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(5.) (-1)2m=1, and (-1)22-=-1; m being any integer. Nature of impossible quantities.

(E. 13951.) (+a)=* = + a. (-a)= +a.(-1)

qəm-i has the same sign as a. (6.) Nature of squares, and square roots. (E. 123–38.) Nature of cubes, and cube roots.

(E. 152_67.) Extraction of the square and cube roots of numerical, and algebraical quantities.

(W. 126—33; E. 317–39.)

SURDS. (7.) Properties of surds.

(W. 239-51; B. 18, 19.) The square root of a quantity cannot be partly rational, and partly, a quadratic surd. If the equation w+ vy=a + vb is generally true, then

x=a, and y=b. The square root of an integer that is not a complete square can neither be expressed by an integer, nor by a rational fraction.

Neither the sum nor difference of the square roots of two quantities prime to each other can be represented by any rational quantity, nor by the square root of any rational quantity, unless each be a complete square.

: The product of two quadratic surds, which cannot be reduced to others having the same irrational part, is irrational.

To extract the square root of a + v6: Assume

a+b=vx+vy; then

=}{a + (a* — b)},

y=}{a-(a' – 6)"}. x and y are rational only when a– is a complete square.

(W. 258; Bour. 118-21.) Method of extracting the nth root of a binomial quadratic surd.

(W. 259.) GREATEST COMMON DIVISOR. (8.) The greatest common divisor of two quantities may be found by arranging both according to the powers of some one letter; then dividing the greater by the less, and the preceding divisor always by the last remainder, until the remainder is equal to nothing; the last divisor will be the greatest common divisor required.

If any divisor contains a factor not contained in both the given quantities, that factor may be rejected.

(Bour. 34-41, 258.—74; W.90; E. 451-60.) (9.) The least common multiple of two quantities is their product divided by their greatest common divisor.

To find the least common multiple of n quantities: find the least common multiple of the first and second; then of the quantity thus obtained and the third, and so on: the result will be the least common multiple required. (W.377–9.)

FRACTIONS. (10.) Nature of fractions.

(E. 68–93.) (11.) Rules for the multiplication and division of fractions.

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(12.) To reduce a fraction to its lowest terms: divide both the numerator and denominator by their greatest common divisor.

To reduce fractions to a common denominator: multiply each numerator by all the denominators except its own, for a new numerator; and all the denominators together, for a new denominator.

(W.96.) If the least common denominator be required; find the least common multiple of all the denominators, for a new denominator; and multiply each numerator by the quotient of the common denominator divided by the corresponding denominator, for a new numerator.

(Bonnycastle, Arith.)

EQUATIONS.

(13.) General rules for the solution of equations.

The same quantity may be added to, or subtracted from, both sides of an equation.

Any quantity may be transposed from one side of an equation to the other, its sign being changed.

The signs of all the terms may be changed.

Every term may be multiplied, or divided, by the same quantity.

Both sides

may

be raised to the same power. The same root of both sides

may

be taken.

(W. 134–41; Bour. 43—5; &c.) (14.) Simple Equations. Every simple equation containing one unknown quantity, when cleared of fractions, and surds, may be reduced to the form

ax=b;

b whence x=

(W. 13442; E. 573—604.)

To clear an equation of fractions: find the least common multiple of all the denominators; and multiply each numerator by the quotient of that quantity divided by the corresponding numerator.

Two simple equations containing two unknown quantities may be reduced to the form

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These two equations may be reduced to one, containing a single unknown quantity, by either of the following methods:

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3rd. if a, =ne, ag=ne,, then
(1) xe - (2) xe, (e,bz-e,b).y=e,C1-C2.

(W. 143; E. 605-12.) The same principles may easily be extended to three or more independent equations containing as many unknown quantities.

(E. 613—22.) General method of deducing the value of n unknown quantities from n simple equations.

(Schweins, Theor. der Diff. 183; F. 111.) (15.) Quadratic Equations. Every quadratic equation may, by clearing it of fractions and surds, and dividing every term by the coefficient of x', be reduced to the form

X* + px=9; adding to each side the square of į the coefficient of the second term, we have

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+9,

whence x =

{{-p+(p? +49)}}. The following are some of the principal forms of equations which may be solved as quadratics :

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a x? + bx + cf* + p.ax+ bx+c=x* +px.

2

a x + b1 = x + px. tp.

0,*+ b,

0,X + b a X + b

All equations containing only one power of the unknown quantity may be solved as simple equations: having found

=a, we have x=a (W. 147–50; E. 638—55.) Various artifices applicable to particular occasions.

(W. 151–6; Bland, Alg. Prob.)

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