Part I. Section I. Chapter 13. “Of Impossible, or Imaginary Quantities, which arise from the same source.”

139 We have already seen that the squares of numbers, negative as well as positive, are always positive, or affected by the sign +; having shown that \(-a\) multiplied by \(-a\) gives \(+aa\), the same as the product of \(+a\) by \(+a\): wherefore, in the preceding chapter, we supposed that all the numbers, of which it was required to extract the square roots, were positive.

140 When it is required, therefore, to extract the root of a negative number, a great difficulty arises; since there is no assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of -4; we here require such a number as, when multiplied by itself, would produce -4: now, this number is neither +2 nor -2, because the square both of +2 and of -2, is +4, and not -4.

141 We must therefore conclude, that the square root of a negative number cannot be either a positive number or a negative number, since the squares of negative numbers also take the sign plus: consequently, the root in question must belong to an entirely distinct species of numbers; since it cannot be ranked either among positive or among negative numbers.

142 Now, we before remarked, that positive numbers are all greater than nothing, or 0, and that negative numbers are all less than nothing, or 0; so that whatever exceeds 0 is expressed by positive numbers, and whatever is less than 0 is expressed by negative numbers. The square roots of negative numbers, therefore, are neither greater nor less than nothing; yet we cannot say, that they are 0; for 0 multiplied by produces 0, and consequently does not give a negative number.

143 And, since all numbers which it is possible to conceive, are either greater or less than 0, or are itself, it is evident that we cannot rank the square root of a negative number amongst possible numbers, and we must therefore say that it is an impossible quantity. In this manner we are led to the idea of numbers, which from their nature are impossible; and therefore they are usually called imaginary quantities, because they exist merely in the imagination.

144 All such expressions, as √-1, √-2, √-3, √-4, etc. are consequently impossible, or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing; which necessarily constitutes them imaginary, or impossible.

145 But notwithstanding this, these numbers present themselves to the mind; they exist in our imagination, and we still have a sufficient idea of them; since we know that by √-4 is meant a number which, multiplied by itself, produces -4; for this reason also, nothing prevents us from making use of these imaginary numbers, and employing them in calculation.

146 The first idea that occurs on the present subject is, that the square of √-3, for example, or the product of √-3 by √-3, must be -3; that the product of √-1 by √-1 is -1; and, in general, that by multiplying \(\surd -a\) by \(\surd -a\), or by taking the square of \(\surd -a\) we obtain \(-a\).

147 Now, as \(-a\) is equal to \(+1\) multiplied by -1, and as the square root of a product is found by multiplying together the roots of its factors, it follows that the root of \(a\) times -1, or \(\surd -a\), is equal to \(\surd a\) multiplied by \(\surd -1\); but \(\surd a\) is a possible or real number, consequently the whole impossibility of an imaginary quantity may be always reduced to √-1; for this reason, √-4 is equal to √4 multiplied by √-1, or equal to 2√-1, because √4 is equal to 2; likewise √-9 is reduced to √9 · √-1, or 3√-1; and √-16 is equal to 4√-1.

148 Moreover, as \(\surd a\) multiplied by \(\surd b\) makes \(\surd ab\), we shall have √6 for the value of √-2 multiplied by √-; and √4, or 2, for the value of the product of √-1 by √-4. Thus we see that two imaginary numbers, multiplied together, produce a real, or possible one. But, on the contrary, a possible number, multiplied by an impossible number, gives always an imaginary product: thus, √-3 by √+5, gives √-15.

149 It is the same with regard to division; for \(\surd a\) divided by \(\surd b\) making \(\frac{\surd a}{\surd b}\), it is evident that √-4 divided by √-1 will make √+4, or 2; that √+3 divided by √-3 will give √-1; and that 1 divided by √-1 gives \(\frac{1}{\surd -1}\), or √-1; because 1 is equal to √+1.1

150 We have before observed, that the square root of any number has always two values, one positive and the other negative; that √4, for example, is both +2 and -2, and that, in general, we may take \(-\surd a\) as well as \(+\surd a\) for the square root of \(a\). This remark applies also to imaginary numbers; the square root of \(-a\) is both \(+\surd -a\) and \(-\surd -a\); but we must not confound the signs + and -, which are before the radical sign √, with the sign which comes after it.

151 It remains for us to remove any doubt, which may be entertained concerning the utility of the numbers of which we have been speaking; for those numbers being impossible, it would not be surprising if they were thought entirely useless, and the object only of an unfounded speculation. This, however, would be a mistake; for the calculation of imaginary quantities is of the greatest importance, as questions frequently arise, of which we cannot immediately say whether they include any thing real and possible, or not; but when the solution of such a question leads to imaginary numbers, we are certain that what is required is impossible.

In order to illustrate what we have said by an example, suppose it were proposed to divide the number 12 into two such parts, that the product of those parts may be 40. If we resolve this question by the ordinary rules, we find for the parts sought 6 + √-4 and 6 - √-4; but these numbers being imaginary, we conclude, that it is impossible to resolve the question.

The difference will be easily perceived, if we suppose the question had been to divide 12 into two parts which multiplied together would produce 35; for it is evident that those parts must be 7 and 5.

Editions

  1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
  2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.

  1. 1 divided by √-1 is in fact -√-1; but indeed both √-1 and -√-1 are square roots of -1. ↩