14. 15. 10. 2. 16. √(3x2) √3. 9.50 -2. 2-6. √24. 612√3. 11. 13. 17. √(ab)· √(− ab3) · √(− ab2). 18. √(− m1n2) · √(− mn3) · √(− m3n3) • √(− m2n). 19. (√3+− 2) (√− 3 + √− 5). 20. (3√−5+4√− 6) (2 √ − 5 — 3√− 6). 21. (√-a+√− b ) ( √ - a - √-b). 22. [√(a+b) + √−b][ √− (a + b) − √− b]. 23. √x2. 24. (x)2. 25. a1. 28. (ab)· √(b − a). 31. 14. 32. ¿101. 38.8-2. 39. 5-352√7. 41. √-α÷√- a2. 43. (√−6+√−8) ÷ √ − 2. 34. ¿35-191 37. 27+√3. 42. √(−ab)÷√-b. 44. (√12 √18)÷√−3. 16. A Complex Number is the algebraic sum of a real and an imaginary number; as, 3+2 i. The general form of a complex number is evidently a + bi, wherein a and b are real numbers. 0, we have any real number. When a = 0, we have any imaginary number. 17. Two complex numbers which differ only in the sign of their imaginary terms are called Conjugate Complex Numbers; as, 2-3 i and 2 + 3 i. 18. Two complex numbers are said to be equal when the real term of one is equal to the real term of the other, and the imaginary term of one is equal to the imaginary term of the other; as, 2+3 i = 2 + 3 i. That is, if then a + bi = c + di, a = c, and bi = di, or b = d. Observe that the preceding statement is a definition of the meaning of the sign of equality between two complex numbers. 19. From the preceding article it follows that, if a + bi = 0 = 0 + 0 i, then a = = 0, b = 0. 20. Addition and Subtraction of Complex Numbers. -The following definition of Addition and Subtraction of Complex Numbers is a natural extension of the definition of these operations for real numbers: Two complex numbers are added or subtracted by adding or subtracting the real parts by themselves and the imaginary parts by themselves. E.g., (2+3)+(−5+6i)=(2−5)+(3i+6 i)= −3+9 i. In general, (a + bi) ± (c + di) = (a + c) + (b ± d)i. 21. The Commutative and Associative Laws hold for algebraic addition of complex numbers. This principle follows immediately from the definition of addition and subtraction. That is, (a + bi) + (c + di) = (c + di) + (a + bi); (a + bi) + (c + di) + (e + fi) = (a + bi) +[(e + fi) + (c + di)]. : 22. The sum or difference of two complex numbers is, in general, a complex number. E.g., (2+3)+(− 4 + 2 i) = (2-4) + (3 + 2) i = −2+5 i. But the sum of two conjugate complex numbers is real. 23. Multiplication of Complex Numbers. We define multiplication by a complex number by assuming that the Distributive Law holds; that is, by the relation or (a+bi) (c+di) = ac + bei + adi+bdi2, (a+bi) (c+di) = (ac - bd) + (bc + ad) i. 24. The Commutative, Associative, and Distributive Laws hold for multiplication of complex numbers. This principle follows from the definition of multiplication. That is, (a+bi) (c + di) = (c +di) (a + bi); (a + bi) (c+di) (e + fi) = (a + bi) [(e + fi) (c+di)] = etc. 25. The product of two complex numbers is, in general, a complex number. E.g., (2+3)(−4+2i)=-8-12i+4i-614-8 i. But the product of two conjugate complex numbers is real and positive. E.g., (-2+3) ( − 2 − 3 i) = ( − 2)2 — (3 i)2 = 4 + 9 = 13. In general, (a + bi) (a — bi) = a2 — (bi)2 = a2 + b2. 26. The square of a complex number is a complex number. E.g., (2 + 3 √ − 1)2 = 4 + 12 √−1 + (3√−1)2 =4+12-1-9 =-5+12√-1=-5+12 i. But the cube of a complex number is sometimes real. - E.g., (-±√−3)3 = − } ± }√−3+} + {√−3 = 1. From these results we see that 3/1 has three values, 1, −1 +1√−3, −1—1√−3. 27. Division of Complex Numbers. The quotient of one complex number by another is in general a complex number. 28. It follows from the preceding article that a fraction whose denominator is a complex number can be expressed as a complex number by multiplying both numerator and denominator by the conjugate of the denominator. Notice that it was necessary to multiply numerator and denominator only by Ex. 2. 1 = √3. 2 + √−3 (2 + √ − 3) (2 − √−3) Complex Factors. 29. A quadratic expression which is the product of two complex factors can be resolved into factors by the method used to resolve a quadratic expression into irrational factors. Completing - 2x to the square of a binomial in x, we have x2-2x+3= x2-2x+1−1+3 30. If then For, from we have √(a − bi)=√x — i√y. √(a+bi)=√x+i√y, a+bi= xy + 2√(xy) · i. Therefore, by Art. 18, a=x-y, and b=2√(xy). |