Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

in which the units increase in the following manner; viz.: counting from the right, 16 units of the lowest denomination make 1 unit of the second; 16 of the second, 1 of the third; 25 of the third, 1 of the fourth; 4 of the fourth, 1 of the fifth; 20 of the fifth, 1 of the sixth. The scale, therefore, for this class of denominate numbers, varies according to the above law.

37. If we take any other class of denominate numbers, as Troy weight, or any of the systems of measures, we shall have different scales for the formation of the different numbers. But in all the formations, we shall recognize the application of the same general principles.

There are, therefore, two general methods of forming the different systems of integral numbers from the unit one. The first consists in preserving a uniform law of relation between the different units. If that law of relation is expressed by 10, we have the system of common numbers.

The second method consists in the application of known, though varying laws of change in the units. These changes in the units, produce different systems of denominate numbers, each of which has its appropriate scale.

INTEGRAL UNITS OF ARITHMETIC.

38. The Integral units of Arithmetic may be arranged into eight classes:

1st. Abstract units: 2d. Units of currency: 3d. Units of length 4th. Units of surface: 5th. Cubic units, or units of volume 6th. Units of weight: 7th. Units of time: 8th. Units of circular measure.

First among the units of arithmetic stands the abstract unit 1 This is the primary base of all abstract numbers, and becomes the base, also, of all denominate numbers, by merely naming, in succession, the particular thing to which it is applied.

37. How many general methods are there of forming numbers from the unit one? What is the first? What is the second!

38. Into how many general

arranged? What are they?

classes may the units of Arithmetic be

OF THE SIGNS.

39. The sign, is called the sign of equality. When placed between two numbers it denotes that they are equal; that is, that each contains the same number of units.

The sign+, is called plus, which signifies more. When placed between two numbers it denotes that they are to be added together: thus, 3 + 2 = 5.

The sign, is called minus, a term signifying less. When placed between two numbers it denotes that the one on the right is to be taken from the one on the left: thus, 6 — 2 = 4. When

The sign X, is called the sign of multiplication. placed between two numbers it denotes that they are to be multiplied together; thus, 12 x 3, denotes that 12 is to be multiplied by 3.

The parenthesis is used to indicate that the sum of two or more numbers is to be multiplied by a single number: thus, (2 + 3 + 5) × 6

shows, that the sum of 2, 3 and 5 is to be multiplied by 6. The parenthesis is also used to denote that the difference between two numbers is to be multiplied by a third; thus,

(5 − 3) × 6,

denotes that the difference between 5 and 3 is to be multiplied by 6.

The sign, is called the sign of division. When placed between two numbers it denotes that the one on the left is to be divided by the one on the right: thus, 45, denotes that 4 is to be divided by 5.

PROPERTIES OF THE 9's.

40. In any number, written with a single significant figure, as 4, 40, 400, 4000, &c., the excess over exact 9's is equal to

39. What is the sign of equality? What is the sign of addition? What of subtraction? What of multiplication? For what is the parenthesis used? What is the sign of division?

40. What will be the excess over exact 9's in any number expressed by a single significant figure? How may the excess over exact 9's be found in any number whatever?

the number of units in the significant figure. For, any such number may be written,

[merged small][merged small][ocr errors][ocr errors][merged small][merged small]

4 = 4.

[blocks in formation]

Each of the numbers 9, 99, 999, &c., contains an exact number of 9's; hence, when multiplied by 4, the several products will contain an exact number of 9's: therefore,

The excess over exact 9's, in each number, is 4; and the same may be shown for each of the other significant figures.

If we write any number, as

6253,

we may read it 6 thousands 2 hundreds 50 and 3. Now, the excess of 9's in the 6 thousands is 6; in 2 hundreds it is 2; in 50 it is 5; and in 3 it is 3: hence, in them all, it is 16, which is one 9 and 7 over: therefore, 7 is the excess over exact 9's in the number 6253.

Hence,

The excess over exact 9's in any number whatever, is found by adding together the significant figures and rejecting the exact 9's from the sum.

NOTE. It is best to reject or drop the 9 as soon as it occurs: thus we say, 3 and 5 are 8 and 2 are 10; then, dropping the 9, we say, 1 to 6 is 7, which is the excess; and the same for all similar operations. 1. What is the excess of 9's in 48701? 2. What is the excess of 9's in 9472021 ? 3. What is the excess of 9's in 87049612?

REDUCTION.

CHANGE OF UNITIES.

In 67498 ?

In 2704962 ?

In 4987051 ?

41. REDUCTION is the operation of changing the unit of a number without altering its value. Thus, if we have 4 yards,

41. What is Reduction? you change feet to inches? you change feet to yards?

How do you change yards to feet?
How do you change inches to feet?

How do

How do

in which the unit is 1 yard, and wish to change to feet, the units of the scale will be 3, since 3 feet make 1 yard: therefore, the number of feet will be

4 x 312 feet.

If it were required to reduce 12 feet to inches, the units of the scale would be 12, since 12 inches make 1 foot. Hence,

4 yards = 4 × 3 = 12 feet 12 x 12 = 144 inches. If, on the contrary, we wish to change 144 inches to feet, and then to yards, we should first divide by 12, the units of the scale in passing from inches to feet; and then by 3, the units of the scale in passing from feet to yards. Hence, Reduction is of two kinds :

1st. To reduce a number from a higher unit to a lower:

Multiply the highest denomination by the number of units in the scale which connects it with the next lower, and then, add to the product the units of that denomination: Proceed in the same manner through all the denominations till the unit is brought to the required denomination.

2d. To reduce a number from a lower unit to a higher:

Divide the given number by the number of units in the scale which connects it with the next higher denomination; and set down the remainder, if there be one. Divide the quotient thus obtained, and each succeeding quotient in the same manner, till the unit is reduced to the required denomination: the last quotient with the several remainders annexed, will be the answer.

EXAMPLES.

1. Reduce £3 14s. 4d. to pence. We first multiply the £3 by 20, which gives 60 shillings. We then add 14, making 74 shillings we next multiply by 12, and the product is 888 pence: to this we add 4d. and we have 892 pence, which are of the same value as £3 14s. 4d.

If, on the contrary, we wished to change 892 pence to pounds shillings and pence, we should first divide by 12: the quotient is 74 shillings, and 4d. over. We next divide by 20, and the

quotient is £3, and 14s. over: hence, the result is £3 14s. 4d., which is equal to 892 pence.

The reductions, in all the denominate numbers, are made in the same manner.

2. In £5 5s., how many shillings, pence, and farthings?

[blocks in formation]

3. In 5040 farthings, how many

pence, shillings, and pounds?
4)5040 farthings.
12)1260 pence.
210)105 shillings.
£5 5s.

In this example, the reduction is from a less to a greater unit.

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

6. In $426, how many cents? How many mills?

7. In 36 eagles 8 dollars and 6 dimes, how many cents?

8. In 8750 mills, how many dollars and cents?

9. In 43 eagles 3 dollars and 5 mills, how many mills?

10. In £37 9s. 8d., how many pence?

11. In 1569 farthings, how many pounds, shillings, pence, and farthings?

2*

« ΠροηγούμενηΣυνέχεια »