Mathematical Task 2.2A

Numbers and Numerals As mentioned earlier, the study of the set of whole numbers, W = {0, 1, 2, 3, 4, …} , is the foundation of elementary school mathematics. But what precisely do we mean by the whole number 3? A number is an idea, or an abstraction, that represents a quantity. The symbols that we see, write, or touch when representing numbers are called numerals. There are three common uses of numbers. The most common use of whole numbers is to describe how many elements are in a finite set. When used in this manner, the number is referred to as a cardinal number. A second use is concerned with order. For example, you may be second in line, or your team may be fourth in the standings. Numbers used in this way are called ordinal numbers. Finally, identification numbers are used to name such things as telephone numbers, bank account numbers, and social security numbers. In this case, the numbers are used in a numeral sense in that only the symbols, rather than their values, are important. Before discussing our system of numeration or symbolization, the concept of cardinal number will be considered.

Ordering Whole Numbers A meaningful way of comparing two whole numbers is to use 1-1 correspondences. We can say that 2 is less than 5 , since any set with two elements matches a proper subset of any set with five elements

Mathematical Task 2.2B

Number Systems

Figure 2.14

Roman Numerals The Roman numeration system, which developed between 500 b.c.e. and 100 c.e. also uses grouping, additivity, and many symbols.

Roman Numerals The Roman numeration system is a subtractive system since it permits simplifications using combinations of basic Roman numerals Thus, when reading from left to right, if the values of the symbols in any pair of symbols increase, group the pair together. The value of this pair, then, is the value of the larger numeral less the value of the smaller. One may wonder if IC is an acceptable way to write 99. The answer is no because of the additional restriction that only I's, X's and C's may be subtracted, but only from the next two larger numerals in each case.

1344, 2993, 249

Mayan Numeration System The Mayan numeration system, which developed between 300 and 900 c.e., was a vertical place-value system, and it introduced a symbol for zero.

Mayan Numerals

Mathematical Task 2.3A

Figure 2.27

Mathematical Task 2.3B

Figure 2.31