ARITHMETIC.

ARITHMETIC explains the properties and relation of numbers, and makes known their practical application.

There are six fundamental operations, with which the scholar must become perfectly familiar before he can advance successfully; viz. Notation, Numeration, Addition, Subtraction, Multiplication, and Division. These operations are called fundamental, because all others are founded upon, or are wrought by the application of one or more of them. They therefore require to be first clearly understood.

NOTATION.

Notation is the art of expressing numbers by numerical characters. The characters employed to express numbers are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, and are called figures. Each of these figures has its own specific, and also its local value, as will be learned from Numeration. Besides these characters, there are others used to express operations.

1st. The sign of addition, viz. +, (or plus, more ;) requiring the numbers between which it is placed to be added : 3 + 2 are 5; that is, 3 added to 2 are 5; usually read, 3 plus 2 are 5.

2d. The sign of subtraction, viz. —, (or minus ;) showing that the number following is to be taken from that which precedes it; thus, 4-2 is 2; that is, 2 taken from 4, 2 remains.

3d. The sign of multiplication, viz. X; requiring the number placed before it to be multiplied by that which follows; thus, 3×4 is 12; that is, 3 multiplied by 4 is 12.

4th. The sign of division, viz. ; requiring the number preceding it to be divided by that which follows; thus, 8÷2 is 4; that is, 8 divided by 2 is 4.

In the use of each of the preceding signs, the figure preceding the sign is to be operated upon by that which follows it.

5th. The signs of proportion, viz. :::; showing that the numbers including and between these dots, are proportionals; thus, 2: 4 :: 6 : 12; that is, 2 bears the same relation to 4 as 6 to 12. The numbers are thus read, 2 is to 4 as 6 is to 12.

6th. The sign of equality, viz. ; expressing the equality of the numbers between which it is placed; or that the numbers on the right equal those on the left; thus, 9+7=20—4.

3 4 5

7th. The characters V, V, V, V, &c. require some root of the number before which they stand to be extracted. The figure placed over the sign always shows what root is required. When the character is used without any figure, it then indicates the square root.

By the use of these characters, any arithmetical operation may be indicated. If it be required to add 9 to 16, from the amount to subtract 5, to divide the remainder by 4, and to multiply the quotient by 6, the operation would be thus expressed: 9+16 54×6=30.

QUESTIONS.-What does Arithmetic explain? What application does it make of numbers? How many are the fundamental operations of Arithmetic? What are they? Why are these called fundamental operations? What is Notation? What are the characters used to express numbers called? What two-fold value has each figure? For what purposes are other characters used? What is the sign of addition, and for what is it used? The sign of subtraction? What does it require? The sign of multiplication? What does it require? The sign of division? What does it require? The signs of proportion ? What do they show? The sign of equality? What does it show? What is the character used to express the extraction of roots called? Ans. The radical sign. What does the figure placed over the radical sign show?

NUMERATION.

The scholar has seen, under Notation, the characters used to express the first nine numbers, viz. ; that to express one whole object or thing, 1 is used; to express two whole things, 2 is employed; and for three whole things, 3 is taken, &c., so that each character has its own specific value; and this it always expresses when it stands alone. But each figure has also a local value, that is, a value depending on the place it occupies ;

Hundreds

thus, the value of 3 differs in each of the following numbers, viz. 003, 030, and 300. In the first number its value is three units, or ones; in the second number it is three tens, or thirty; and in the last, it is three hundreds. It will therefore be readily perceived, that the position of a figure materially affects its value. Numeration teaches how to determine what this value is; and thus it also enables us to determine the total value of any number of figures. It will be found that any figure is increased in a ten-fold ratio by having a single figure placed on the right of it; thus, 6 alone, is 6 units; but if another figure be placed on the right of this, its value is ten times as great as before; thus, in 63, the 6 is six tens, equal to 60, and the 3 is three units. This value is increased a hundred-fold by having two figures placed on the right; thus, 600; and a thousand-fold by having three figures on the right of it; thus, 6000. In the first of the last two examples, the value of 6 is six hundred, and in the second it is six thousand. Hence, the scholar will see the necessity of terms by which to designate this local value of figures, and will also readily see the appropriateness of those used, viz. Units, Tens, Hundreds, Thousands, Tens of Thousands, Hundreds of Thousands, Millions, Tens of Millions, Hundreds of Millions, &c. These nine terms are sufficient to express any number in common practice. The higher denominations are Billions, Tens of Billions, Hundreds of Billions; Trillions, Tens of Trillions, Hundreds of Trillions; Quadrillions, Tens of Quadrillions, Hundreds of Quadrillions; Quintillions, Tens of Hundreds of; Sextillions, Tens of Hundreds of; Septillions, Tens of-Hundreds of; Octillions, Tens of Hundreds of―; Nonillions, Tens of Hundreds of-, &c.

It will be observed that as the first three figures, reckoning from the right, are units, tens, and hundreds, so every suc ceeding three are appropriated to the units, tens, and hundreds of the succeeding higher denominations. The following table will serve as an illustration:

369, 342, 900, 976, 368, 265, 371, 502, 634, 436.

This table will enable the scholar to see at a glance, that the names and value of figures are entirely dependent on their location. If they be counted from the right hand towards the left, the first figure in any line of figures is units; the second is tens; the third, hundreds; the fourth, thousands; the fifth, tens of thousands, &c.; and whatever station or place any figure may occupy, its value becomes ten times as great by being moved one degree farther to the left.

QUESTIONS.-What are the characters used to express the first nine numbers? Give an example. When does each figure express its own specific value? What other value has each figure? On what does the local value of a figure depend? Give an illustration. What, therefore, does Numeration teach us to do? What does it enable us to do? In what ratio is the value of any figure increased by having a single figure placed on the right? In what ratio is it increased by having two figures on the right of it? In what by having three placed on its right? In what ratio do numbers continue to increase from the right to the left? Ans. In a ten-fold ratio. What are the terms by which the local value of figures are expressed? To what are each three successive figures in any number appropriated? Ans. To the units, tens, and hundreds of each denomination. In tracing the figures from the right to the left, what is the first figure called? The second? The third? The fourth? &c. What effect is produced on the value of any figure by moving it one place to the left?

Enumerate the following numbers, viz. 6; 27; 467; 568; *4269; 13786; 27599; 367595; 1729567; 67596422; 586; 379872689; 278063; 596402606; 295; 336003; 300300303; 505050505; 467327986427; 585950876; 688001; 100000000; 99999999999; 6398742913; 4678.

The scholar should be taught to read these figures accurate. ly; for example, suppose he be required to enumerate the last number, viz. 4678; let him commence and repeat thus: eight units, seven tens, six hundreds, and four thousands; and then unite them, thus: four thousand six hundred and seventy-eight. Let him also be required to give the value of any figure as it may vary by being written at different points under any line of figures.

After the scholar has become familiar with the preceding exercise, he may write the following numbers on his slate in figures, taking care to express each number accurately :— 1. Thirty-five. 2. Three hundred and seventy-five. 3. Three hundred and five. 4. Seven thousand six hundred and thirtyfive. 5. Seven thousand and thirty-five. 6. Seventy-five thousand four hundred and sixteen. 7. Seventy-five thousand and sixteen. 8. Seventy-five thousand and six. 9. Seventy

five thousand. 10. Three hundred and thirty-three thousand three hundred and thirty-three. 11. Three hundred thousand and three. 12. Three hundred thousand three hundred and three. 13. Five millions and five. Six millions and seventyfive. One hundred and sixty millions. Forty-seven millions, one hundred and five thousand and sixty. 14. One hundred millions, one hundred and one. 15. One hundred and seven millions, one hundred and seven thousand, one hundred and seven. 16. Two billions, three hundred and three millions, five hundred and five thousand and six. 17. Seven hundred and seven trillions, six hundred and seventy-two billions, nine millions, three hundred and five thousand, six hundred and nine.

There is yet another method of expressing numbers; viz. the Roman method; in which the letters of the alphabet are used, as may be seen from the following table.

It is highly important that the scholar obtain clear and distinct views of the nature of what he has to perform. He is therefore recommended to make himself familiar with the fol