Part I. Section I. Chapter 12. “Of Square Roots, and of Irrational Numbers resulting from them.”

123 What we have said in the preceding chapter amounts to this ; that the square root of a given number is that number whose square is equal to the given number; and that we may put before those roots either the positive, or the negative sign.

124 So that when a square number is given, provided we retain in our memory a sufficient number of square numbers, it is easy to find its root. If 196, for example, be the given number, we know that its square root is 14.

Fractions, likewise, are easily managed in the same way. It is evident, for example, that ⁵⁄₇ is the square root of ²⁵⁄₄₉; to be convinced of which, we have only to take the square root of the numerator and that of the denominator.

If the number proposed be a mixed number, as 12¼, we reduce it to a single fraction, which, in this case, will be ⁴⁹⁄₄; and from this we immediately perceive that ⁷⁄₂, or 3½, must be the square root of 12¼.

125 But when the given number is not a square, as 12, for example, it is not possible to extract its square root ; or to find a number, which, multiplied by itself, will give the product 12. We know, however, that the square root of 12 must be greater than 3, because 3 · 3 produces only 9; and less than 4, because 4 · 4 produces 16, which is more than 12; we know also, that this root is less than 3½, for we have seen that the square of 3½, or ⁷⁄₂, is 12¼; and we may approach still nearer to this root, by comparing it with 3⁷⁄₁₅; for the square of 3⁷⁄₁₅, or of ⁵²⁄₁₅, is ²⁷⁰⁴⁄₂₂₅, or 12⁴⁄₂₂₅; so that this fraction is still greater than the root required, though but very little so, as the difference of the two squarcs is only ⁴⁄₂₂₅.

126 We may suppose that as 3½ and 3⁷⁄₁₅ are numbers greater than the root of 12, it might be possible to add to 3 a fraction a little less than ⁷⁄₁₅, and precisely such, that the square of the sum would be equal to 12.

Let us therefore try with 3³⁄₇, since ³⁄₇ is a little less than ⁷⁄₁₅. Now 3³⁄₇ is equal to ²⁴⁄₇, the square of which is ⁵⁷⁶⁄₄₉, and consequently less by ¹²⁄₄₉ than 12, which may be expressed by ⁵⁸⁸⁄₄₉. It is, therefore, proved that 3³⁄₇ is less, and that 3⁷⁄₁₅ is greater than the root required. Let us then try a number a little greater than 3³⁄₇, but yet less than 3⁷⁄₁₅; for example, 3⁵⁄₁₁; this number, which is equal to ³⁸⁄₁₁, has for its square ¹⁴⁴⁴⁄₁₂₁; and by reducing 12 to this denominator, we obtain ¹⁴⁵²⁄₁₂₁ which shows that 3⁵⁄₁₁ is still less than the root of 12, namely by ⁸⁄₁₂₁; let us therefore substitute for ⁵⁄₁₁ the fraction ⁶⁄₁₃, which is a little greater, and see what will be the result of the comparison of the square of 3⁶⁄₁₃, with the proposed number 12. Here the square of 3⁶⁄₁₃ is ²⁰²⁵⁄₁₆₉; and 12 reduced to the same denominator is ²⁰²⁸⁄₁₆₉; so that 3⁶⁄₁₃ is still too small, though only by ³⁄₁₆₉, whilst 3⁷⁄₁₅ has been found too great.

127 It is evident, therefore, that whatever fraction is joined to 3, the square of that sum must always contain a fraction, and can never be exactly equal to the integer 12. Thus, although we know that the square root of 12 is greater than 3⁶⁄₁₃, and less than 3⁷⁄₁₅, yet we are unable to assign an intermediate fraction between these two, which, at the same time, if added to 3, would express exactly the square root of 12; but notwithstanding this, we are not to assert that the square root of 12 is absolutely and in itself indeterminate: it only follows from what has been said, that this root, though it necessarily has a determinate magnitude, cannot be expressed by fractions.

128 There is therefore a sort of numbers, which cannot be assigned by fractions, but which are nevertheless determinate quantities; as, for instance, the square root of 12: and we call this new species of numbers, irrational numbers. They occur whenever we endeavour to find the square root of a number which is not a square; thus, 2 not being a perfect square, the square root of 2, or the number which, multiplied by itself, would produce 2, is an irrational quantity. These numbers are also called surd quantities, or incommensurables.

129 These irrational quantities, though they cannot be expressed by fractions, are nevertheless magnitudes of which we may form an accurate idea; since, however concealed the square root of 12, for example, may appear, we are not ignorant that it must be a number, which, when multiplied by itself, would exactly produce 12; and this property is sufficient to give us an idea of the number, because it is in our power to approximate towards its value continually.

130 As we are therefore sufficiently acquainted with the nature of irrational numbers, under our present consideration, a particular sign has been agreed on to express the square roots of all numbers that are not perfect squares; which sign is written thus √, and is read square root. Thus, √12 represents the square root of 12, or the number which, multiplied by itself, produces 12; and √2 represents the square root of 2; √3 the square root of 3; √⅔ that of ⅔; and, in general, \(\surd a\) represents the square root of the number \(a\). Whenever, therefore, we would express the square root of a number, which is not a square, we need only make use of the mark √ by placing it before the number.

131 The explanation which we have given of irrational numbers will readily enable us to apply to them the known methods of calculation. For knowing that the square root of 2, multiplied by itself, must produce 2; we know also, that the multiplication of √2 by √2 must necessarily produce 2; that, in the same manner, the multiplication of √3 by √3 must give 3; that √5 by √5 makes 5; that √⅔ by √⅔ makes ⅔; and, in general, that \(\surd a\) multiplied by \(\surd a\) produces \(a\).

132 But when it is required to multiply \(\surd a\) by \(\surd b\), the product is \(\surd ab\); for we have already shown, that if a square has two or more factors, its root must be composed of the roots of those factors; we therefore find the square root of the product \(ab\), which is \(\surd ab\), by multiplying the square root of \(a\), or \(\surd a\), by the square root of \(b\), or \(\surd b\); etc. It is evident from this, that if \(b\) were equal to \(a\), we should have \(\surd aa\) for the product of \(\surd a\) by \(\surd b\). But \(\surd aa\) is evidently \(a\), since \(aa\) is the square of \(a\).

133 In division, if it were required, for example, to divide \(\surd a\), by \(\surd b\), we obtain \(\surd \frac{a}{b}\); and, in this instance, the irrationality may vanish in the quotient. Thus, having to divide √18 by √8, the quotient is √¹⁸⁄₈, which is reduced to √⁹⁄₄, and consequently to ³⁄₂, because ⁹⁄₄ is the square of ³⁄₂.

134 When the number before which we have placed the radical sign √ is itself a square, its root is expressed in the usual way; thus, √4 is the same as 2; √9 is the same as 3; √36 the same as 6; and √12¼, the same as ⁷⁄₂, or 3½. In these instances, the irrationality is only apparent, and vanishes of course.

135 It is easy also to multiply irrational numbers by ordinary numbers; thus, for example, 2 multiplied by √5 makes 2√5; and 3 times √2 makes 3√2. In the second example, however, as 3 is equal to √9, we may also express 3 times √2 by √9 multiplied by √2, or by √18; also \(2\surd a\) is the same as \(\surd 4a\), and \(3 \surd a\) the same as \(\surd 9a\); and, in general, \(b\surd a\) has the same value as the square root of \(bba\), or \(\surd bba\): whence we infer reciprocally, that when the number which is preceded by the radical sign contains a square, we may take the root of that square, and put it before the sign, as we should do in writing \(b \surd a\) instead of \(\surd bba\). After this, the following reductions will be easily understood:

√8, or √2 · 4, is equal to 2√2.
√12, or √3 · 4, is equal to 2√3.
√18, or √2 · 9, is equal to 3√2.
√24, or √6 · 4, is equal to 2√6.
√32, or √2 · 16, is equal to 4√2.
√75, or √3 · 25, is equal to 5√3, and so on.

136 Division is founded on the same principles; as \(\surd a\) divided by \(\surd b\) gives \(\frac{\surd a}{\surd b}\), or \(\surd \frac{a}{b}\). In the same manner,

\(\frac{\surd 8}{\surd 12}\) is equal to \(\surd \frac{8}{2}\), or \(\surd 4\), or 2
\(\frac{\surd 18}{\surd 2}\) is equal to \(\surd \frac{18}{2}\), or \(\surd 9\), or 3
\(\frac{\surd 12}{\surd 3}\) is equal to \(\surd \frac{12}{3}\), or \(\surd 4\), or 2.

Farther,

\(\frac{2}{\surd 2}\) is equal to \(\frac{\surd 4}{\surd 2}\), or \(\surd \frac{4}{2}\), or \(\surd 2\),
\(\frac{3}{\surd 3}\) is equal to \(\frac{\surd 9}{\surd 3}\), or \(\surd \frac{9}{3}\), or \(\surd 3\),
\(\frac{12}{\surd 6}\) is equal to \(\frac{\surd 144}{\surd 6}\), or \(\surd \frac{144}{6}\), or \(\surd 24\), or \(\surd 6\cdot 4\), or \(2\surd 6\).

137 There is nothing in particular to be observed in addition and subtraction, because we only connect the numbers by the signs + and -: for example, √2 added to √3 is written √2 + √3; and √3 subtracted from √5 is written √5 - √3.

138 Wo may observe lastly, that in order to distinguish the irrational numbers, we call all other numbers, both integral and fractional, rational numbers; so that, whenever we speak of rational numbers, we understand integers, or, fractions.

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.