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THE COLUMBIAN CALCULATOR.

ARITHMETIC.

ARITHMETIC is a part 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. The knowledge of this science is so universally necessary, that scarcely anything in life, and nothing in trade, can be dono without it.

NOTATION.

Notation teaches to express words, or numbers, by ten Arabic characters, or digits, namely, 1, one ; 2, two; 3, three ; 4, four ; 5, five ; 6, six ; 7, seven ; 8, eight ; 9, nine ; 0, cipher ; by the use of which, all numbers are expressed, and increase in value from right to left, in a tenfold proportion ; thus, 1824, the figure (4) in the place of units, denotes only its simple value, 4; that in the second place, or place of tens (2), is ten times its simple value, 24; that in the third place, or place of hundreds (8), one hundred times its simple value, 824 ; that in the fourth place, or place of thousands (1), one thousand times its simple value, or 1824, one thousand, eight hundred and twenty-four.

A cipher when alone is of no value, but when placed to the right of a figure, increases the value of that figure in a tenfold proportion ; thus, 5 alone is only five; but annex a cipher to the right, thus, 50, and it increases the value tenfold, or fifty.

EXAMPLES.

Write the following numbers in figures : twenty-five ; seventy-six ; ninety-one ; eighty-four ; sixty-five, nineteen ; eleven; one hundred; one hundred and sixty-seven; two hundred ; four hundred ; one thousand ; two thousand, nine hundred and ninety-nine.

In the Roman method of 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 following table, which the pupil should com

mit to memory

ROMAN METHOD, OR NOTATION BY LETTERS.
One
I
Ninety

XC
Two
II

One hundred с
Three III

Two hundred CC
Four

IIII, or IV Three hundred CCC
Five
V

Four hundred СССС
Six
VI
Five hundred

D, or 10
Seven
VII

Six hundred DC
Eight VIII Seven hundred DCC
Nine

VIIII, or IX Eight hundred DCCC Ten х

Nine hundred DCCCC Twenty XX

One thousand M, or Clot
Thirty XXX Five thousand

XL
Ten thousand

CCIɔɔ, or X Fifty

L
Fifty thousand

Iാാാ
Sixty

LX Hundred thousand CCCI., or C Seventy LXX One million M Eighty LXXX Two millions MM

MDCCCXLIV=A. D. 1844

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Forty

•15 is used instead of D, to represent five hundred, and for every additional y annexed at the right, the number is increased ten times.

#CIU is used to represent one thousand, and for every C and a sut at each end, the number is increased ten times.

A line over any number increases its value one thousand times.

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NUMERATION.

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, and more correctly and perfectly exemplified by the following tables :

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1 ono. 12 twelve. 123 1 hundred 23. 1234 I 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

To enumerate figures where the numbers are large, it will be found convenient to divide, or separate them, into periods of three figures, the first being hundreds, the second, hundreds of thousands, &c. Then begin at the right hand, or place of units, and read toward the left, as in table ist; thus 1 is in the place of units, 2 is in the place of tens, and 3 in the place ef hundreds, which three figures, taken in connexion, would express (321) three hundred and twenty one.

In table 2d, the numbers may be enumerated in the same way, and then read from left to right, in the order in which they stand.

TABLE III.

I units. 12 tens. 123 hundreds. 1234 thousands. 12345 tens of thrusands. 123456 hundreds thousands. 1234567 millions. 12345678 tens of millions. 123456789 hundreds of millions. 1234567891 thousands of millions. 12345678912 tens of thousands of millions. 123456789123 hundreds of thousands of millions. 1234567891234 billions. 12345678912345 tens of billions. 123456789123456 hundreds of billions. 1234567891234567 thousands of billions. 12345678912345678 tens of thousands of billions. 123456789123456789 hundreds of thousands of billions.

Quintillions. Quadrillions. Trillions.

555,555 555,555 555,555 Nonillions. Octillions. Septillions. Sectillions.

555,555 555,555 555,555 555,555 The third table is given to express the higher powers, and can be read and applied in the same manner as the two preceding tables.

When even hundreds, thousands, &c., are to be written, the vacant places of units, tens, hundreds, &c., are to be supplied by ciphers; thus, to write one thousand (1000), place three ciphers to the right of the 1, and so in all cases of a similar nature.

EXAMPLES.

Write down in words, the following numbers : 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 150, 476, 891, 999, 1001. Let the following numbers be expressed in figures : twenty, thirty, forty-five, seventy-six, eighty-three, ninety-one, one hundred and one, one hundred and nine, one hundred and fifteen, one hundred and thirty, one hundred and ninety-four, two hundred and sixty, three hundred and forty-five, five hundred and eighty-five, six hundred and ninety-one, one hundred and fifty thousand.

O

The preceding is the English method of Numeration ; the following is the French. The teacher can use either at his pleasure: Quadrillions. Trillious. 'Billions. Millions. Thousands. Units. 321 321 321

321

321 321

REVIEW.

What is Arithmetic ? By what means are operations in Arithmetic performed ? -Name them. What is Notation! What is Numeration? How must figures be enumerated ? In what manner should figures be read? Why do you enumerate from right to left? In what proportion do they increase in value? Recite the Numeration Table. Write down seventeen millions :

seventeen hundred thousand.-Eleven billions and nine hundred thousand.

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ADDITION is the first primary rule in Arithmetic, the use of which is to ascertain the amount or sum total of two or more numbers, when put or added together; as, 5+5=10: that is 5 and 5 make 10.

RULE.

Set the given numbers under each other, with units under units, tens under tens, hundreds under hundreds, &c. Then draw a line under the lowest number, and begin at the right hand column, or place of units, and add (upward) all the column together; set down the sum when less than ten; if ten or more, set down the right hand figure, and add the left hand figure to the next column; and thus proceed to the last column, and set down the whole amount of it.

PROOF.

Perform the operation a second time, agreeably to the Rule; but in this case, begin at the top ; or reserve one of the given numbers, find the sum of the rest, and thereto add the number reserved.

Note.-The reason why you carry one for every ten, is this · in the place of units, it requires ten to make one in the place of tens; and in the place of tens, it requires ten to make one in the place of hundreds; therefore, you always carry one from one denomination to another, as it requires of that denomination to make one in the next.

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