Manual of Algebra (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1875 edition. Excerpt: ...the second; the seventh, the same as the third; the eighth, the same as the fourth; the ninth, again, the same as the first; and so on indefinitely, as shown in the table, n being any whole number. To show the use of this table, let it be required to find the continued product of V--4, V--3, V--2, V--7, and V--8. Reducing these expressions to the proper form, and indicating the multiplication, we have, 2V-1 x VsV-T x A/2VT x A/7v1 x 2a/2a/t. Changing the order of the factors, (2 x V3 x A/2 x A/7 x 2a/2) (V1)5. Hence, the product is equal to, 8a/21 X V--1=8V--21. EXAMPLES. Perform the multiplications indicated below: 1. A/1 x A/2.. Ans. a x (Vl)2 =--ab. From what precedes, it follows that the only radical parts of any power of an expression of the form, a bV--1, will be of the form cV--1. Properties of Imaginary Quantities. 138. 1. A quantity of the form, aV--1, cannot be equal to the sum of a rational quantity and a quantity of the form, b V--1 For, if so, let us have the equality, aV--1 = x + bV--1; squaring both members, we have, --a2 = x2 + %bxf1--b2; transposing, and dividing by 2bx, --=-b2--a2--x2 VX = Ux' an equation which is manifestly absurd, for the first member is imaginary, and the second real, and no imaginary quantity can be equal to a real quantity; hence, the hypothesis isabsurd; and, consequently, the principle enunciated is true. In the same way, it may be shown that no radical of the second degree can be equal to an entire quantity plus a radical of the second degree. 2. If, a + b V--1 = x + yV--1, then a = x, and b = y For, by transposition, we have, b V--1 = (x--a) + yV--1; but from the preceding principle, this equation can only be true when x--a = 0, -or x = a; making this supposition, and dividing both members of the...