THE

YOUTH'S COLUMBIAN CALCULATOR.

ARITHMETIC.

1. ARITHMETIC is a branch of the science of mathematics, and is the art of computing by numbers, by the operation of six rules, namely: Notation, Numeration, Addition, Subtraction, Multiplication, and Division; two of which may be considered primary rules-namely, Addition and Subtraction-and the other four secondary, as they naturally arise from the operation of the former.

2. NUMBER is that which is composed of one or more units.

NOTATION.

3. NOTATION teaches us to write or express numbers or words by the ten Arabic characters, or digits, so called from numbering or counting on the fingers, before the use of figures was known :—

eight,

One, two, three, four, five, six, seven, eight, nine, cipher. 1 2 3 4 5 6 7 9 0

4. By the use of those nine figures and ciphers, all numbers may be expressed, and their value or amount depends upon their place, or method of writing them; thus I is a unit, or one: as one apple, one book.

5. A CIPHER, when alone, is of no value; but when placed on the right of a figure or figures, it increases their value in a tenfold proportion; thus 1 and 0 are 1, but when joined (10) ten, in this manner, they become ten, which is ten ones (1111111111, 10). Annex or place another cipher on the right of 10, and it is increased ten times, which is (100) one hundred, for ten tens are one hundred; then place another cipher on the right of 100, and it is increased ten times, and becomes (1000) one thousand, for ten hundred are one thousand.

6. THE VALUE of all the figures increases in the same manner: thus, 1847, the figure (7) in the place of units, denotes only its simple value (7) seven; that in the second place, or place of tens (4), is ten times its simple value, and the two figures (47) forty-seven; that in the third place, or place of hundreds (8), one hundred times its simple value (847) eight hundred and forty-seven; that in the fourth place, or place of thousands (1), one thousand times its simple value, or (1847) one thousand eight hundred and forty

seven.

REVIEW.

1. What is Arithmetic ? 2. Of number? 3. Notation ? 4. Of the use of figures? 5. A cipher? 6. Of the value What is knowledge? There is another method of Notation by letters, viz. :

of figures? What is science?

Explanation.-In notation by letters, I represents one; V five; X ten; L fifty; C one hundred; D five hundred; M one thousand, &c.

As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number, placed before one representing a greater-then, the less number is taken from the greater: thus IV represents four; IX nine, &c., as will be seen in the above table, which the pupil should commit to memory.

NUMERATION.

1. By Numeration, we are taught to read any number of figures, and ascertain their relative value, when taken in connexion with each other, which is determined by the situation in which they are placed, which may be easily learned from the following tables :

:

123 1 hundred and 23.

1234 1 thousand 2 hundred and 34. 12345 12 thousand 3 hundred and 45.

123456 123 thousand 4 hundred and 56.

1234567 1 million 234 thousand 5 hundred and 67.

12345678 12 millions 345 thousand 6 hundred and 78.

123456789 123 millions 456 thousand 7 hundred and 89.

1234567891 thousands of millions.

12345678912 tens of thousands of millions.

123456789123 hundreds of thousands of millions.

1234567891234 billions.

::

2. To enumerate where the numbers are large, it will be convenient to divide or separate them into periods of three figures the first period being hundreds; the second, hundreds of thousands; the third, hundreds of millions, &c.: thus, 987,654,321. Then begin at the right, or place of units, and read toward the left as in table first, which is nine hundred and eighty-seven millions, six hundred and fifty-four thousand, three hundred and twenty-one.

3. When even hundreds, thousands, &c., are to be written, place ciphers on the right of 1, thus: (100), one hundred; (1,000), one thousand; (10,000), ten thousand; (100,000), one hundred thousand, &c.

Write in words the following numbers: 21, 47, 52, 86, 79, 84, 96, 109, 150, 192, 704, 879, 1001, 10004, 2468791. Write in figures the following numbers: seventy-four; eighty-three; five hundred; six hundred and ninety-nine; seven hundred and forty-two; nine thousand, seven hundred; one million, two hundred thousand; five millions, three hundred and twenty thousand, four hundred.

REVIEW.

1. What is Numeration? Repeat the table, first beginning with units. Read Table 2, as far as the ninth line of figures. 2. How do you enumerate figures? Why do you enumerate from right to left? Ans. Because they increase in a tenfold proportion. How do you read figures to express their value? Ans. From left to right, because they decrease from left to right, in the same manner as they increase in value from right to left by enumeration. 3. How will you express or write even hundreds, thousands, &c.? Write in figures one million, two hundred thousand. Enumerate the following numbers: 10011, 70204, 600302, 5400891.

ADDITION.

1. ADDITION is the first primary rule in Arithmetic, the use of which is to ascertain the amount or sum total of two or more given numbers when put or added together.

2. Numbers to be added must be of the same kind or denomination. You can not add yards and dollars, but dollars can be added to dollars, and yards to yards; because 5

yards added to 2 dollars would make 7, it would express no meaning, for it is still 5 yards and 2 dollars; but 5 yards and 2 yards make 7 yards, and 5 dollars and 2 dollars make 7 dollars.

3. The sign used in addition is the cross (+), which means, when placed between two or more numbers, that all those numbers are to be added together; it is also called plus, and means more; it is sometimes used to signify a remainder, or something more.

L

4. This sign (=) signifies equal, or equality: as, 10 mills are equal to 1 cent, and when placed between two numbers denotes that they are equal to each other; thus 5+5=10, five added to five make ten, or are equal to ten; 7+6=13; 4+3+2=9; 7+8+5=20; 4+5+1+3+5=18: last example, 4 and 5 are 9 and 1 is 10 and 3 are 13, and 5 are 18.

1. Which is the first primary rule in Arithmetic? What is the use of Addition? 2. When numbers are to be added, of what kind must they be? 3. What sign is used in addition? What does the sign signify besides addition? 4. What do you understand by the sign of equality? Can you give examples of the signs of addition and equality?

ADDITION TABLE.

2 and 2 are 413 and 9 are 1215 and 9 are 14 8 and 8 are 16

To read the table, say 2 and 2 are 4; 3 and 3 are 6, &c. (Repeat the table.)