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A R I T H M E T I C

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 efficient calculation.

First, we need a plan for naming numbers with words, and then for 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

We can only count things of the same kind, that is, that have the same name.

"One apple, two apples, three apples."

"One ten, two tens, three tens."

"One thing, two things, three things" -- on the table.

Whatever has that name is the unit. It is what we are calling one. It is what we are counting.

When we measure, the unit is the standard we adopt to 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 distinct, indivisible units:

When we measure, we need the names of fractions or decimals. And so we speak of whole number arithmetic and whole number numeration; it does not include fractions or decimals.

Numerals

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

(They are also called the natural numbers. )

The symbols for the counting-numbers are "1, 2, 3, 4," and so on. They are called numerals. The student is no doubt 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.

A numeral -- 5 -- like any symbol is not the thing. The thing is the number.

Thus we speak of a number of children. That number is what we actually count, not the name or numeral that we count with. In different languages -- "cinco," "cinq," "fünf" -- the different names, after all, refer to the same number.  It has become common nevertheless 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 the number with a symbol.

The sequence of counting-names

The English name of the first is One.    This is its numeral: 1.

The name of the next is Two.  2.

The name of the next is Three.  3.

(Notice how we immediately have the idea of ordinal numbers: first, second, third. For that is how we learn the sequence of the names. The next, the next, the next.)

Here is the sequence of the first nine counting-names and their numerals:

 One 1 Four 4 Seven 7 Two 2 Five 5 Eight 8 Three 3 Six 6 Nine 9

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

Thus Five is one more than Four. We have added one to Four to produce Five.

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

So much for counting. Apart from that, each actual number is an autonomous whole. Your five fingers did not come about by adding one to four.

Tens.  The name of the number one more than Nine 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:

 1 Ten 10 2 Tens are called Twenty 20 3 Tens are called Thirty 30 4 Tens are called Forty 40 5 Tens are called Fifty 50 6 Tens are called Sixty 60 7 Tens are called Seventy 70 8 Tens are called Eighty 80 9 Tens are called Ninety 90

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

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

Here are the numbers between 30 and 40:

 31 Thirty-one (Which means: "One more than Thirty.") 32 Thirty-two 33 Thirty-three 34 Thirty-four And so on.

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

 11 Eleven 12 Twelve 13 Thirteen 14 Fourteen 15 Fifteen And so on.

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:

 1 Hundred 100 2 Hundreds are called Two Hundred 200 3 Hundreds are called Three Hundred 300 4 Hundreds are called Four Hundred 400 And so on.

Numbers between two consecutive Hundreds.  To compose the numbers between two consecutive Hundreds -- between 300 and 400, for example -- successively add the first ninety-nine numbers to the lower Hundred. 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.  In the next Lesson, we will see 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