Math L3 Module 1

Mathematics NQF Level 3 *see terms and conditions Downloaded from hlayiso.com Mathematics NQF Level 3

Complex numbers Topic 1 Downloaded from hlayiso.com Mathematics NQF Level 3

Represent complex numbers in a form appropriate to the context Module 1 Downloaded from hlayiso.com Mathematics NQF Level 3

Writing imaginary numbers in i-notation Figure 1.4: A graphical representation of the number system with complex number Downloaded from hlayiso.com Unit 1.1

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Learning activity 1.1 Downloaded from hlayiso.com Mathematics NQF Level 3

8 4 20 2. Simplify 60 j + 10 j − 20 j and write in the rectangular form a ± bi: Remember, i and j are both used to 60 j 8 + 10 j 4 − 20 j 20 represent imaginary numbers = 60( j 4 ) 2 + 10( j 4 ) − 20( j 4 )5 2 2 2 2 2 2 2 5 = 60( j j ) + 10( j j ) − 20( j j ) 2 5 = 60[(−1)(−1)] + 10[(−1)(−1)] − 20[(−1)(−1)] Substitute j 2 = −1 Downloaded from hlayiso.com Learning activity 1.1

2. Continued = 60[(−1)(−1)]2 + 10[(−1)(−1)] − 20[(−1)(−1)]5 = 60(1) 2 + 10(1) − 20(1)5 = 60 + 10 − 20 = 50 Downloaded from hlayiso.com Learning activity 1.1

4. Solve for x and write the answers in i- notation: 1) −3 − 81 −1 Subsitute i 2 = −1 = 3 3 (−1 − 3i ) = 3 =−1 − 3i Downloaded from hlayiso.com Learning activity 1.1

Learning activity 1.2 Downloaded from hlayiso.com Mathematics NQF Level 3

Simplify: 6) 3 2i × i 7 13) = 2i10 = 2(i 2 )5 2 5 = = 2(−1) −1 = −2 Subsitute i 2 = −1 Subsitute i 2 = −1 Downloaded from hlayiso.com Learning activity 1.2

Learning activity 1.3 Downloaded from hlayiso.com Mathematics NQF Level 3

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The relationship between the rectangular and polar forms Figure 1.11: The relationship between the rectangular and polar forms Downloaded from hlayiso.com Unit 1.4

Learning activity 1.4 Downloaded from hlayiso.com Mathematics NQF Level 3

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1. Write the following complex number in polar form: 1.3) Modulus = r a 2 + b2 Argand diagram Figure 1.9 Downloaded from hlayiso.com Learning activity 1.4

1. Write the following complex number in polar form: a and b without signs 1.3) 2 − 2i b Acute angle α = tan −1 a Figure 1.9 Downloaded from hlayiso.com Learning activity 1.4

1. Write the following complex number in polar form: 4th 1.3) quadrant Figure 1.9 Downloaded from hlayiso.com Learning activity 1.4

1. Write the following complex number in polar form: 1.3) Polar form: z = r cis θ or r θ Figure 1.9 Downloaded from hlayiso.com Learning activity 1.4

2. Write the following complex number in rectangular form (a + bi): 2.5) 5 [cos 240° + i sin 240°] = 5 [−0,5 + i (−0,866)] Rectangular form: z = r cis θ z= a + bi =z r (cos θ + i sin θ ) Downloaded from hlayiso.com Learning activity 1.4

Summative assessment Module 1 Downloaded from hlayiso.com Mathematics NQF Level 3

1. Simplify the following to the a + bi form 17 16 1.1) i 1.2) i 2 8 2 8 = (i ) i = (i ) 8 = (−1) i 8 = (−1) =i =1 Subsitute i 2 = −1 Subsitute i 2 = −1 Downloaded from hlayiso.com Summative assessment – Module 1

1. Simplify the following to the a + bi form ( 3i ) × i 3 6 1.3) 1.4) i 2 + 3i 3 – 2i 4 = 27i × i 3 6 =−1 + 3(i 2 )i − 2(i 2 ) 2 9 = 27i =−1 + 3(−1)i − 2(−1) 2 2 4 = 27(i ) i =−1 − 3i − 2 = 27(−1) 4 i =−3 − 3i = 27i Downloaded from hlayiso.com Summative assessment – Module 1

1. Simplify the following to the a + bi form 1.5) 3i 4 1.6) i ÷ 2i 6 3 9i 1 = ( i ) 2 2 = 2 2 (i )2 3 3i 1 1 = = 3(−1) 2(−1)3 1 1 = − = − 3 2 Downloaded from hlayiso.com Summative assessment – Module 1

2. Simplify without using a calculator and write in standard form: 2.1) = 2i (2 − i + 3i ) = 2 2i − 2i 2 + 6i 2 = 2,828i − 2 (−1) + 6 (−1) = 2,828i + 1, 414 − 2, 449 = −1, 035 + 2,828i Downloaded from hlayiso.com Summative assessment – Module 1

2. Simplify without using a calculator and write in standard form: 2.2) 6−4 j = 2 10 j + 6 j − 15 − 9 j 6−4 j = j + 6(−1) − 15 6 − 4 j −21 − j = × −21 + j −21 − j Downloaded from hlayiso.com Summative assessment – Module 1

2. Simplify without using a calculator and write in standard form: 2.2) Continued −126 − 6 j + 84 j + 4 j 2 = 441 + 21 j − 21 j − j 2 −126 + 78 j + 4(−1) = 441 − (−1) −130 + 78 j = 442 = −0, 294 + 0,176 j Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: Rectangular form 3.1) −2 + 4i z= a + bi z =−2 + 4i 2 2 Modulus = r a +b ∴ a =−2 ; b =4 ( ) 2 2 = −2 + 4 Argand diagram = 20 = 4, 472 Figure 1.13 Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: a and b 3.1) −2 + 4i without signs b Acute angle α = tan −1 a −1 4 = tan 2 = 63, 435° Figure 1.13 Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: 2nd 3.1) −2 + 4i quadrant = 180° − 63, 435° = 116,565° Figure 1.13 Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: 3.1) −2 + 4i Polar form: z = r cis θ or r θ =∴ z 4, 472 cis 116,565° = 4, 472 116,565° Figure 1.13 Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: 3.2) 4 − 7i =r a 2 + b2 4 + ( −7 ) 2 2 = = 8, 062 b α = tan −1 a 7 −1 = tan 4 Figure 1.14 = 60, 255° Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: 3.2) 4 − 7i = 360° − α θ = 360° − 60, 255° = 299, 745° z = r cis θ =∴ z 8, 062 cis 299,745° = 8, 062 299, 745° Figure 1.14 Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: 3.3) −1 − 3i =r a 2 + b2 ( −3) + ( −1) 2 2 = = 10 = 3,162 b α = tan −1 a −1 3 = tan 1 Figure 1.15 = 71,57° Downloaded from hlayiso.com Summative assessment – Module 1

3. Write the following complex numbers in polar form: 3.3) −1 − 3i = 180° + α θ = 180° + 71,57° = 251,57° z = r cis θ =∴ z 3,162 cis 251,57° = 3,162 251,57° Figure 1.15 Downloaded from hlayiso.com Summative assessment – Module 1

4. Write the following complex numbers in rectangular form: 4.1) 2 cis 100° 2(cos 100° + i sin 100°) 2(− = 0.173648.... + i 0.9848....) = −0,347 + 1,97i Rectangular form: z = r cis θ z= a + bi =z r (cos θ + i sin θ ) Downloaded from hlayiso.com Summative assessment – Module 1

4. Write the following complex numbers in rectangular form: 4.2) 4 cis 300° 4(cos 300° + i sin 300°) 4(cos 300° + i sin 300°) = 2 − 3, 464i Downloaded from hlayiso.com Summative assessment – Module 1

4. Write the following complex numbers in rectangular form: 4.3) 12 30° 12[cos 30° + i sin 30°] 12(0.866 + 0.5i ) = 10,392 + 6i Downloaded from hlayiso.com Summative assessment – Module 1

4. Write the following complex numbers in rectangular form: 4.4) 12 40° 12(cos 40° + i sin 40°) 12(0, 766 + i.0, 643) = 9,192 + 7, 716i Downloaded from hlayiso.com Summative assessment – Module 1

5. Represent z = −2 + 4i on an Argand diagram Figure 1.16: Argand diagram of z = −2+ 4i Downloaded from hlayiso.com Summative assessment – Module 1

6. Given z = 2 + 3i: 6.1) Represent z graphically in an Argand diagram. Figure 1.17: Argand diagram of z = 2 + 3i Downloaded from hlayiso.com Summative assessment – Module 1

6. Given z = 2 + 3i: 6.2) Calculate the modulus (r) and the argument (θ) of z The modulus ( r ) : The argument (θ ) : 2 r= 22 + 32 Downloaded from hlayiso.com Summative assessment – Module 1

6. Given z = 2 + 3i: 6.3) Express z in polar form Polar form Downloaded from hlayiso.com Summative assessment – Module 1

7. Study the argand diagram and answer the questions that follow: 7.1) The rectangular form of the complex number z z= a + bi z=−2 + 4i Figure 1.19: Argand diagram for Question 7 Downloaded from hlayiso.com Summative assessment – Module 1

7. Given the diagram below determine: 7.2) The modulus (r) =r a 2 + b2 ( ) 2 2 r= −2 + 4 = 20 = 4, 472 Figure 1.19: Argand diagram for Question 7 Downloaded from hlayiso.com Summative assessment – Module 1

7. Given the diagram below determine: 7.3) The value of the argument (θ) 4 tan α = 2 4 ∴α =tan −1 2 = 63, 43° θ = 180° − 63, 43° = Figure 1.19: Argand diagram for Question 7 116,57° Downloaded from hlayiso.com Summative assessment – Module 1

7. Given the diagram below determine: 7.4) Write the answer in polar form z=r θ =z 20 116,57° = 4, 472 116,57° Figure 1.19: Argand diagram for Question 7 Downloaded from hlayiso.com Summative assessment – Module 1

Mathematics NQF Level 2 Downloaded from hlayiso.com Mathematics NQF Level 3