Lesson 1

NUMERATION OF THE
WHOLE NUMBERS

Units

Numerals

The sequence of counting-names

Tens

Numbers between two consecutive Tens

Hundreds

Numbers between two consecutive Hundreds

ARITHMETIC is the science that studies numbers. Its object is to express the relationships between them ("5 is half of 10"), and clarify the operations between them: addition, subtraction, multiplication and division. All of arithmetic is based ultimately on counting; on a sequence of names. Arithmetic therefore must begin with a system for naming numbers with words, and then writing them with symbols. That is called a system of numeration. Numeration is the foundation upon which arithmetic is built and expressed.

The current system, which is in worldwide use, is the decimal system. That means it is based on what are called the powers of 10 (Lesson 2). Decem in Latin means 10.

Units

A unit is whatever we call one. One apple, one orange, one inch.  We count a collection of units. And each unit we count must have the same name. (We can count "boys." We can count "girls." But we cannot count boys and girls -- at least not until we call them "children.")

When we measure, the unit is the standard to which we relate quantities of the same kind.  One meter, for example, for measuring length; one second for measuring time; one pound for measuring weight; one liter for measuring volume. And so on.

A unit is that form in accordance with which each thing that is we call one.

(After Euclid, Book VII, Definition 1.)

There are several classes of numbers; fractions, decimals, mixed numbers. A whole number is composed of indivisible units. We count them using the names of the whole numbers: "One, two, three, four" and so on.  We call those the counting-names; sometimes called the counting-numbers.

The units we count are actually called the natural numbers. But it is conventional to speak of whole number arithmetic and whole number numeration;
it does not include fractions or decimals.

Numerals

A numeral is a symbol for a number.  'V' is the Roman numeral for this number:

'5' is the  Arabic numeral.  For it was the Arab mathematicians who introduced them into Europe from India, where their forms evolved. "Five" is the English word.

Again, a number in arithmetic, as when we speak of a number of children, is what we actually distinguish and count, not the names or symbols that we count with.  Nevertheless, it has become common to call the numerals themselves -- 1, 2, 3, 4, and so on -- "numbers."

The sequence of counting-names

The English name of the first is One. This is its numeral: 1.  The name of the second is Two:  2.  Here is the sequence of the first nine names and their numerals:

Starting with One, we say that each number is "one more" than the previous number.  We say that we have "added one" to the previous number.

To count those figures means to match each figure with the sequence of counting-names.

Five, we say, is one more than Four, and that we have added one to Four to reach Five.

Not only is that the first lesson in addition. It links the sequence (ordinality) of the counting-names with their cardinality: how many.

Thus "Five" means five units, even though we do not say or write the word units. For there is no "Five" or "5" apart from that name and symbol for how many, and its position in the sequence of numbers.

Tens.  Upon adding one to Nine, the name of that number is Ten: 10.
Its numeral is 1 followed by 0 (zero).

Ten is a collection of ten Ones.

If we now let Ten be the unit, then we count by Tens and name them as follows:

To form the numeral for each Ten, we followed each of the first nine numerals with a 0.

Numbers between two consecutive Tens.  To name the numbers between two consecutive Tens -- between 30 and 40, for example -- add successively to the lower Ten the first nine numbers.  To write their numerals, successively replace the 0 of the lower Ten with each numeral of the first nine.

Here are the numbers between 30 and 40:

The numbers between 10 and 20, however, have unique names:

We now have named and constructed the numerals for the numbers 1 through 99.

Hundreds.  A collection of ten Tens form the number One Hundred. Its numeral is 100.  Upon letting One Hundred be the unit, we count those Hundreds and name them as follows:

Numbers between two consecutive Hundreds.  To name the numbers between two consecutive Hundreds -- between 300 and 400, for example -- add successively to the lower Hundred the first ninety-nine numbers. To write their numerals, successively replace the two 0's of the lower Hundred with the numerals of the first ninety-nine numbers.

For example:  Three Hundred One (301), Three Hundred Two (302), Three Hundred Three (303), . . . , Three Hundred Ninety-eight (398), Three Hundred Ninety-nine (399).

In this way we name and construct the numerals for the numbers 100 through 999.

We have now then constructed the names and the numerals for all the numbers 1 through 999.  By continuing to add one, there is no limit to the numbers of numbers. We will see in the next Lesson that to name any whole number, however large, it is sufficient to know the names through 999.

Also in the next Lesson we will analyze our system of numeration in terms of place value.

The student should begin mastering Elementary Addition and the Multiplication Table.

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Lesson:  The Powers of 10

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