College Algebra Volume 2 (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. 1919 Excerpt: ...x and y, and may be represented graphically by a point in a plane. Two lines, X'X, Y' Y, are drawn perpendicular to each other and intersecting at 0, Fig. 18. To represent the number 2 + 3 i, measure off on X'X to the right the distance 2, and up the distance 3. In general, the graph of the number x + iy is the point whose coordinates are (x, y). The line X'X is often called the axis of "reals" and the line Y'Ythe axis of imaginaries. See Klein, Famous Problems in Elementary Geometry, translation by Beman and Smith, p. 68. It is often convenient to represent complex numbers by, r another method. Connect the point which represents x + iy with the origin as in Fig. 19. Let the length of this x+iy iine be r. The point can then be represented by giving the length r and the angle 6. From the figure ." x = r cos 8, y=r sin 8, Fw-19 x +? = r. Hence, the number x-f-iy may be written in the form x + iy = r (cos 8 + i sin 8). This form is called the polar form of a complex number. The angle 8 is called the argument or amplitude, the length r the modulus or absolute value of the complex number. It should be noted that the complex numbers include all real numbers. In Fig. 18, the real numbers are represented by points on the line X'X. The pure imaginary numbers are represented by points on the line TY. 80. Equal complex numbers. If two complex numbers a + hi and c + di are equal, then a = c and b = d. For, if a + bi = c + di, (1) by transposing, a--c = (d--b)i. (2) Unless a--c = d--6 = 0, we should have a--c, a real number, equal to (d--b)i, an imaginary number. Conversely, if a = c, and b = d, a + bi = c + di. Hence, when any two expressions containing imaginary and real terms are equal to each other, we may equate the real parts and the imaginary parts...