Does order count in addition? How can you tell just by looking at the "shape" of the + table?
How is the location of 5 + 8 related to the location of 8 + 5 in the table? Close Tip
Problem C8
Go back to the context (units digits or remainders by 10) and explain your answer to Problem C7.
Problems C7 and C8 show that this algebraic structure is commutative under addition; that is, the order of the object being added does not matter.
Problem C9
Does order count in multiplication? How can you tell just by looking at the "shape" of the * table?
Problem C10
Go back to the context (units digits or remainders by 10) and explain your answer to Problem C9.
Problems C9 and C10 show that this structure is commutative under multiplication; that is, the order of the objects being multiplied does not matter.
Problem C11
Is it true of this new addition that adding 0 doesn't change a number? Explain.
Since 0 does not change a number under addition, it is the identity element of this structure; that is, adding zero to a number does not change the value of that number.
The opposite or negative of a number is the number you have to add to it to get 0. In ordinary arithmetic, the opposite of 4 is -4 (think of a thermometer). In ordinary arithmetic, every number has a negative (what's the negative of 0?). In our little algebraic structure above, units arithmetic, the opposite of 4 is 6, because 4 + 6 = 0.
Problem C12
Does every number have an opposite in this system? Explain.
You can use the addition table above to find opposites. Close Tip
Under addition, 4 and 6 are additive inverses, because 4 + 6 = 0. Numbers which are inverses under addition are more typically referred to as opposites.
Problem C13
Give a rule for determining the opposite of a number, if it has one.
Video Segment This short video segment describes how to find opposites using the table of addition in mod 10.
You can find this segment on the session video, approximately 13 minutes and 43 seconds after the Annenberg Media logo.
