Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For example, x + x 1 0 has solutons x + 4 4 () ( 1) () or or x B B 4AC A x 4 4 () ( 1) () These expressons smplfy to x 1 or x 1 When we apply the same formula to x + x + 5 0 we arrve at These smplfy to x ± 16 x ± 4 4 (1) (5) (1) ± 16 1 ± 4 1 but we cannot go further because there s no real number whose square s 1, (snce the square of every non-zero real number s postve). The only way forward s to ntroduce more numbers. We take the bold step of ntroducng a new number, to be denoted by, wth the property that 1. Then 1 ± Now we can solve the quadratc equaton x + x + 5 0 to get These may be smplfed as follows: x + 4 + 4 or x ( 1 + ) 4 1 + and 4 ( 1 ) respectvely. It turns out that ntroducng ths new number enables us to solve any gven quadratc equaton. Examples (a) The quadratc equaton x 4x + 1 0 has solutons x 4 + 16 4(1)(1) These may be smplfed to + or respectvely. (b) The quadratc equaton x x + 7 0 has solutons x + 9 56 4 ± 6 ± 1 Snce s not a perfect square, we leave the expressons + Exercse 1 Solve the followng quadratc equatons: 1. x 4x + 8 0 and as they are. 1
. x + x + 10 0. x + 7x + 6 0 4. x x + 0 Expressons lke +, 1, are called complex numbers. (The term "complex numbers" should not ntmdate you. There s nothng complex about them.) They all have the form a + b, called the standard form, where a and b are real numbers and s the new number we have ntroduced wth the property that 1. Some Standard Terms Gven a complex number a + b, the number a that s not multpled by s called the real part of a + b. The number b that s multpled by s called the complex part of a + b. Thus the real part of 1 s 1 and ts complex part s. In the case of, the real part s and the complex part s. A number lke whch has no real part s called a purely complex number. Lkewse, a number lke 6 whch has no complex part s called a purely real number. Manpulatng complex numbers Addng/subtractng complex numbers To add complex numbers, smply add the real parts then and do the same thng to the complex parts. Thus ( + ) + (5 6) ( + ) + ( 6) 5 4 and ( 5 + ) + ( 1 9) + 6 6 To subtract (c + d) from a + b, you do what you would expect: Subtract c from a, to get the real part of the dfference, then subtract d from b to get the complex part of the dfference. In other words (a + b) (c + d) (a c) + (b d). Thus ( ) (4 5) + and 5 ( + ) (1 4) 4 + 7 Exercse 1. Carry out the followng operatons: () 7 + (4 5) () (4 ) + ( + 1 ) () 7 + (6 5) + ( 1 + 4). Fnd the sum of the two solutons of the quadratc equaton x + 4x + 8 0. Carry out the followng operatons: () 5 6 (4 ) () ( ) ( + 1 ) () 7 (6 5) ( 1 + 4) 4. Determne the two solutons of the quadratc equaton 4x + 4x + 5 0 then subtract one of them from the other. 5. Show that f you add a complex number to ts complex conjugate then the result must be a purely real number. What s the result of subtractng a complex number from ts complex conjugate? 6. Show that f the solutons of a quadratc equaton Ax + Bx + C 0 are complex then one of them s a complex conjugate of the other.
Multplyng complex numbers To multply a + b and c + d, smply multply as usual then collect lke terms. But along the way use the fact that 1. Examples: ( ) ( + 4) 6 + 1 4 8 6 + 8 8 ( 1) 14 + 8 ( ) (5 ) 10 6 5 + 10 11 + ( 1) 7 11 ( ) ( + ) 9 6 + 6 4 9 4 ( 1) 1 Exercse ( 1 + ) ( ) 1 1 + 4 9 ( 1) 1 + 4 9 1 9 1. Do the multplcaton and wrte the result n the standard form a + b. () ( 4) (5 6) () ( + 5) ( 5) () 5 (1 + ) ( + ). Fnd the product of the two solutons of the quadratc equaton 4x 4x + 5 0. Note that 1,, 4 ( 1) ( 1) 1, 5 4,... Use these observatons to put the followng phrases n ther correct postons below. The phrases are (a) n s dvsble by 4, (b) n leaves remander 1 when t s dvded by 4, (c) n leaves remander when t s dvded by 4, (d) n leaves remander when t s dvded by 4. 1 f n 1 f f f Dvdng by a complex number Before dong the dvson, we have to ntroduce the concept of a complex conjugate. Here are three examples: The complex conjugate of + s the number. The complex conjugate of 7 5 s the number 7 + 5. The complex conjugate of 4 9 s the number 4 + 9. In general, the complex conjugate of a gven complex number s obtaned by smply changng the sgn of the complex part of the gven number. If t s postve, you change t to a negatve. If t s negatve, you change t to a postve. In the table below, we gve the complex conjugate of each gven number then form the product of the two numbers, (.e. the product of the number and ts conjugate). Remember that 1. Number Its conjugate The product of the number and ts conjugate 5 5 + (5 ) (5 + ) 5 + 10 10 + 5 + 1 + 1 1 1 ( 1 + 1 ) ( 1 1 ) ( 1) + 1 1 + ( ) 1 ( 1) + ( ) 1 4 + 4 ( 4 + ) ( 4 ) ( ) 4 8 9 + 8 9 + ( ( ) 4 ( ) + ) a + b a b (a + b) (a b) a ab + ab + b a + b u v u + v (u v) (u + v) u + uv uv + v u + v
The man pont to take away from the above table s that the product of a number and ts complex conjugate s a purely real number. Ths s the property we use to move a complex number from the denomnator to the numerator n a gven quotent. Example 4 To dvde + 4 by 1 and wrte the result n the standard form a + b of a complex number. We start wth the expresson + 4 1 and proceed to remove the complex number from the denomnator. As suggested by the above observatons regardng a complex number and ts conjugate, smply multply the denomnator of the above fracton by the complex conjugate of 1, whch s 1 +. But we must avod changng the gven fracton, therefore we must also multply the numerator by the same complex conjugate. The result s + 4 1 ( + 4) (1 + ) + 6 + 4 + 1 + 10 + 1( 1) 10 + 10 (1 ) (1 + ) 1 + 9 1 9( 1) 10 10( 1 + ) 1 + 10 Example 5 To dvde 7 by. Thus we want to wrte 7 n the standard form a + b of a complex number. We multply the numerator and denomnator of 7 by the conjugate of, (whch s ), then smplfy. The result s Example 6 To wrte 7 + 1 (7 ) ( ) () ( ) 1 6 9 7 n the standard form a + b. One way to do ths s to remove the complex numbers from both numerators. Thus multply the numerator and denomnator of and then multply the numerator and denomnator of Exercse 7 + 1 1. Show that 6 4 1 + 1 5. Dvde + 5 by 1.. Perform the operatons: 1 by (1 + ). The result s ( ) ( + ) ( ) (1 + ) (1 ) (1 + ) + 1 + 4 10 5 1 10 5 + 10 4 5 10 1 + by ( ) (a) + (a 4) (b) (4 ) ( + 4 ) (c) ( 5) + ( + 7) (4 ) (d) (4 ) (e) (6 + 7) (5 4) (f) (1 + ) (1 ) ( + ) (g) 5 10 1 + (h) 6 1 + () 10 (j) 1 1 + (k) + 1 + (l) + 4 + 4
4. You are gven the quadratc equaton x + 6x + 10 0 (a) Determne ts two solutons. (b) Show that when you multply the two solutons the result s 10. 5. You are gven the quadratc equaton x + bx + c where b and c are constants. (a) Determne ts two solutons. (b) Show that when you add the two solutons, the result s b and when you multply the two solutons the result s c. 6. Wrte the followng n standard form a. 1 + 1 b. + c. a + a a a d. b b b + b 7. Show that f the real part of a complex number z s equal to ts complex part then the real part of z s zero. What s the complex part of z equal to? 8. The real part of a complex number z s the negatve of ts complex part. What can you conclude about: (a) the real part of z? (b) the complex part of z? 5