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× Multiplied by.

Divided by.

2537

63

= Equal.

is, so is.

7-2+5=10

9-2+5=2

The Sign of Addition; as 4+4=8; that is, 4 added to 4 more is equal to 8.

The Sign of Subtraction; as 8-2=6; that is, 8
lessened by 2 is equal to 6.

The Sign of Multiplication; as 4×6=24;
4 multiplied by 6 is equal to 24.

that is,

The Sign of Division; as 8÷÷2=4; that is, 8 divided by 2 is equal to 4.

Numbers placed like a fraction, do likewise denote Division; the upper number being the dividend, and the lower the divisor.

The Sign of Equality; as 6+4=10; or, 4 qrs=1 cwt. signifies, that 4 qrs. are equal to 1 cwt.

The Sign of Proportion; as 2:4::8; 16; that is, as 2 is to 4 so is 8 to 16.

Shows that the difference between 2 and 7, added to 5, is equal to 10.

Signifies that the sum of 2 and 5 taken from 9 is equal to 2.

radical sign. Signifying that the quantity before which it is placed is to have its square root extracted.

s. d. q.

Thus, 2, or

number 2.

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43

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denotes the square root of the

denotes the cube root of the number 4. Also 82 denotes that the number 8 is to be squared, and 93 denotes that the number 9 is to be cubed. denotes therefore.-. denotes because.

£. stands for pounds; s. for shillings; d. for penee; and 9. for farthings.

£. s. d. q. are initials of Latin words of the same signification, viz.: Libræ, Solidi, Denarii, and Quadrantes.

An Aliquot Part is a number which is contained in a greater an exact number of times; thus 5 is an aliquot part of 25, but not of 26, as it is contained exactly 5 times in the former, and in the latter 5 times and 1 over. An Integer......is any whole number; as a pound, a mile, &c., 1, 2, 4, 6, 9, &c.

Minuend.

.is the greater number in Subtraction. Subtrahend.. .....is the less number.

Multiplicand ...in Multiplication is the number to be multiplied or

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repeated.

...is the number by which we multiply, or which expresses how often the multiplicand is to be repeated. the sum or result of the operation in Multiplication. The Multiplicand and Multiplier are called factors of the Product.

ARITHMETIC.

PART I

ARITHMETIC IN WHOLE NUMBERS.

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INTRODUCTION.

ARITHMETIC is that part of Mathematical Science which explains the properties, and shows the uses of numbers. It comprises both theory and practice.

The theory considers the nature and quality of numbers, and demonstrates the reason of practical numbers. The practice is that which shows the method of working by numbers, so as to be the most useful and expeditious for business, and has five principal or fundamental rules for the operations, viz.:

NOTATION, or NUMERATION, ADDITION, SUBTRACTION, MULTIPLICATION, and DIVISION.

PRELIMINARY EXERCISES IN NUMERATION.

All numbers are expressed by the ten following figures:one, or unit, two, three, four, five, six, seven,

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5 6

7

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The first nine are called digits; the last is often called zero, 0, which signifies nothing.

The first Exercise to which a child should be put is that of counting Units, which may be represented by the following objects:

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To give the young Pupil an idea of the simple operations of numbers, a slate, or black board, or the Numerical Frame (consisting of balls moving on horizontal wires) will be found of the utmost service.

2. Write upon the Slate, or Black Board, |, and ask, How many are here ?-Proceed in the same way with ││, | | |, | | | |, &c., &c.

3. Write the Sign or Figure of the Number, as 1, or 2, or 3, or 4, &c., saying, What is this? If the Pupil cannot tell, write besides, or under the figure, thus: What is this? 2, ; or this? 3, | | | ; 4, | | | |, &c. 4. Write first one stroke, and then another, thus, , ask, How many are one and one? 5. How many are one and two, | proceed with, two and two, | | |||||; two and four, | |

? and thus ; two and three,

||, &c., &c.

simplest manner, and

Thus Addition may be taught in the this appeal to visible objects will be readily comprehended by the young Pupil.

Subtraction and Division also may by this method be simply taught; thus:-How many are one and one | ; Ans. 2, writing that figure on the Slate as soon as the child answers. What is the half of 2? Ans. 1. If I take 1 from 2, how many will remain? Ans. 1. And thus proceed with the higher numbers.

6. Holding up 2 fingers of one hand, and 1 of the other, ask, How many are two and one? How many are two and three? two and four ? two and five? &c.

7. On my right hand I have five fingers, I put two down, how many remain? On both hands I have ten; if I put down two, or three, &c., how many remain ?

8. In this class are six boys; if I take one, or two, &c., how many remain ?

away,

9. How many are two nuts and three nuts? four nuts and two nuts? five nuts and three nuts? In my hand I

have six nuts; if I take one away, how many will remain ? if I take two away, &c. Tell the half of six nuts?

10. Here are five crosses +++++ and four more

++++, how many in all? How many must you take from nine to make five? How many to make four ?

11. Here are eight stars ********; if I take two, or three, &c., how many remain? If I divide these stars into two parts, how many will be in each part?

12. This boy has two hands; how many two boys? three boys, &c.

hands have

13. How many legs have one horse and two cows? How many legs have two boys and one man? How many legs have two dogs and two cats? How many ears have three horses? How many eyes have four boys? &c.

NOTATION AND NUMERATION.

NOTATION is the writing of numbers by figures; and NUMERATION is the art of reading figures correctly, or of expressing such numbers in words.

The value of figures depends upon the place in which they stand: as in the following table :—

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7.000,000.-Millions

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7 figures

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13 figures

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19 figures

8.000,000.000,000.-Billions

9.000,000.000,000.000,000.-Trillions

A figure placed singly, or on the right hand of other figures, signifies so many units. In the second place, it represents so many tens; and in the third place, so many hundreds.

The cipher serves to bring figures to their proper places by supplying vacant places. Thus 5, five; 50, fifty; 550, five hundred and fifty; and 503, five hundred and three.

When the number consists of more than three places.

RULE. Divide it from the right hand into parcels of three figures, by placing a comma at the left hand of the first three, a period after the next three, and so on alternately. Then, beginning at the left hand read each parcel by itself as before, and call the periods millions, and the commas thousands, as below.

Hundreds of Thousands of Trillions.
Tens of Thousands of Trillions.
Thousands of Trillions.

Hundreds of Trillions.

∞Tens of Trillions.

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Trillions, or Millions three times repeated.
Hundreds of Thousands of Billions.

Tens of

of Thousands of Billions.

of Billions.

Thousands of

Hundreds of

Billions.

Tens of Billions.

Hundreds of

Billions, or

Billions

Millions of Millions.

Thousands of Millions.

Tens of thousands of Millions.

Thousands of Millions.

Hundreds

of Millions.

Tens of Millions.

Millions.

Hundreds of Thousands.

Tens of Thousands.

Thousands.

Hundreds.

coTens. Units.

Figures.

Half Periods. CX M, CX U. CX M, CX U. CX M, CX U. CX M, CXU.

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The foregoing table is read thus,-Five hundred forty-six thousand seven hundred and eighty-three trillions, Nine hundred twenty-five thousand one hundred and fortytwo billions, Seven hundred thirteen thousand eight hundred and sixty-two millions, Five hundred ninety-four thousand six hundred and thirty eight.

If the student wish to carry the above table to a greater number of figures, he may extend it to any length by continually prefixing a period of six figures towards the left hand, and writing the words quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, &c., under their respective places.

To express in words the Numbers denoted by figures.

RULE. (1.) Begin at the right hand side; divide the given figures into periods of three figures each, till not more than three remain. (2.) Then the first period towards the right hand contains units or ones; the second, thousands; the third, millions, &c., as in the Numeration Table: and therefore commencing at the left hand side, annex to the value expressed by the figures of each period, except that of the units, the name of the period.

Thus the expression 48063705, becomes by division into periods 48.068,705, and is read forty-eight millions, sixty-three thousand, seven hundred and five, the term units or ones being omitted, i. e., not terminating with, and five units.

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