Maths Level 4 Module 1

Mathematics NQF Level 4 *see terms and conditions Downloaded from hlayiso.com Mathematics NQF Level 4

Complex numbers Topic 1 Downloaded from hlayiso.com Mathematics NQF Level 4

Work with complex numbers Module 1 Downloaded from hlayiso.com Mathematics NQF Level 4

Powers of i Figure 1.1: The powers of i form a cycle that repeats as the powers increase Downloaded from hlayiso.com Introduction

Learning activity 1.1 Downloaded from hlayiso.com Mathematics NQF Level 4

Simplify: NOTE: The rules of exponents still apply when we work with imaginary numbers −31 46 12. 5i + 4i 5 = 31 + 4 ( i ) 2 23 (a m ) n = a mn  i 5 = 30 + 4 ( −1) 23 i 2 = −1 i ⋅i 5 = + 4 ( −1) (i ) ⋅ i 2 15 5 −4 i 2 = −1 ( −1) ⋅ i 15 Downloaded from hlayiso.com Learning activity 1.1

Simplify: −31 46 12. 5i + 4i Continued  5 i  i  =  × −4  Multiply by i = 1  −1i i   5i   −1i 2  − 4  5i  = −4 i 2 = −1  −1( −1)  = 5i − 4 =−4 + 5i Downloaded from hlayiso.com Learning activity 1.1

Learning activity 1.2 Downloaded from hlayiso.com Mathematics NQF Level 4

Simplify and write your answers in standard form. 12. ( − −25 + 3) + ( 3 + 7i ) − ( 2 −1 + 2i ) 25 30 (a m ) n = a mn  ( ) ( )( =− 25i + 3 + 3 + 7 ( i ) i − 2i + 2 ( i ) 2 12 2 15 ) i=  −1  = ( ) ( ( −5i + 3) + 3 + 7 ( −1) i − 2i + 2 ( −1) 12 15 ) i 2 = −1 =( −5i + 3) + ( 3 + 7 (1) i ) − ( 2i + 2 ( −1) ) ( 5i + 3) + ( 3 + 7i ) − ( 2i − 2 ) =− Downloaded from hlayiso.com Learning activity 1.2

Simplify and write your answers in standard form. 12. ( − −25 + 3) + ( 3 + 7i 25 ) − ( 2 −1 + 2i 30 ) Continued =−5i + 3 + 3 + 7i − 2i + 2 = 8 + 0i [Standard form: a + bi ] NOTE: Rectangular form is also called standard form Downloaded from hlayiso.com Learning activity 1.2

Learning activity 1.3 Downloaded from hlayiso.com Mathematics NQF Level 4

Simplify and write the answers in standard form. 3. (1 − i )(1 + i ) We are multiplying a complex number 2 = 1+ i − i − i by its conjugate = 1 − (−1) =2 [ Always results in a REAL number ] = 2 + 0i [ In standard form] Downloaded from hlayiso.com Learning activity 1.3

Learning activity 1.6 Downloaded from hlayiso.com Mathematics NQF Level 4

7. (4 cis 120°) × ( 2 cis 30°) 2 2 4 VIDEO (3 cis 20°) Click above to play this video. Downloaded from hlayiso.com Learning activity 1.6

Learning activity 1.10 Downloaded from hlayiso.com Mathematics NQF Level 4

Click above to play this video. Downloaded from hlayiso.com Learning activity 1.10

Summative assessment Module 1 Downloaded from hlayiso.com Mathematics NQF Level 4

1. Simplify 2i4 − 3i3 + 2i2 2i 4 − 3i 3 + 2i 2 2 2 2 = 2(i ) − 3(i )i + 2(−1) (a m ) n = a mn  2 = 2(−1) − 3(−1)i + 2(−1) i 2 = −1 = 2 + 3i − 2 = 3i Downloaded from hlayiso.com Summative assessment – Module 1

2. Simplify and write the answer in rectangular form: 2.1) (2 + 3i) − (5 − 9i) = 2 + 3i − 5 + 9i =−3 + 12i [Standard form: a + bi ] 2.2) (2 – 3i )(1 – 2i) = 2 − 4i − 3i + 6i 2 =−4 − 7i Downloaded from hlayiso.com Summative assessment – Module 1

2-i 2.3) 4+i  4−i  2−i 4−i  Multiply by  = ×  4−i  4+i 4−i (conjugate of denominator)  2 8 − 2i − 4i + i = i 2 = −1 16 − 4i + 4i − i 2 7 − 6i = 17 7 6 = − i 17 17Downloaded from hlayiso.com Summative assessment – Module 1

2.4) ( -1 + 2i )( 7 - 5i ) + ( -3 + 4i ) 2 =−7 + 5i + 14i − 10i − 3 + 4i [ FOIL] 2 =−10 + 23i − 10i =−10 + 23i − 10(−1) i 2 = −1 = 23i Downloaded from hlayiso.com Summative assessment – Module 1

3. Simplify and write the answer in polar form: 3.1) ( 2 cis 80°)4 Abbreviated polar form: 2 2 cis (4 × 80°) z = r cisθ 4 cis 320°  De Moivre's Theorem: (r cis θ ) n = r n cis nθ  Downloaded from hlayiso.com Summative assessment – Module 1

3.2) 4 90° × 2 60° 3 50° 4⋅2 r1.r2 90° + 60° − 50° [ θ1 + θ 2 − θ3 ] 3 r3 8 100° 3 Downloaded from hlayiso.com Summative assessment – Module 1

3.3) (2 cis 210°)(4 cis 90°) = 2 × 4 cis (210° + 90°) [r1.r2 cis (θ1 + θ 2 )] 8 cis 300° Downloaded from hlayiso.com Summative assessment – Module 1

4. Simplify 5 150° + 4 80° and write the answer in standard form 5 150° + 4 80° 5cos150° + 5i sin150° + 4 cos 80° + 4i sin 80° = −4,330 + 2,5i + 0, 695 + 3,939i = −3, 635 + 6, 439i z = θ  = r cos θ + ( r sin θ )i    Downloaded from hlayiso.com Summative assessment – Module 1

5. Simplify ( 3 - 4i ) (1 − i ) and write the 5 answer in standard form Convert 3 − 4i to polar form: z r (mod) = = 32 + 42 = 5 4 tan α = 3 α = tan −1 1,333 ∴θ = 360° − 53,123° = 306,877° Figure 1.23: Question 5 Downloaded from hlayiso.com Summative assessment – Module 1

5. Continued ∴ (3 − 4 = i ) 5 cis 306,877° 5 (3 − 4i ) (1 − i ) ( 5 cis 306, 877° ) (1 − i) 5 3 125 cis 5 ( 306, 877° )  (1 − i ) [ De Moivre] ( 3 125 cis 1 534,385° ) (1 − i) ( 3 125 cos1 534,385° + 3 125i sin 1 534,385° ) (1 − i) [ r cis = θ r (cos θ + i sin θ ] Downloaded from hlayiso.com Summative assessment – Module 1

5. Continued (− = 238,93 + 3 115,85i )(1 − i ) = −238,93 + 238,93i + 3 115,85i − 3115,85i 2 2 876,92 + 3 354, 78i Downloaded from hlayiso.com Summative assessment – Module 1

Mathematics NQF Level 2 Downloaded from hlayiso.com Mathematics NQF Level 4