Question: What common properties can be found in mathematical systems?

We previously defined multiplication modulo n. Now we are going to define addition modulo n. In this addition, the sum of two numbers is the remainder when the ordinary sum of the two numbers is divided by n. For example, in addition modulo 8,

6 +₈ 5 = 3 since 6 + 5 = 11 and the remainder when 11 is

divided by 8 is 3

1. Find all sums modulo 8 of the numbers less than 8. What observation can you make? Include 0 in the set of numbers less than 8.

2. Similar types of addition can be defined for other numbers. Write a definition for addition modulo 5. Repeat #1 for sums modulo 5, sums modulo 6, and sums modulo 7. What observations can you make now?

3. An addition table can be made for each of the types of addition you just computed. Make an addition table for addition modulo 5, addition modulo 6, addition modulo 7 and addition modulo 8.

Notice that with the definition of addition modulo n, we have another mathematical system. That is, a set of elements together with a rule for combining those elements. In this case, the elements are numbers and the operation is addition modulo n.

4. Calculate the following sums.

9 +₁₂ 5      17 +₂₀ 14   37 +₆₀ 49     14 +₁₆ 10

A matrix is a rectangular array of numbers, i.e., a systematic arrangement of numbers in rows and columns enclosed by brackets. Below is an example.

The numbers inside the brackets are called the elements of the matrix. Because it has two rows and four columns, it is referred to as a 2 by 4 matrix, written 2 × 4. The number of rows and the number of columns are the dimensions of the matrix.

5. Find all of the 2 × 2 matrices with elements 0 and 1.

6. Make a definition of addition for matrices.

7. Does your definition of addition work for the following problems? If so, find the sum.

The sum of two matrices having the same dimensions is found by adding the corresponding elements of the two matrices and placing the sum in the same position in the resulting matrix.