Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Thus, aXb shows that the quantit is to be multiplied by that represented

a into b.

ed by a

and is read

The multiplication of simple quantities is also frequently denoted by a point, or by joining the letters together in the form of a word.

Thus, aXb, a. b, and ab, all signify the product of a and b also, 3Xa, or 3a, is the product of 3 and a ; and is read 3 times a.

by, the sign of division; of the two quantities betwee divided by the latter.

signifying that the former which it is placed is to be

Thus, ab shows that the quantity represented by a is to be divided by that represented by b; and is read a by b, or a divided by b.

Division is also frequently denoted by placing one of the two quantities over the other, in the form of a fraction.

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

=

equal to, the sign of equality; signifying that the quantities between which it is placed are equal to each other.

Thus, x=a+b shows that the quantity denoted by x isual to the sum of the quantities a and b ; and is read xqual to a plus b.

= identical to, or the sign of equivalence; signifying that the expressions between which it is placed are of the same value, for all values of the letters of which they are composed.

Thus, (x+a) x (x− a) = x2-a2, whatever numeral values may be given to the quantities represented by x and a.

greater than, the sign of majority; signifying that the former of the two quantities between which it is placed is greater than the latter.

Thus, a=b shows that the quantity represented by a is greater than that represented by b; and is read a greater than b.

= less than, the sign of minority; signifying that the former of the two quantities between which it is placed is less than the latter.

Thus, ab shows that the quantity represented by a is less than that represented by b; and is read a less than b.

as, or to, and :: so is, the signs of an equality of ratios; signifying that the quantities between which they are placed are proportional.

Thus, ab :: c d denotes that a has the same ratio to b that c has to d, or that a, b, c, d, are proportionals; and is read, as a is to b so is c to d, or a is to b as c is to d.

the radical sign, signifying that the quantity before which it is placed is to have some root of it extracted.

Thus, a is the square root of a;a is the cube root of a; and a is the fourth root of a; &c.

The roots of quantities, are also represented by figures placed at the right hand corner of them, in the form of a fraction.

Thus, a is the square root of a; a3 is the cube root

1

of a; and a is the nth root of a, or a root denoted by any number n.

In like manner, a is the square of a; a3 is the cube of a; and am is the mth power of a, or any power denoted by the number m.

is the sign of infinity, signifying that the quantity

standing before it is of an unlimited value, or greater than any quantity that can be assigned.

The coefficient of a quantity is the number or letter which is prefixed to it.

3

3

Thus, in the quantities 36, -6, 3 and - are the coefficients of b; and a is the coefficient of x in the quantity ax.

[ocr errors]

A quantity without any coefficient prefixed to it is supposed to have 1 or unity; and when a quantity has no sign before it, is always understood.

Thus, a is the same as +or+ 1a; and a is the same as 1a.

[ocr errors]

A term is any part or member of a compound quantity, which is separated from the rest by the signs +

er-.

Thus, a and b are the terms of a + b ; and 3a, and+5cd, are the terms of 3a

2b+5cd.

[ocr errors][merged small]

In like manner, the terms of a product, fraction, or proportion, are the several parts or quantities of which they are composed.

a

b

Thus, a and b are the terms of ab, or of and a, b, ., d, are the terms of the proportion a : b :: c: d. A factor is one of the terms, or multipliers which form the product of two or more quantities.

Thus, a and b are the factors of ab; also, 2, a, and ba, are the factors of 2ab2; and a -x and b. -x are the factor of the product (a−x) × (b-x).

A composite number, or quantity, is that which is produced by the multiplication of two or more terms or factors.

Thus, 6 is a composite number, formed of the factors 2 and 3, or 2×3; and 3abc is a composite quantity, the factors of which are 3, a, b, c.

Like quantities, are those which consist of the same letters or combinations of letters; as a and 3a, or 5ab and 7ab, or 2aab and 9a2b.

Dr

Unlike quantities, are those which consist of

erent

letters, or combinations of letters; as a and b, or 3c and a2, or 5ab2 and 7a2b.

Given quantities, are such as have known values, and are generally represented by some of the first letters of the alphabet; as a, b, c, d, &c.

Unknown quantities, are such as have no fixed values, and are usually represented by some of the final letters of the alphabet; as x, y, z.

Simple quantities, are those which consist of one term only; as 3a, 5ab, -8a2b, &c.

Compound quantities, are those which consist of several terms; as 2a+b, or 3a-2c, or a+2b-3c, &c.

Positive, or affirmative quantities, are those which are to be added; as a, or +a, or +3ab, &c.

Negative quantities, are those which are to be subtracted; asa, or -3ab, or -7ab2, &c.

Like signs, are such as are all positive, or all negative ; s+and+, or - and

Unlike signs, are when some are positive and others negative; as + and or and +.

[blocks in formation]

-

consisting of one term only:

A binomial, is a quantity consisting of two terms; as a+b, or a-b; the latter of which is, also, sometimes called a residual quantity.

A trionomial, is a quantity consisting of three terms, as a+2b-3c; a quadrinomial of four, as a-2b + 3*.. d; and a polynomial, or multinomial, is that which has many terms.

The power of a quantity, is its square, cube, biquadrate, &c.; called also its second, third, fourth power, &c.; as a2, a3, a1, &c.

The index, or exponent of a quantity, is the number which denotes its power or root.

Thus, 1 is the index of a1, 2 is the index of a2,

of ajora.

and

When a quantity appears without any index, or exponent, it is always understood to have unity, or 1.

Thus, a is the same as a1, and 2x is the same as 2x1; the 1, in such cases, being usually omitted.

A rational quantity, is that which can be expressed in finite terms, or without any radical sign, or fractional index; as a, ora, or 5a &c.

An irrational quantity, or surd, is that which has no exact root, or which can only be expressed by means of the radical sign, or a fractional index ; as 2 or 2a, Va2

[merged small][ocr errors]

A square or cube number, &c. is that which has an exact square or cube root, &c.

Thus, 4 and 2 are square numbers; and 64 and a3 are cube numbers, &c.

A measure of any quantity, is that by which it can divided without leaving a remainder.

Thus, 3 is a measure of 6, 7a is a measure of 35a, and 9ab of 27 a2b2.

Commensurable quantities, are such as can be each divided by the same quantity, without leaving a remainder.

are

Thus, 6 and 8, 2√2 and 3 √2, 5a2b and 7ab2, commensurable quantities; the common divisors being 2,2 and ab.

Incommensurable quantities, are such as have no common measure, or divisor, except unity.

Thus, 15 and 16, 2 and 3, and a + b and a2 + b2, are incommensurable quantities.

A multiple of any quantity, is that which is some exact number of times that quantity.

Thus, 12 is a multiple of 4, 15a is a multiple of 3a, and 20a3b2 of 5ab.

« ΠροηγούμενηΣυνέχεια »