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; the relationships between them and the operations with them. Arithmetic is the art of counting.

First, we need a plan 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 axiom, one ten. We count units, which is to say, we must be able to call each one by the same name.

"One apple, two apples, three apples."

"One ten, two tens, three tens."

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. And so on.

A whole number is composed of units,

and not parts of a unit: not half of one, or a third, or a millionth. Those are called fractions. And so we speak of whole number arithmetic and whole number numeration; it does not include fractions or decimals.

Numerals

To do arithmetic, of course, the numbers must have names and symbols. The names (in English) for the whole numbers are "One, two, three, four" and so on. We call them the counting-names; sometimes the counting-numbers.

The symbols that represent the numbers are "1, 2, 3, 4," and so on. They are called numerals. The student is perhaps familiar with Roman numerals.  '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.

0 (zero) is also called a whole number. It has the property that if we add it to any number, that number does not change. 2 + 0 = 2.

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

Children often learn the concepts of arithmetic with manipulatives, which are actual numbers -- physical units -- such as matchsticks or blocks. They enable
the child to grasp the idea of a number, and thus
eventually represent numbers with symbols.

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.  

The name of the third is Three.  3.  

(Notice how the ordinal numbers are unavoidable.)

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 -- those units -- 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.

That is the first lesson in addition. It links the sequence of the counting-names with their cardinality: how many.

Tens.  Upon adding one to Nine, the name of that number is Ten: 10.

Ten is composed of ten Ones.

Its numeral is 1 followed by 0 (zero).

Let Ten now be the unit.  On counting the Tens, here are their names and their numerals:

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 numbers.

Here are the numbers between 30 and 40:

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

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

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 the numbers 100 through 999, and construct their numerals.

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.  And in Lesson 3, we will extend our system to decimals.

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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