning Math: Patterns, Functions, and Algebra,Session 9.
