TopicName Test



WorkSHEET 2.2 Number systems: complex numbers

Name: ___________________________

| |Write [pic]. | |4 |

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| |Express [pic] in General / Cartesian form. | |3 |

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| |Find the modulus and Argument of z = (7 + 7i. | |3 |

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| |(a) Sketch z = 5 – 5i. | |3 |

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| |(b) Express z = 5 – 5i in polar form. | | |

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| |Sketch the following complex numbers and express them in polar form: | |6 |

| |z = 2 + 3i | | |

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| |w = –4 – 2i | | |

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| |Convert z = 4 cis[pic] and w = 6 cis[pic] into General form. | |5 |

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| |Perform the following operations using | |7 |

| |z = 4 cis[pic] and w = 6 cis[pic]. | | |

| |Convert into General form: | | |

| |z + w | | |

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| |w – z | | |

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| |z × w | | |

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| |[pic] | | |

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| |[pic] | | |

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| |Sketch the following complex numbers and express them in General form.| |9 |

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| |Use common trigonometric ratios and special triangles. | | |

| |z = 2 cis[pic] | | |

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| |z = 4 cis[pic] | | |

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| |z = 2 cis[pic] | | |

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| |Express [pic] in General and Polar form. | |7 |

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SOLUTIONS

Name: ___________________________

| |Write [pic]. |[pic] |4 |

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| |Express [pic] in General / Cartesian form. |[pic] |3 |

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| |Find the modulus and Argument of z = (7 + 7i. |[pic] |3 |

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| |(a) Sketch z = 5 – 5i. |(a) |3 |

| | |[pic] | |

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| | |(b) | |

| | |[pic] | |

| | |z = 5[pic]cis(([pic]) | |

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| |(b) Express z = 5 – 5i in polar form. | | |

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| |Sketch the following complex numbers and express them in polar form: | |6 |

| |z = 2 + 3i | | |

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| | |[pic] | |

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| | |[pic] | |

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| |w = –4 – 2i | | |

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| |Convert z = 4 cis[pic] and w = 6 cis[pic] into General form. | |5 |

| | |[pic] | |

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| | |z = 4 cis[pic] | |

| | |z = 4(cos[pic] + isin[pic]) | |

| | |z = 4([pic]+ i[pic]) | |

| | |z = 2 + 2[pic]i | |

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| | |w = 6 cis[pic] | |

| | |w = 6(cos[pic] + isin[pic]) | |

| | |w = 6(([pic] + i[pic]) | |

| | |w = (3 + 3[pic]i | |

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| |Perform the following operations using | |7 |

| |z = 4 cis[pic] and w = 6 cis[pic]. | | |

| |Convert into General form: | | |

| |z + w |Using the answers from the previous question: | |

| | |[pic] | |

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| |w – z |[pic] | |

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| | |[pic] | |

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| | |[pic] | |

| |z × w | | |

| | |[pic] | |

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| |[pic] | | |

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| |[pic] | | |

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| |Sketch the following complex numbers and express them in General form.| |9 |

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| |Use common trigonometric ratios and special triangles. | | |

| |z = 2 cis[pic] |(a) z = 2 cis[pic] | |

| | |z = 2(cos[pic] + isin[pic]) | |

| | |z = 2(([pic]+ [pic]i) | |

| | |z = ([pic] + [pic]i | |

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| | |[pic] | |

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| | |(b) z = 4 cis[pic] | |

| | |z = 4(cos[pic] + isin[pic]) | |

| | |z = 4([pic]+[pic]i) | |

| | |z = 2[pic] + 2i | |

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| | |[pic] | |

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| |z = 4 cis[pic] |(c) z = 2 cis[pic] | |

| | |z = 2(cos[pic] + isin[pic]) | |

| | |z = 2(([pic]([pic]i) | |

| | |z = (1 ( [pic]i | |

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| | |[pic] | |

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| |z = 2 cis[pic] | | |

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| |Express [pic] in General and Polar form. |Let [pic] = a + bi |7 |

| | |2 + 5i = a2 + 2abi – b2) | |

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| | |Therefore 2 = a2 – b2 (1) | |

| | |and 5 = 2ab (2) | |

| | |From (2): a = [pic] | |

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| | |Substituting a into (1): | |

| | |2 = [pic] – b2 | |

| | |2 = [pic] – b2 | |

| | |8b2 = 25 – 4b4 | |

| | |0 = 4b4 + 8b2 – 25 | |

| | |b2 = [pic] | |

| | |= [pic] | |

| | |( –3.7 or 1.7 | |

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| | |(Ignore b2 = –3.7 because b must be real) | |

| | |Therefore, b2 = 1.7 and b ( ± 1.3. | |

| | |For b = (1.3, a = (1.9. For b = 1.3, a = 1.9. | |

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| | |Therefore [pic] = ±(1.9 + 1.3i) | |

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| | |In polar form: (always sketch the number first) | |

| | ||[pic]| = [pic] | |

| | |= 2.3 | |

| | |[pic] = tan–1 [pic] | |

| | |= 0.6 | |

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| | |Argument = 0.6 or (2.5 | |

| | |[pic] = 2.3 cis 0.6 or 2.3 cis (–2.5) | |

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| | |[pic] | |

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