{"id":91,"date":"2014-10-27T21:20:57","date_gmt":"2014-10-27T21:20:57","guid":{"rendered":"https:\/\/blue-mathbelt.marjoriesayer.com\/?page_id=91"},"modified":"2017-07-23T09:08:04","modified_gmt":"2017-07-23T09:08:04","slug":"week-3-complex-numbers-answers","status":"publish","type":"page","link":"https:\/\/blue-mathbelt.marjoriesayer.com\/?page_id=91","title":{"rendered":"Week 3: Complex Numbers – Answers"},"content":{"rendered":"

Week 3: Complex Numbers – Day 5<\/strong><\/p>\n

\"FullSizeRender-2\"<\/a><\/p>\n

Week 3: Complex Numbers – Day 4<\/strong><\/p>\n

1. The non-real number i raised to what power is one? (Of course, when raised to the zeroth power it is one, but there are other powers as well.)<\/p>\n

i2<\/sup> = -1
\ni3<\/sup> = -1 x i = -i
\ni4<\/sup> = -i x i = -(-1) = 1<\/p>\n

The fourth power of i is one. i is a fourth root of one.<\/p>\n

2. Compute the square of:2-1\/2<\/sup>(1 + i)<\/p>\n

The square is: (2-1\/2<\/sup>)2<\/sup>(1 + i)2<\/sup><\/p>\n

= (1\/2) x (1 + i) x (1 + i) = 1\/2 x (1 + 2i – 1) = 1\/2 x 2i = i<\/p>\n

The square is i.<\/p>\n

3. Compute the cube of: 2-1\/2<\/sup>(1 + i)<\/p>\n

The cube is: \u00a0i x 2-1\/2<\/sup>(1 + i)
\nwhich is: \u00a02-1\/2<\/sup>(i – 1)
\nwhich is: \u00a02-1\/2<\/sup>(-1 + i)<\/p>\n

4. What power of 2-1\/2<\/sup>(1 + i) is one? In other words, 2-1\/2<\/sup>(1 + i) raised to what power is one?<\/p>\n

Because the square is i, and i4<\/sup> = 1, then:<\/p>\n

((2-1\/2<\/sup>(1 + i))2<\/sup>)4<\/sup> = 1<\/p>\n

2-1\/2<\/sup>(1 + i) raised to the 8th power is 1.<\/p>\n

5. Compute the square of: 2-1\/2<\/sup>(1 – i)<\/p>\n

The square is 1\/2 x (1 – i) x (1 – i) = 1\/2 x (1 – 2i +(-1)) = 1\/2 x (-2i)
\n= -i.<\/p>\n

\n

 <\/p>\n

Week 3: Complex Numbers – Day 3<\/strong><\/p>\n

1. 1 \/ i<\/p>\n

complex conjugate of i is -i.<\/p>\n

= 1*(-i) \/ i*(-i) = -i \/1 = -i<\/p>\n

(1 divided by i).<\/p>\n

2. (2 + 3i)\/i<\/p>\n

=(2 + 3i)*(-i) \/ i*(-i) = 3 – 2i<\/p>\n

3. (1 + i) \/ (1 – i)<\/p>\n

= (1 + i)*(1 + i) \/ (1 – i)*(1 + i) = (1 +2i – 1) \/ (1 – i + i + 1) = 2i\/2 = i<\/p>\n

4. i \/ (1 – i)<\/p>\n

= i*(1 + i) \/ (1 – i)*(1 + i) = (i – 1) \/ (1 – i + i + 1) = -1\/2 + i\/2<\/p>\n

5. (3 – 2i) \/ (3 + 2i)<\/p>\n

= (3 – 2i)*(3 – 2i) \/ (3 + 2i)*(3 – 2i) = (9 – 6i – 4) \/ (3 + 6i – 6i + 4) = (5 – 6i) \/ 7 = 5\/7 – 6i\/7<\/p>\n

\n

 <\/p>\n

Week 3: Complex Numbers – Day 2<\/strong><\/p>\n

Complex\u00a0Solutions<\/p>\n

1. What are the solutions of x2<\/sup> + 9 = 0?<\/p>\n

x2<\/sup> = -9
\nTake square roots of both sides:
\nx = 3i or -3i<\/p>\n

2. What are the solutions of (x + 1)2<\/sup> + 4 = 0?<\/p>\n

(x + 1)2<\/sup> = -4
\nTake square roots of both sides:
\nx + 1 = 2i or -2i
\nSolutions are -1 + 2i or -1 – 2i<\/p>\n

3. What are the solutions of x2<\/sup> – 2x + 10 = 0?<\/p>\n

Completing the square: x2<\/sup> – 2x + 1 – 1 + 10 = 0<\/p>\n

(x – 1)2<\/sup> + 9 = 0
\n(x – 1)2<\/sup> = -9
\nTake square roots of both sides:
\nx – 1 = 3i or -3i<\/p>\n

Solutions: 1 + 3i or 1 – 3i<\/p>\n

4. What are the solutions of x3<\/sup> + 25x = 0?<\/p>\n

Factoring: x(x2<\/sup> + 25) = 0<\/p>\n

Solutions are x = 0 and x2<\/sup> + 25 = 0.
\nSolving the second equation:<\/p>\n

x2<\/sup> = -25
\nTake square roots of both sides:
\nx = 5i or -5i
\nThree solutions are: 0, 5i, and -5i<\/p>\n

5. What are the solutions of (x – 1)3<\/sup> + 36(x – 1) = 0?<\/p>\n

Factoring: (x – 1)((x-1)2<\/sup> + 36) = 0<\/p>\n

Solutions: x – 1 = 0 (x = 1), and solutions of (x-1)2<\/sup> + 36 = 0<\/p>\n

Solving the second equation:
\n(x – 1)2<\/sup> = -36
\nTake square roots of both sides:
\n(x – 1) = 6i or -6i<\/p>\n

x = 1 + 6i or 1 – 6i<\/p>\n

\n

 <\/p>\n

Week 3: Complex Numbers – Day 1<\/strong><\/p>\n

Simplify these expressions. The final result should be a number of the form A + Bi.<\/p>\n

1. (-1 + i) + (3 + 2i)<\/p>\n

= -1 + 3 + i + 2i = 2 + 3i<\/p>\n

2. (7 + 2i) + 10<\/p>\n

= 17 + 2i<\/p>\n

3. (2 + 2i) x 3i<\/p>\n

= 6i + 6x(-1) = -6 + 6i<\/p>\n

4. (5 + 3i) x (5 – 3i)<\/p>\n

= 25 + 15i – 15i + 9 = 34 = 34 + 0i<\/p>\n

5. (4 + i) x (1 + 6i)<\/p>\n

= 4 + 24i + i – 6 = -2 + 25i<\/p>\n","protected":false},"excerpt":{"rendered":"

Week 3: Complex Numbers – Day 5 Week 3: Complex Numbers – Day 4 1. The non-real number i raised to what power is one? (Of course, when raised to the zeroth power it is one, but there are other powers as well.) i2 = -1 i3 = -1 x i = -i i4 = […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":10,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-91","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/91","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=91"}],"version-history":[{"count":8,"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/91\/revisions"}],"predecessor-version":[{"id":124,"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/91\/revisions\/124"}],"up":[{"embeddable":true,"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/10"}],"wp:attachment":[{"href":"https:\/\/blue-mathbelt.marjoriesayer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=91"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}