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Unformatted text preview: INVESTIGATIONS ON THE IMAGINARY ROOTS OF EQUATIONS Leonhard Euler 1. Every algebraic equation which has been freed of fractions and radical signs always reduces to this general form x n + Ax n- 1 + Bx n- 2 + Cx n- 3 + Dx n- 4 + + N = 0 , where the letters A , B , C , D , . . . , N indicate constant real quantities, either positive or negative, not excluding zero. The roots of such an equation are the values which when put for x produce an identity equation 0 = 0. Now if x + is a divisor or a factor of the given formula, the other factor being indicated by X , so that the equation has this form ( x + ) X = 0 , then it is clear that this happens when x + = 0 , or x =- . From this we see that the roots of an equation are found by looking for the divisors or factors of this same equation; and all the roots of an equation are derived from all the simple divisors of the form x + . 2. So to find all the roots of a given equation, we have only to look for all the simple factors of the quantity x n + Ax n- 1 + Bx n- 2 + Cx n- 3 + Dx n- 4 + + N, and if we set these factors: ( x + )( x + )( x + )( x + ) Recherches sur les racines imaginaires des equations. M emoires de lacad emie des sciences de Berlin  (1749), 1751 p. 222288. Number 170 in the Enestrom index. Translation Copyright c 2005 Todd Doucet. All Rights Reserved. Comments & queries: email@example.com 2 INVESTIGATIONS ON THE then it is immediately clear that the number of these factors must be equal to the exponent n ; and therefore the number of all the roots, which will be x =- , x =- , x =- , x =- , . . . , will also equal this same exponent n , since a product such as ( x + )( x + )( x + )( x + ) cannot become equal to zero unless one of its factors vanishes. Every equation then, of whatever degree, will always have as many roots as the exponent of its highest power contains units. 3. Now it very often happens that not all of these roots are real quantities, and that some, or perhaps all, are imaginary quantities. We call a quantity imaginary when it is neither greater than zero, nor less than zero, nor equal to zero. This will be then something impossible, as for example - 1, or in general a + bi , since such a quantity is neither positive, nor negative, nor zero. So in this equation x 3- 3 xx + 6 x- 4 = 0 which has these three roots x = 1 , x = 1 + - 3 , x = 1- - 3 , the last two are imaginary, and there is only one real root, x = 1. From this we see that if we wish to include under the name of roots only those which are real, their number would often be much smaller than the highest exponent in the equation. And therefore when we say that every equation has as many roots as its degree exponent indicates, that must be understood to include all the roots, both real and imaginary....
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