If we cannot tell at a glance how many objects there are in a group we can count the objects. When we have finished counting the last number we said tells the number of objects in the group. A number like “five” is used to tell how many objects there are in a group of objects is called a cardinal number. If we wish to describe the position of the last object in the group, we would use the number word “fifth.” A number like “fifth” which is used to describe the position of an object when it is in a group of objects is called an ordinal number.
THE NUMBER OF A SET
Beginning with one, we see that one and one more is two, two and one more is three and one more is four, etc. These arranged numbers are the ones we use in counting. They are called counting numbers, or natural numbers. 4,5,6,7,etc. When we refer to all the counting numbers and zero,
we call this set of numbers the whole numbers, or integers.
1. USING NUMBER SYMBOLS AND NUMBER NAMES
Thousands of years ago, man began to represent numbers by the use of number symbols. Primitive man kept a record of a number by making scratches on a stone or in the dirt, by cutting notches in a stick, or by tying knots in a rope. He also kept a count of his animals by placing in a pile one pebble for each animal.
In each of these methods of recording numbers one mark stood for one object and there were as many marks as there were objects. In those early days of recording numbers, there was no one mark to stand for several objects.
After many years, man learned to represent several objects by using a single symbol. This was a great step forward in the development of number symbols.
Different peoples in ancient civilizations created number sybols of various kinds. Zero, the empty set, or something to hold a place came along much later. When I play Sudoku I realize that Zero was a hard concept for the Asian to understand for a long time. Kakuro is another game of numbers that does not use Zero.
The number symbol or number name that is used to represent a number is called a numeral.
We must be aware of the fact that a numeral and the number it represents are not the same thing. The numeral is only the symbol for the number, not the number itself. we know for example that the number 4 is larger than the number 3 even though we might see a larger printed 3 and a small printed 4.
In our mathematics work we shall add, subtract, multiply and divide numbers, not numerals. At times however to avoid awkward and cumbersome language expressions we will use the word number when we should really use the word numeral. For example, we may say “write the number” instead of saying “write the numberal that represents the number.” This is commonly accepted practice.