SURD

, in Arithmetic, denotes a number or quantity that is incommensurate to unity; or that is inexpressible in rational numbers by any known way of notation, otherwise than by its radical sign or index.—This is otherwise called an irrational or incommensurable number, as also an imperfect power.

These Surds arise in this manner: when it is proposed to extract a certain root of some number or quantity, which is not a complete power or a true figurate number of that kind; as, if its square root be demanded, and it is not a true square; or if its cube root be required, and it is not a true cube, &c; then it is impossible to assign, either in whole numbers, or in fractions, the exact root of such proposed number. And whenever this happens, it is usual to denote the root by setting before it the proper mark of radicality, which is √, and placing above this radical sign the number that shews what kind of root is required. Thus, √²2 or √2 signifies the square root of 2, and √³10 signifies the cube root of 10; which roots, because it is impossible to express them in numbers exactly, are properly called Surd roots.

There is also another way of notation, now much in use, by which roots are expressed by fractional indices, without the radical sign: thus, like as x², x³, x⁴, &c, denote the square, cube, 4th power, &c, of x; so x^1/2, x^1/3, x^1/4, &c, denote the square root, cube root, 4th root, &c, of the same quantity x.—The reason of this is plain enough; for since √x is a geometrical mean proportional between 1 and x, so 1/2 is an arithmetical mean between 0 and 1; and therefore, as 2 is the index of the square of x, 1/2 will be the proper index of its square root, &c.

It may be observed that, for convenience, or the sake of brevity, quantities which are not naturally Surds, are often expressed in the form of Surd roots. Thus √4, √(9/4), √27, are the same as 2, 3/2, 3.

Surds are either simple or compound.

Simple Surds, are such as are expressed by one single term; as √2, or √³a, &c.

Compound Surds, are such as consist of two or more simple Surds connected together by the signs + or -; as √3 + √2, or √3 - √2, or √³(5 + √2): which last is called an universal root, and denotes the cubic root of the sum arising by adding 5 and the root of 2 together.

1. Such Surds as √2, √3, √5, &c. though they are themselves incommensurable with unity, according to the definition, are commensurable in power with it, because their powers are integers, which are multiples of unity. They may also be sometimes commensurable with one another; as √8 and √2, which are to one another as 2 to 1, as is found by dividing them by their greatest common measure, which is √2, for then those two become √4 = 2, and 1 the ratio.

2. To reduce Rational Quantities to the form of any proposed Surd Roots.—Involve the rational quantity according to the index of the power of the Surd, and then prefix before that power the proposed radical sign.

Thus,

And in this way may a simple Surd fraction, whose radical sign refers to only one of its terms, be changed into another, which shall include both numerator and denominator. Thus, √2/5 is reduced to √(2/25), and 5/√³4 to √³(125/4): thus also the quantity a reduced to the form of x^1/n or √ⁿx, is (―aⁿ)^1/n or √ⁿaⁿ. And thus may roots with rational coefficients be reduced so as to be wholly affected by the radical sign; as

3. To reduce Simple Surds, having different radical signs (which are called heterogeneal Surds) to others that may have one common radical sign, or which are homogeneal: Or to reduce roots of different names to roots of the same name.—Involve the powers reciprocally, each according to the index of the other, for new powers; and multiply their indices together, for the common index. Otherwise, as Surds may be considered as powers with fractional exponents, reduce these fractional exponents to fractions having the same value and a common denominator.

Thus, by the 1st way, √ⁿa and √^mx become √^mna^m and √^mnxⁿ; | and, by the 2d way, a^1/n and x^1/m become (―a^m))^1/(mn) and (―xⁿ))^1/(mn).

Also √3 and √³2 are reduced to √⁶27 and √⁶4, which are equal to them, and have a common radical sign.

4. To reduce Surds to their most simple expressions, or to the lowest terms possible.—Divide the Surd by the greatest power, of the same name with that of the root, which you can discover is contained in it, and which will measure or divide it without a remainder; then extract the root of that power, and place it before the quotient or Surd so divided; this will produce a new Surd of the same value with the former, but in more simple terms. Thus, √(16a²x), by dividing by 16a², and prefixing its root 4a, before the quotient √x, becomes 4a√x; in like manner, √12 or √(4 X 3), becomes 2√3; And .

5. To Add and Subtract Surds.—When they are reduced to their lowest terms, if they have the same irrational part, add or subtract their rational coefficients, and to the sum or difference subjoin the common irrational part.

Thus, .

Or such Surds may be added and subtracted, by first squaring them (by uniting the square of each part with double their product), and then extracting the root universal of the whole. Thus, for the first example above, .

If the quantities cannot be reduced to the same irrational part, they may just be connected by the signs + or -.

6. To Multiply and Divide Surds.—If the terms have the same radical, they will be multiplied and divided like powers, viz, by adding their indices for multiplication, and subtracting them for division.

Thus, .

If the quantities be different, but under the same radical sign; multiply or divide the quantities, and place the radical sign to the product or quotient.

Thus, .

But if the Surds have not the same radical sign, reduce them to such as shall have the same radical sign, and proceed as before.

Thus, .

If the Surds have any rational coefficients, their product or quotient must be prefixed.

Thus, .

7. Involution and Evolution of Surds.—Surds are involved, or raised to any power, by multiplying their indices by the index of the power; and they are evolved or extracted, by dividing their indices by the index of the root.

Thus, the square of .

Or thus: involve or extract the quantity under the radical sign according to the power or root required, continuing the same radical sign.

So the square of √³2 is √³4; and the square root of √³4, is √³2.

Unless the index of the power is equal to the name of the Surd, or a multiple of it, for in that case the power of the Surd becomes rational. Thus, the square of √3 is 3, and the cube of √³a² is a².

Simple Surds are commensurable in power, and by being multiplied by themselves give, at length, rational quantities: but compound Surds, multiplied by themselves, commonly give irrational products. Yet, in this case, when any compound Surd is proposed, there is another compound Surd, which, multiplied by it, gives a rational product.

Thus, √a + √b multiplied by √a - √b gives a - b; and √³a - √³b mult. by √³a² + √³(ab) + √³b² gives a - b. The finding of such a Surd as multiplying the proposed Surd gives a rational product, is made easy by three theorems, delivered by Maclaurin, in his Algebra, pa. 109 &c.

This operation is of use in reducing Surd expressions to more simple forms. Thus, suppose a binomial Surd divided by another, as √20 + √12 by √5 - √3, the quotient might be expressed by ; but this will be expressed in a more simple form, by multiplying both numerator and denominator by such a Surd as makes the product of the denominator become a rational quantity: thus, multiplying them by √5 + √3, the fraction or quotient becomes. . To do this generally, see Maclaurin's Alg. p. 113.

When the square root of a Surd is required, it may be found nearly, by extracting the root of a rational quantity that approximates to its value. Thus, to find the | square root of 3 + 2√2; first calculate √2 = 1.41421; hence 3 + 2√2 = 5.82842, the root of which is nearly 2.41421.

In like manner we may proceed with any other proposed root. And if the index of the root be very high, a table of logarithms may be used to advantage: thus, to extract the root √⁷(5 + √¹³17); take the logarithm of 17, divide it by 13, find the number answering to the quotient, add this number to 5, find the log. of the sum, and divide it by 7, and the number answering to this quotient will be nearly equal to √⁷(5 + √¹³17).

But it is sometimes requisite to express the roots of Surds exactly by other Surds. Thus, in the first example, the square root of 3 + 2√2 is 1 + √2, for . For the method of performing this, the curious may consult Maclaurin's Algeb. p. 115, where also rules for trinomials &c may be found. See also the article Binomial Roots, in this Dictionary.

For extracting the higher roots of a binomial, whose two members when squared are commensurable numbers, we have a rule in Newton's Arith. pa. 59, but without demonstration. This is supplied by Maclaurin, in his Alg. p. 120: as also by Gravesande, in his Matheseos Univers. Elem. p. 211.

It sometimes happens, in the resolution of cubic equations, that binomials of this form a ± b√-1 occur, the cube roots of which must be found; and to these Newton's rule cannot always be applied, because of the impossible or imaginary factor √-1; yet if the root be expressible in rational numbers, the rule will often yield to it in a short way, not merely tentative, the trials being confined to known limits. See Maclaurin's Alg. p. 127. It may be farther observed, that such roots, whether expressible in rational numbers or not, may be found by evolving the quantity a + b√-1 by Newton's binomial theorem, and summing up the alternate terms. Maclaurin, p. 130.

Those who are desirous of a general and elegant solution of the problem, to extract any root of an impossible binomial a + b√-1, or of a possible binomial a + √b, may have recourse to the appendix to Saunderson's Algebra, and to the Philos. Trans. number 451, or Abridg. vol. 8, p. 1. On the management of Surds, see also the numerous authors upon Algebra.

SURDESOLID. See Sursolid.