Is it true of this new multiplication that multiplying by 1 doesn't change a number? Explain.
Under multiplication, 3 and 7 are inverses, because 3 * 7 = 1. Numbers which are inverses under multiplication are more typically referred to as reciprocals. The reciprocal of a number is the number you have to multiply it by to get 1. In ordinary arithmetic, the reciprocal of 3 is 1/3, because 3 * 1/3 = 1. In ordinary arithmetic, every number except 0 has a reciprocal. In our algebraic structure above, the reciprocal of 3 is 7, because 3 * 7 = 1.
Problem C15
Which numbers have a reciprocal in this system?
You can use the multiplication table above to find reciprocals. Not all numbers have reciprocals in this system! Close Tip
What's common among all the numbers that have reciprocals? What's common among all the numbers that don't have reciprocals?
Think about the types of numbers which do not have reciprocals, then see if you can decide why. Close Tip
Video Segment This video segment describes how to find reciprocals using the table of multiplication in mod 10, followed by a short discussion of why some numbers do not have reciprocals. Watch this video segment after you have completed Problem C16.
You can find this segment on the session video, approximately 14 minutes and 43 seconds after the Annenberg Media logo.
Problem C17
Which of the properties found in units digit arithmetic are also true in ordinary integer arithmetic?
Problem C18
In ordinary arithmetic, if the product of two numbers is 0, one of them must be 0. Is that true here? What makes it work in ordinary arithmetic that doesn't carry over to our new system?
Is there more than one whole number that is equivalent to 0 in this structure? How is this different from ordinary arithmetic? Close Tip
Problem C19
Find, describe, and explain at least two patterns in each table that you haven't yet used.
Is there any relationship between rows of numbers that are inverses? Close Tip
During the 19th century, mathematicians expanded number systems into larger sets. This led to the development of the concepts of groups and fields, which you can examine in the following optional problems.
