67. THEOREM. If a number of ratios are equal, the sum of any number of antecedents is to any antecedent as the sum of the corresponding consequents is to the corresponding consequent.

Divide the sum of the first three equations by the last and we get a + c + e b + d + f

68. Mean proportion. The mean proportional between two numbers a and c is the number b, such that

10. Find the mean proportional between a2 + c2 and b2 + d2.

11. Find the mean proportional between a2 + b2 + c2 and b2 + c2 + d2.

CHAPTER VI

IRRATIONAL NUMBERS AND RADICALS

69. Existence of irrational numbers. We have seen that in order to solve any linear equation or set of linear equations with rational coefficients we need to make use only of the operations of addition, subtraction, multiplication, and division. When, however, we attempt to solve the equation of the second degree, x2 = 2, we find that there is no rational number that satisfies it.

ASSUMPTION. A factor of one member of an identity between integers is also a factor of the other member.

Thus let 2. a= b, where a and b are integers. Then since 2 is a factor of the left-hand member, it must also be contained in b.

THEOREM. No rational number satisfies the equation x2 = 2.

a

b

Suppose the rational number be a fraction reduced to its lowest terms which satisfies the equation. Then

Thus, by the assumption, 2 is contained in a2, and hence in a.

that is, 2 must also be contained in b, which contradicts the

a

hypothesis that is a fraction reduced to its lowest terms.

b

= 2 has no rational solution is

The fact that the equation x2 analogous to the geometrical fact that the hypotenuse of an isosceles right triangle is incommensurable with a leg.

70. The practical necessity for irrational numbers. For the practical purposes of the draughtsman, the surveyor, or the machinist, the introduction of this irrational number is superfluous, as no measuring rule can be made exact enough to distinguish between a length represented by a rational number and one that cannot be so represented. As the draughtsman does not use a mathematically perfect triangle, but one of rubber or wood, it is impossible to see in the fact of geometrical incommensurability just noted a practical demand from everyday life for the introduction of the irrational number. In fact the irrational number is a mathematical necessity, not a necessity for the laboratory or draughting room, as are the fraction and the negative number. We need irrational numbers because we cannot solve all quadratic equations without them, and the practical utility of those numbers comes only through the immense gain in mathematical power which they bring.

71. Extraction of square root of polynomials. This process, from which a method of extracting the square root of numbers is immediately deduced, may be performed as follows:

RULE. Arrange the terms of the polynomial according to the powers of some letter.

Extract the square root of the first term, write the result as the first term of the root, and subtract its square from the given polynomial.

Divide the first term of the remainder by twice the root already found, and add this quotient to the root and also to the trial divisor, thus forming the complete divisor.

Multiply the complete divisor by the last term of the root and subtract the product from the last remainder.

If terms of the given polynomial still remain, find the next term of the root by dividing the first term of the remainder by twice the first term of the root, form the complete divisor, and proceed as before until the desired number of terms of the root have been found.

8. 49a4-42 a3b+37 a2b2 – 12 ab3+4b4. 9. 2 ab 2 ac- 2 bc + a2 + b2 + c2.

10. u1w2 + v1u2 + w¥v2 + 2 u3v2w + 2 v3w2u + 2 w3u2v.

72. Extraction of square root of numbers. We have the following

RULE. Separate the number into periods of two figures each, beginning at the decimal point. Find the greatest number whose square is contained in the left-hand period. This is the first figure of the required root.

Subtract its square from the first period, and to the remainder annex the next period of the number.

Divide this remainder, omitting the right-hand digit, by twice the root already found, and annex the quotient to both root and divisor, thus forming the complete divisor.

Multiply the complete divisor by the last digit of the root, subtract the result from the dividend, and annex to the remainder the next period for a new dividend.

Double the whole root now found for a new divisor and proceed as before until the desired number of digits in the root have been found.

In applying this rule it often happens that the product of the complete divisor and the last digit of the root is larger than the dividend. In such a case we must diminish the last figure of the root by unity until we obtain a product which is not greater than the dividend.