## An Elementary Treatise on Algebra: To which are Added Exponential Equations and Logarithms |

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Página 27

Let the greatest of the two quantities be A , and the

their quotient after division be Q , and the remainder R ; and let the greatest

Greatest Common Divisor . common divisor of A and B CH . II . § 1. ] 27

REDUCTION ...

Let the greatest of the two quantities be A , and the

**least**B ; let the entire part oftheir quotient after division be Q , and the remainder R ; and let the greatest

Greatest Common Divisor . common divisor of A and B CH . II . § 1. ] 27

REDUCTION ...

Página 37

To find the

quantities are decomposed into their simplest factors , as is the case with

monomials , their

equal to ...

To find the

**least**common multiple of given quantities . Solution When the givenquantities are decomposed into their simplest factors , as is the case with

monomials , their

**least**common multiple is readily obtained ; for it is obviouslyequal to ...

Página 217

Number of Real Roots of an Equation of an Odd Degree . 284. Theorem . Every

equation of an uneven degree , has at

contrary to that of its last term , and the number of all its roots is uneven . Proof .

Number of Real Roots of an Equation of an Odd Degree . 284. Theorem . Every

equation of an uneven degree , has at

**least**one real root affected with a signcontrary to that of its last term , and the number of all its roots is uneven . Proof .

Página 234

When an uneven number ( m ) of successive terms is wanting in an equation , the

number of imaginary roots must be at

preceding the deficient terms has the same sign with the term following them ;

and ...

When an uneven number ( m ) of successive terms is wanting in an equation , the

number of imaginary roots must be at

**least**as great as ( m + 1 ) , if the termpreceding the deficient terms has the same sign with the term following them ;

and ...

Página 235

When an even number of successive terms is wanting in an equation , the

number of imaginary roots must be at

deficient terms . Proof . Let the place of the first deficient term be supplied by zero

affected ...

When an even number of successive terms is wanting in an equation , the

number of imaginary roots must be at

**least**as great as the number of thesedeficient terms . Proof . Let the place of the first deficient term be supplied by zero

affected ...

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An Elementary Treatise on Algebra: To which are Added Exponential Equations ... Benjamin Peirce Visualização integral - 1842 |

An Elementary Treatise on Algebra: To which are Added Exponential Equations ... Benjamin Peirce Visualização integral - 1860 |

An Elementary Treatise on Algebra: To which are Added Exponential Equations ... Benjamin Peirce Visualização integral - 1858 |

### Palavras e frases frequentes

affected approximate values arithmetical becomes body called coefficient consequently contained continued fraction continued product Corollary corresponding decimal denominator denote derivative difference Divide dividend division Elimination equal roots equation 23 EXAMPLES exponent expression Extract factor figure Find Find the greatest Find the square Find the sum follows fourth fraction Free function gallons given equation gives greater greatest common divisor Hence imaginary increased infinite integral known last term least less letter limit logarithm means method monomials multiplied negative number of real number of terms obtained places polynomial positive Problem progression Proof proportion putting quotient ratio real roots reduced remainder result reverse row of signs Solution Solve the equation square root substitution subtracted successive suppressed Theorem third true unity unknown quantity variable whence wine

### Passagens conhecidas

Página 46 - In any proportion the terms are in proportion by Composition and Division ; that is, the sum of the first two terms is to their difference, as the sum of the last two terms is to their difference.

Página 190 - One hundred stones being placed on the ground in a straight line, at the distance of 2 yards from each other, how far will a person travel who shall bring them one by one to a basket, placed at 2 yards from the first stone ? Ans.

Página 266 - The logarithm of the quotient is equal to the logarithm of the dividend, diminished by the logarithm of the divisor.

Página 61 - A term may be transposed from one member of an equation to the other by changing its sign.

Página 184 - I = the last term, r = the common difference, n = the number of terms, S = the sum of all 'the terms.

Página 53 - There is a number consisting of two digits, the second of which is greater than the first, and if the number be divided by the sum of its digits, the quotient is 4...

Página 30 - The 2d line of col. 1 is the 1st line multiplied by 7 in order to render its first term divisible by the first term of the new divisor ; the remainder of the division is the 4th line of col.

Página 125 - Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend. III. Double the root already found and place it on the left for a divisor.

Página 230 - Rule. 324. An equation of any degree whatever cannot have a greater number of positive roots than there are variations in the signs of Us terms, nor a greater number of negative roots than there are permanences of these signs.

Página 45 - Likewise, the sum of the antecedents is to their difference, as the sum of the consequents is to their difference. Ratio of Sum of two first Terms to that of two last. Moreover, in finding...