Introduction to Counting Numbers

=== by Bob Sutherland ===

An explaination of the different names Mathematics teachers use to refer to the simple numbers we learned to count with.

Our Counting Numbers

There are some unique things about a plain list of our counting numbers. The first item that I notice is that you will probably spend more time during your lifetime learning to count using these numbers than any of my other lists of numbers on this website. You may still be learning. Can you count up to a googol yet? Yes there is such a number and most adults still have not learned how to count that high. It is time we sent them all back to school to learn a bit more.

The second unique thing about our counting numbers is how many unique names parents and teachers in school will use to refer to a plain list of our numbers. It is no wonder that students get confused when we keep changing the name of something so fundamental as our list of numbers.

Preschool children and early elementary pupils will hear their parents, teachers and other people in their environment refer to our numbers as the Counting Numbers. In high school the Mathematics teacher will inform the students that this ordinary list of numbers is from now on to be called the set of Whole Numbers with the list continuing on to infinity. Then the Mathematics teacher will tell the students that if the number zero is missing from the list the name changes and this list of numbers becomes the set of Natural Numbers.

A name change just because we are missing a zero? That is a big change for a number that numerically represents nothing.

The name calling does not end there though. Just in case there might still be some students in the Mathematics class who are not totally confused the teacher will again rename this list of numbers in a few days as the set of Positive Integer Numbers. Before the students can figure out what happened the high school Mathematics teacher will be calling this list of numbers the Base 10 Decimal Numbers and also the Arabic Numerals.

So what is going on? Why does a Mathematics teacher use so many different names when referring to a simple list of numbers? Well all high school teachers are responsible for introducing students to the vocabulary of the subject they are teaching. In Mathematics a subtle little change in the context in which you are referring to our Counting Numbers changes the name of the list of numbers.

Counting Numbers
I am sure that parents and elementary school teachers are more concerned with teaching children to count than with the name of the number system. Counting Numbers is a simple, self-explanatory, natural name for the parents and teachers to use in their conversations when teaching the children to count.
Whole Numbers
At some point in history someone with an interest in public relations, school image and personal reputations became concerned over the issue that university Mathematics professors and high school Mathematics teachers were using such childish sounding phrases as Counting Numbers. The term Whole Numbers was therefore invented because it sounded so much more sophisticated and distinguished than Counting Numbers. (This is my theory, do you have a better one?)

Similar to a whole apple or a whole pie a whole number is one that has not been sliced up into fractions or decimal fractions.

Natural Numbers
As a child when you were learning how to count you started at the number one. If I were to ask you to count today you would probably start at the number one. Everyone has usually started counting at number one going back in history for many centuries to the caveman's invention of numbers. The number zero is a new number that Mathematicians only invented a few hundred years ago to make subtraction and arithmetic easier. Natural Numbers is a term we use for the numbers we naturally count with starting at number one.
Positive Integer Numbers
The term Positive Integer Numbers includes all of the numbers with a value greater than or equal to zero. The numbers may or may not have an optional positive (plus) sign in front of the number. Integer Numbers do not include any numbers that have a decimal point or fractional part.

Use the term Positive Integer Numbers when comparing the numbers with Negative Integer Numbers and Real Numbers.

Negative Integer Numbers
The term Negative Integer Numbers includes all of the numbers with a value less than zero, a negative (minus) sign in front of the number, and no decimal point or fractional part.
Real Numbers
Real numbers can be thought of as including all of the numbers along a number line from negative infinity through zero to positive infinity. Real numbers include all of the Whole Numbers. Real Numbers include all of the positive and negative Integer numbers. Real Numbers include all of the numbers with a decimal point or fraction as part of the number.

In computer programming languages numbers are usually classified as being one of two types: either integer numbers because they do not have a decimal point, or real numbers with a decimal point.

Arabic Numerals
Use the term Arabic Numerals when comparing our number system to a number system that has different symbols for the digits, such as the Roman Numerals.
Base 10 Decimal Numbers
Use the term Decimal Numbers when contrasting our base 10 numbers to a number system that has a different base, such as Binary Numbers (base 2), Octal Numbers (base 8), Hexadecimal Numbers (base 16) or a base 26 system that uses the letters of the alphabet as digits.