At any point in the process of extracting the square root of a number before the exact square root is found, the square of the result already obtained is less than the original number. If the last digit of the result be replaced by the next higher one, the square of this number is greater than the original number.

There are always two values of the square root of any number. Thus √‡ = + 2 or — 2, since (+ 2)2 = (− 2)2 = 4. The positive root of any positive number or expression is called the principal root. When no sign is written before the radical, the principal root is assumed.

73. Approximation of irrational numbers. In the preceding process of extracting the square root of 2 we never can obtain a number whose square is exactly 2, for we have seen that such a number expressed as a rational (i.e. as a decimal) fraction does not exist. But as we proceed we get a number whose square differs less and less from 2.

Though we cannot say that 1.414 is the square root of 2, we may say that 1.414 is the square root of 2 correct to three decimal places, meaning that

(1.414)2 < 2 < (1.415)2.

74. Sequences. The exact value of the square root of moșt numbers, as, for instance, 2, 3, 5, cannot be found exactly in decimal form and so are usually expressed symbolically. By means of the process of extracting square root, however, we can find a number whose square is as near the given number as we may desire. We may, in fact, assert that the succession or sequence of numbers obtained by the process of extracting the square root of a number defines the square root of that number. Thus the sequence of numbers (1, 1.4,.1.41, 1.414, ...) defines the square root of 2.

75. Operations on irrational numbers. Just as we defined the laws of operation on the fraction and negative numbers (pp. 2-4), we should now define the meaning of the sum, difference, product, and quotient of the numbers defined by the sequence of numbers obtained by the square-root process. To define and explain completely the operations on irrational numbers is beyond the scope of this chapter. It turns out, however, that the number defined by a sequence is the limiting value of the rational numbers that constitute that sequence, that is, it is a value from which every number in the sequence beyond a certain point differs by as little as we please. We may, however, make the following statement regarding the multiplication of irrational numbers: In the sequence defining the square root of 2, namely, (1, 1.4, 1.41, 1.414, ...) we saw that we could obtain a number very nearly equal to 2 by multiplying 1.414 by itself. In general, we multiply numbers defined by sequences by multiplying the elements of these sequences; the new sequence, consisting of the products, defines the product of the original numbers.

Thus (1, 1.4, 1.41, 1.414, ...) (1, 1.4, 1.41, 1.414, ...)

=(1, 1.96, 1.9881, 1.999396, ...).

The numbers in this sequence approach 2 as a limit, and hence the sequence may be said to represent 2.

76. Notation. We denote the square root of a (where a represents any number or expression) symbolically by Va, and assert √a. √a =(√a)2 = a,

that

or, more generally,

Similarly,

√a. √b = √a.b.

√a ÷ √b = √a÷b.

EXERCISES

1. Form five elements of a sequence defining √3. 2. Form five elements of a sequence defining √5. 3. Form five elements of a sequence defining √6.

4. Form, in accordance with the rule just given, four elements of the sequence √2. √3. Compare the result with the elements obtained in Ex. 3. 5. Form similarly the first four elements of product √2. √5 with the first four elements obtained by extracting the square root of 10.

77. Other irrational numbers. The cube root and higher roots of numbers could also be found by processes analogous to the method employed in finding the square root, but as they are almost never used practically, they will not be included here. It should be kept in mind, however, that by these processes sequences of numbers may be derived that define the various roots of numbers precisely as the sequences derived in the preceding paragraphs define the square root of numbers.

The nth root of any expression a is symbolized by Va. Here n is sometimes called the index of the radical. The principle for the multiplication and division of radicals with any integral index is given by the following

ASSUMPTION. The product (or quotient) of the nth root of two numbers is equal to the nth root of the product (or quotient) of the numbers.

78. Reduction of a radical to its simplest form. A radical is in its simplest form when the expression under the sign is integral (§ 11) and contains no factor raised to a power which equals the index of the radical; in other words, when no factor can be removed from under the radical sign and still leave an integral expression. We may reduce a quadratic radical to its simplest form by the following

RULE. If the expression under the radical sign is fractional, multiply both numerator and denominator by some expression that will make the denominator a perfect square.

Factor the expression under the radical into two factors, one of which is the greatest square factor that it contains.

Take the square root of the factor that is a perfect square, and express the multiplication of the result by the remaining factor under the radical sign.

If the radical is of the nth index, the denominator must be made a perfect nth power, and any factor that is to be taken from under the radical sign must also be a perfect nth power.

79. Addition and subtraction of radicals. Radicals that are of the same index and have the same expression under the radical sign are similar. Only similar radicals can be united into one term by addition and subtraction. We add radical expressions by the following

RULE. Reduce the radicals to be added to their simplest form. Add the coefficients of similar radicals and prefix this sum as the coefficient of the corresponding radical in the result.

A rule precisely similar is followed in subtracting radical expressions.

8. 7 √4x+4√9x + 3 √45 x − 5 √36 x − 2 √80 x.

9. Va−b+ √16 a − 16 b + √ax2

bx2 − √(9a − b).

10. 4 √a2x - 3 √b2x + 2 √c2x + √d2x − 2 √(b + d)2x.

11. 6 √x + 3 √√2 x − 5 √3 x − 2 √4 x + √12 x − √ 18 x.