ADDITION AND SUBTRACTION OF IMAGINARIES 336. In the addition and subtraction of imaginary and complex numbers, the laws established for other irrational numbers apply. Hence, to add such numbers, reduce them to the form a√1, or a + b√-1 if complex, and then add the real parts and the imaginary parts separately, writing the results as an algebraic sum. A similar method applies in subtraction. 1. Find the sum of √– 4, √−9, and √– 3. √ - 3 = √3 x − 1 = √3√−1 .. the sum = (5 + √3) √ − 1 2. Simplify -9+V-16-√9. V-9= 3√ 1 √ 16 = 4√-1 9 = 3√-1 :. √ − 9 + √ − 16 – √ − 9 = 3√ − 1 + 4√ − 1 − 3√ − 1 = 4√ — 1. 3. Add 2-2 − 1, 5 +√ −3, and 3+√16. Simplify each of the following expressions: 4. 2√12+3√ = 27. 5. 7√ − 81 + 5√ − 144. 6. √27+√48+√75. 9. (3+√-4)+(5—2√−9). 10. (3-2-5)-(2-3√-5). MULTIPLICATION OF IMAGINARIES 337. To find the product when imaginary or complex numbers are involved, first reduce them to the form a√-1, or a+b√ −1, and then multiply as in other radicals, but observing that √-1× √−1 = −1. 1. Find the product of V-5 by √-3. 6 +2√2 √−1 - 3 √5 √ − 1 − √10( − 1) 6 + (2 √2 − 3 √5) √ − 1 + √10 3. Raise V-1 to the 2d, 3d, 4th, and 5th powers. Observe that the 5th power is the same as the first, the first four powers being respectively + √ − 1, − 1, − √ − 1, and + 1. It will be found that for higher powers these results recur in the same order. Verify by reference to the figure in Art. 329, It is also to be observed that any even power is real, and any odd power is imaginary. Find the products : 4. 4-27 x 2√3. 5. √−2× √ ÷ 50. 6. V-5x-3√ - 10. 7. 5V 2x3√-2. 8. (−3√-a)(5√ −4). 9. (√-3-2-1). 10. (−√3)o. 15. (2+i√3) (2 – i√3). 16. (2+3)(3+V-2). 17. (3+2-1) (3-2√-1). 18. (V-2+V−3)(√−4−√−5). 19. (2√−4-4√−3)(5√=4+7√ — 3). 20. (√3-2√− 2) (√3 + 2 √ − 2). 21. (a+b√1)(c + d√ − 1). 23. (2√−3+3 √ −6) (3√ −3− 5 √ − 2). DIVISION OF IMAGINARIES 338. The quotients of imaginary or complex numbers may be found by first reducing them to the form a√-1, or a+b√ − 1, as the case may be, and then dividing as in other radicals, observing the principles used in multiplication. In practice it is best to write the dividend and divisor as a fraction, then rationalize the denominator, and simplify the result. 9. 2-6÷3√3. 10. (√12 + √3) ÷ √ − 3. 14. (8 −√ −4) ÷ (4 + 2 √ − 1). 18. (a+b√1) ÷ (a — b√− 1). 19. (2√3-3√ − 2) ÷ (3√3 +2 √ − 2). QUADRATIC EQUATIONS WITH ONE UNKNOWN NUMBER 339. An integral equation in which the highest power of the unknown number is a square is called a Quadratic Equation. It may contain only the square of the unknown number, or it may contain both the second and first powers. = Thus, x2 12, x2+2x= 3, and ax2 + bx c are quadratic equations. The term "quadratic" is derived from the Latin quadratus, meaning squared. Squaring both members of x = 3, we have x2 = 9, or x2 – 9 = 0, a quadratic equation containing only the second power of x. Squaring x − 2 = 3, we have x2 - 4x + 4 = 9, another quadratic containing both the first and second powers of x. 340. Every quadratic equation may be made to assume the form ax2+ bx + c = 0, where a is the unknown number, and a, b, and c are known numbers. The latter are called the coefficients of the equation. The third term c is called the constant term. = 341. When b 0, the second term, ba, is 0, and the equation contains only the second power of a; it is therefore said to be an Incomplete or a Pure Quadratic. When b is not 0, the second term remains, and the equation contains both the first and second powers of a; it is therefore called a Complete or an Affected Quadratic. 342. When we consider an equation like a2 = 25, we find that it can be solved by merely extracting the square root of both members; that is, if = 25, x = ± 5, two roots. |