NUMERATION OF THE
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.
To count things, they must have the same name.
"One apple, two apples, three apples."
"One ten, two tens, three tens."
Whatever has that name is the unit. It is what we are calling one.
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 units, that is, of ones:
-- and not a part of one: not half of one, or a third, or a millionth. Those parts of a continuous unit we name with fractions or decimals. And so we speak of whole number arithmetic and whole number numeration; it does not include fractions or decimals.
To do arithmetic, the numbers obviously 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.
(They are also the names of what are called the natural numbers.)
The symbols that refer to the 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.
Again, a number, as when we speak of a number of children, is what we actually distinguish and count, not the name or numeral that we count with; the name in each language, after all -- "cinco," "cinq," "fünf" -- refers 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 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 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:
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:
|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.")|
|And so on.|
The numbers between 10 and 20, however, have unique names:
|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:
|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.
At this point, please "turn" the page and do some Problems.
Continue on to the next Lesson: The Powers of 10
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Copyright © 2014 Lawrence Spector
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