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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.83,0:00:06.24,Default,,0000,0000,0000,,In this unit, went to look at\Nhow to multiply 2 complex
Dialogue: 0,0:00:06.24,0:00:09.24,Default,,0000,0000,0000,,numbers together. And\Nmultiplying two complex numbers
Dialogue: 0,0:00:09.24,0:00:13.34,Default,,0000,0000,0000,,simply requires us to be able to\Nmultiply out brackets to collect
Dialogue: 0,0:00:13.34,0:00:17.11,Default,,0000,0000,0000,,together like turns, and to\Nremember that I is the special
Dialogue: 0,0:00:17.11,0:00:21.21,Default,,0000,0000,0000,,number whose property is that I\Nsquared is equal to minus one.
Dialogue: 0,0:00:21.86,0:00:24.28,Default,,0000,0000,0000,,So let's look at how that\Nall works in an example.
Dialogue: 0,0:00:25.83,0:00:29.25,Default,,0000,0000,0000,,So here we have two complex\Nnumbers 4 + 7 I.
Dialogue: 0,0:00:29.76,0:00:34.11,Default,,0000,0000,0000,,And 2 + 3 I and we're going to\Ndo is going to multiply these
Dialogue: 0,0:00:34.11,0:00:35.27,Default,,0000,0000,0000,,two complex numbers together.
Dialogue: 0,0:00:36.08,0:00:40.70,Default,,0000,0000,0000,,And the first thing we do is we\Njust multiply out the brackets
Dialogue: 0,0:00:40.70,0:00:44.96,Default,,0000,0000,0000,,so we each term in the first\Nbracket must multiply each term
Dialogue: 0,0:00:44.96,0:00:46.38,Default,,0000,0000,0000,,in the SEC bracket.
Dialogue: 0,0:00:47.06,0:00:50.99,Default,,0000,0000,0000,,So we have 4 * 2 which is 8.
Dialogue: 0,0:00:51.77,0:00:57.11,Default,,0000,0000,0000,,Four times plus three I, which\Nis plus 12 I.
Dialogue: 0,0:00:58.10,0:01:02.04,Default,,0000,0000,0000,,Plus Seven I times two is
Dialogue: 0,0:01:02.04,0:01:09.61,Default,,0000,0000,0000,,plus 49. And plus Seven\NI times plus three. I is
Dialogue: 0,0:01:09.61,0:01:11.41,Default,,0000,0000,0000,,21. I squared.
Dialogue: 0,0:01:13.40,0:01:17.27,Default,,0000,0000,0000,,Now we see straight away that we\Nwould be able to combine these
Dialogue: 0,0:01:17.27,0:01:20.25,Default,,0000,0000,0000,,two terms because they're both\Nterms with eyes in them.
Dialogue: 0,0:01:20.94,0:01:28.35,Default,,0000,0000,0000,,So the the front is going to\Nstay the same plus 12 I plus
Dialogue: 0,0:01:28.35,0:01:32.05,Default,,0000,0000,0000,,14. I gives us plus 26 I.
Dialogue: 0,0:01:33.38,0:01:38.77,Default,,0000,0000,0000,,Now over this term, on the end,\Nwhich is 21, I squared. Now we
Dialogue: 0,0:01:38.77,0:01:44.16,Default,,0000,0000,0000,,have to remember what we know\Nabout I. I is the square root of
Dialogue: 0,0:01:44.16,0:01:49.16,Default,,0000,0000,0000,,minus one or said another way I\Nsquared is equal to minus one.
Dialogue: 0,0:01:49.16,0:01:54.17,Default,,0000,0000,0000,,So in this term 21 I squared we\Ncan replace the isquared by
Dialogue: 0,0:01:54.17,0:01:57.64,Default,,0000,0000,0000,,minus one. So that's plus 21\Ntimes minus one.
Dialogue: 0,0:01:58.82,0:02:05.80,Default,,0000,0000,0000,,Now 21 times minus one is minus\N21, so we have 8 - 21. You can
Dialogue: 0,0:02:05.80,0:02:07.54,Default,,0000,0000,0000,,combine those two terms.
Dialogue: 0,0:02:08.07,0:02:11.49,Default,,0000,0000,0000,,8 - 21 is
Dialogue: 0,0:02:11.49,0:02:19.31,Default,,0000,0000,0000,,minus 13. Plus the\N26 I was already there.
Dialogue: 0,0:02:19.31,0:02:23.23,Default,,0000,0000,0000,,And so our answer, when we\Nmultiply these two complex
Dialogue: 0,0:02:23.23,0:02:27.93,Default,,0000,0000,0000,,numbers together, is this new\Ncomplex number minus 13 + 26? I
Dialogue: 0,0:02:27.93,0:02:31.85,Default,,0000,0000,0000,,OK? We're going to look at\Nanother example, two different
Dialogue: 0,0:02:31.85,0:02:36.56,Default,,0000,0000,0000,,complex numbers. This time the\Ncomplex numbers minus 2 + 5. I
Dialogue: 0,0:02:36.56,0:02:42.05,Default,,0000,0000,0000,,first one on 1 - 3 I is the\Nsecond one, exactly the same
Dialogue: 0,0:02:42.05,0:02:46.75,Default,,0000,0000,0000,,principles before, but we have\Nto be a bit more careful 'cause
Dialogue: 0,0:02:46.75,0:02:49.10,Default,,0000,0000,0000,,we got lots of minus signs
Dialogue: 0,0:02:49.10,0:02:54.19,Default,,0000,0000,0000,,floating about. So we multiply\Nout the brackets minus 2 * 1.
Dialogue: 0,0:02:54.77,0:02:56.02,Default,,0000,0000,0000,,Is minus 2.
Dialogue: 0,0:02:56.78,0:03:02.36,Default,,0000,0000,0000,,Minus two times minus three I\Ngives us plus 6I.
Dialogue: 0,0:03:03.03,0:03:06.61,Default,,0000,0000,0000,,Plus 5I Times one gives us
Dialogue: 0,0:03:06.61,0:03:13.99,Default,,0000,0000,0000,,plus 5I. And plus 5I\NTimes minus three I years, minus
Dialogue: 0,0:03:13.99,0:03:15.73,Default,,0000,0000,0000,,15 I squared.
Dialogue: 0,0:03:17.60,0:03:23.83,Default,,0000,0000,0000,,Now we combine together our\Nitems so we have minus 2 + 6
Dialogue: 0,0:03:23.83,0:03:27.66,Default,,0000,0000,0000,,I plus 5I, giving us plus 11 I.
Dialogue: 0,0:03:28.90,0:03:33.87,Default,,0000,0000,0000,,And in this term, remember that\NI squared is minus one, so this
Dialogue: 0,0:03:33.87,0:03:37.69,Default,,0000,0000,0000,,is minus 15 times minus one,\Nwhich is plus 15.
Dialogue: 0,0:03:38.54,0:03:45.06,Default,,0000,0000,0000,,And so the final thing we do is\Ncombine the minus two and the
Dialogue: 0,0:03:45.06,0:03:50.19,Default,,0000,0000,0000,,plus 15 to get plus 13 and then\Nplus 11 I.
Dialogue: 0,0:03:50.19,0:03:54.34,Default,,0000,0000,0000,,And so our answer, we multiply\Nthese two complex numbers
Dialogue: 0,0:03:54.34,0:03:58.08,Default,,0000,0000,0000,,together is the complex number\N13 + 11 I.
Dialogue: 0,0:03:59.93,0:04:02.81,Default,,0000,0000,0000,,So that's how we multiply\Ntogether to complex numbers
Dialogue: 0,0:04:02.81,0:04:06.65,Default,,0000,0000,0000,,in the next unit. We're going\Nto look at a property that
Dialogue: 0,0:04:06.65,0:04:08.89,Default,,0000,0000,0000,,complex numbers have called\Nthe complex conjugate.