CHAPTER XX COMPLEX NUMBERS. THEORY OF QUADRATICS. CUBE ROOT. BINOMIAL THEOREM. 206. The roots of the equation x2 - 2x+2=0 are 1 +√1 and 1-√-1. These numbers are new. The square of a real number is not negative. On the other hand, the square root of a negative number, for example 16, is not real. It is known as a pure imaginary. The unit of imaginaries is √−1. It is denoted by i. The defining property of i is 22 1. =- ī It is subject to all the laws of algebra. The standard way of writing √-a, a being a positive real number, is iva. For (i√a)2 = i2 · a = — a, and (√— a)2=-a. (Def. of sq. root.) Illustrations. √=4=i√4= 2 i, √ — 17 = i√17. 207. The sum of a pure imaginary and a real number is called a complex number. Its standard type is a+ib. (a and b real.) If b=0, a + ib becomes a. Hence a + ib includes the totality of all real numbers. = 23 = 12. i = -1.i = 1412.2 = 208. As 2-1 and i is subject to the laws of algebra, ¿1=i, 12= ¿2 = -1, -1.-1=1, ¿5i, ¿6=-1, -i, 281. 26 " = The powers of i recur, the period being 4. 279 = (¿4) 19 . ¿3 = — i, ¿46 = (¿4) 11 . ¿2 — — 1. 209. Graphical representation of complex numbers. If a point P has for its coördinates a and b, then the point P represents the number a + ib. Consequently every point in the plane represents some complex number. The points of the x-axis represent all real numbers. The totality of all real b= 1, then a + ib becomes i. The number i is represented by a point on OY, one unit's distance from O. Similarly the number in is a point on OY, n units distant from 0. All pure imaginaries are therefore represented by points on the line YOY'. Example 1. V-2.√-8=i√2. i√8 = 2√16 =-1.44. Example 2. 6 ÷√ 46÷2 i = 6 i÷2¿2 = 6 i ÷ − 2 =- 3 i. 1+i 1+i 1+i 1+2i+22 1+2 i-1 = i. 1-i 14+5 i 4- i 1-i 1+i 1-2 1-(-1) ... (2x-5)2=-19, i.e. x2-5x+11= 0. 27. (a+ib)2. 28. (√x+√y)2. 29. (2+√3)2. 30. (3-V-2)2. 32. (-1-√3)3. 33. (1+√−1)2. 31. (3+√−4)(3−√−4). 38. √-a2 ÷ √ — b2. 39. 1÷i. + 40. Find the value of x2-4x+5, when x=2 ± i. 41. Find the value of x2 + 2x + 4, when x= −1 ± i√3. 42. Multiply a + ib by a - ib, and from the result determine the rationalizing factor of 5-4 i and of 1+ √ — 3. 43. Find the value of x2 x + 1, when x = = {(1−√−3). 54. Construct the equation whose roots are 1±i; 3'+V-59 If b2 4 ac, the roots are real. If 62 = 4 ac, the roots are equal and real. If b2 4 ac, the roots are imaginary. 4 ac is a perfect square, the roots are rational. If 62 If the absolute term is zero, one root of a quadratic is zero. Cor. 2. If b= 0, the roots become +V-4 ac If the coefficient of x is zero, the roots of a quadratic are equal numerically, but opposite in sign. i.e. 211. The equation having m and n for its roots is If m and n are the roots of ax2 + bx + c = 0, then x2 - (m + n)x+mn=0, and (1) NOTE. These results may be obtained directly from § 210. |