1
00:00:00,906 --> 00:00:04,667
In this video, I want to introduce you to the number i,

2
00:00:04,667 --> 00:00:10,333
which is sometimes called the imaginary, imaginary unit

3
00:00:10,333 --> 00:00:12,562
What you're gonna see here, and it might be a little bit difficult,

4
00:00:12,562 --> 00:00:17,333
to fully appreciate, is that its a more bizzare number than

5
00:00:17,333 --> 00:00:20,178
some of the other wacky numbers we learn in mathematics,

6
00:00:20,178 --> 00:00:25,983
like pi, or e. And its more bizzare because it doesnt have a tangible value

7
00:00:25,983 --> 00:00:29,002
in the sense that we normally, or are used to defining numbers.

8
00:00:29,002 --> 00:00:35,667
"i" is defined as the number whose square is equal to negative 1.

9
00:00:35,667 --> 00:00:44,002
This is the definition of "i", and it leads to all sorts of interesting things.

10
00:00:44,002 --> 00:00:46,324
Now some places you will see "i" defined this way;

11
00:00:46,324 --> 00:00:50,968
"i" as being equal to the principle square root of negative one.

12
00:00:50,968 --> 00:00:55,240
I want to just point out to you that this is not wrong, it might make sense to you,

13
00:00:55,240 --> 00:00:57,800
you know something squared is negative one, then maybe

14
00:00:57,800 --> 00:01:00,674
its the principle square root of negative one.

15
00:01:00,674 --> 00:01:02,671
And so these seem to be almost the same statement,

16
00:01:02,671 --> 00:01:04,900
but I just want to make you a little bit careful, when you do this

17
00:01:04,900 --> 00:01:06,733
some people will even go so far as to say this is wrong,

18
00:01:06,733 --> 00:01:09,467
and it actually turns out that they are wrong to say that this is wrong.

19
00:01:09,467 --> 00:01:12,841
But, when you do this you have to be a little bit careful about what it means

20
00:01:12,841 --> 00:01:16,649
to take a principle square root of a negative number, and it being defined

21
00:01:16,649 --> 00:01:19,900
for imaginary, and we'll learn in the future, complex numbers.

22
00:01:19,900 --> 00:01:22,918
But for your understanding right now, you dont have to differentiate them,

23
00:01:22,918 --> 00:01:26,587
you don't have to split hairs between any of these definitions.

24
00:01:26,587 --> 00:01:31,333
Now with this definition, let us think about what these different powers of "i" are.

25
00:01:31,333 --> 00:01:33,321
because you can imagine, if something squared is negative one,

26
00:01:33,321 --> 00:01:37,667
if I take it to all sorts of powers, maybe that will give us weird things.

27
00:01:37,667 --> 00:01:40,566
And what we'll see is that the powers of "i" are kind of neat,

28
00:01:40,566 --> 00:01:45,302
because they kind of cycle, where they do cycle, through a whole set of values.

29
00:01:45,302 --> 00:01:50,225
So I could start with, lets start with "i" to the zeroth power.

30
00:01:50,225 --> 00:01:53,569
And so you might say, anything to the zeroth power is one,

31
00:01:53,569 --> 00:01:57,005
so "i" to the zeroth power is one, and that is true.

32
00:01:57,005 --> 00:01:59,978
And you could actually derive that even from this definition,

33
00:01:59,978 --> 00:02:03,646
but this is pretty straight forward; anything to the zeroth power, including "i" is one.

34
00:02:03,646 --> 00:02:07,000
Then you say, ok, what is "i" to the first power,

35
00:02:07,000 --> 00:02:11,667
well anything to the first power is just that number times itself once.

36
00:02:11,667 --> 00:02:14,002
So that's justgoing to be "i".

37
00:02:14,002 --> 00:02:16,324
Really by the definition of what it means to take an exponent,

38
00:02:16,324 --> 00:02:18,000
so that completely makes sense.

39
00:02:18,000 --> 00:02:19,993
And then you have "i" to the second power.

40
00:02:19,993 --> 00:02:22,933
"i" to the second power, well by definition,

41
00:02:22,933 --> 00:02:29,002
"i" to the second power is equal to negative one.

42
00:02:29,002 --> 00:02:33,000
Lets try "i" to the third power ill do this in a color i haven't used.

43
00:02:33,000 --> 00:02:41,634
"i" to the third power, well that's going to be "i" to the second power times "i"

44
00:02:41,634 --> 00:02:44,838
And we know that "i" to the second power is negative one,

45
00:02:44,838 --> 00:02:48,067
so its negative one times "i" let me make that clear.

46
00:02:48,067 --> 00:02:51,067
This is the same thing as this, which is the same thing as that,

47
00:02:51,067 --> 00:02:52,965
"i" squared is negative one.

48
00:02:52,965 --> 00:02:58,492
So you multiply it out, negative one times "i" equals negative "i".

49
00:02:58,492 --> 00:03:01,325
Now what happens when you take "i" to the fourth power,

50
00:03:01,325 --> 00:03:06,897
I'll do it up here. "i" to the fourth power.

51
00:03:06,897 --> 00:03:10,659
Well once again this is going to be "i" times "i" to the third power.

52
00:03:10,659 --> 00:03:14,235
So that's "i" times "i" to the third power. "i" times "i" to the third power

53
00:03:14,235 --> 00:03:21,990
Well what was "i" to the third power? "i" to the third power was negative "i"

54
00:03:21,990 --> 00:03:28,306
This over here is negative "i". And so "i" times "i" would get negative one,

55
00:03:28,306 --> 00:03:31,650
but you have a negative out here, so its "i" times "i" is negative one,

56
00:03:31,650 --> 00:03:35,000
and you have a negative, that gives you positive one.

57
00:03:35,000 --> 00:03:37,966
Let me write it out. This is the same thing

58
00:03:37,966 --> 00:03:43,167
as, so this is "i" times negative "i", which is the same thing as

59
00:03:43,167 --> 00:03:46,975
negative one times, remember multiplication is commutative,

60
00:03:46,975 --> 00:03:49,251
if you're multiplying a bunch of numbers you can just switch the order.

61
00:03:49,251 --> 00:03:52,316
This is the same thing as negative one times "i" times "i".

62
00:03:52,316 --> 00:03:55,984
"i" times "i", by definition, is negative one.

63
00:03:55,984 --> 00:03:59,839
Negative one times negative one is equal to positive one.

64
00:03:59,839 --> 00:04:02,672
So "i" to the fourth is the same thing as "i" to the zeroth power.

65
00:04:02,672 --> 00:04:05,319
Now lets try "i" to the fifth.

66
00:04:05,319 --> 00:04:09,266
"i" to the fifth power. Well that's just going to be "i" to to the fourth

67
00:04:09,266 --> 00:04:14,653
times "i". And we know what "i" to the fourth is. It is one.

68
00:04:14,653 --> 00:04:19,669
So its one times "i", or it is one times "i",

69
00:04:19,669 --> 00:04:21,341
or it is just "i" again. So once again it is exactly the same thing

70
00:04:21,341 --> 00:04:23,012
as "i" to the first power.

71
00:04:23,012 --> 00:04:25,067
Lets try again just to see the pattern keep going.

72
00:04:25,067 --> 00:04:27,006
Lets try "i" to the seventh power.

73
00:04:27,006 --> 00:04:28,400
Sorry, "i" to the sixth power.

74
00:04:28,400 --> 00:04:35,319
Well that's "i" times "i" to the fifth power, that's "i" times "i" to the fifth,

75
00:04:35,319 --> 00:04:38,570
"i" to the fifth we already established as just "i",

76
00:04:38,570 --> 00:04:44,235
so its "i" times "i", it is equal to, by definition,"i" times "i" is negative one.

77
00:04:44,235 --> 00:04:47,997
And then lets finish off, well we could keep going on this way

78
00:04:47,997 --> 00:04:50,598
We can keep putting high and higher powers of "i" here.

79
00:04:50,598 --> 00:04:52,502
An we'll see that it keeps cycling back.

80
00:04:52,502 --> 00:04:56,263
In the next video I'll teach you how taking an arbitrarily high power of "i",

81
00:04:56,263 --> 00:04:58,333
how you can figure out what that's going to be.

82
00:04:58,333 --> 00:05:00,489
But lets just verify that this cycle keeps going.

83
00:05:00,489 --> 00:05:06,667
"i" to the seventh power is equal to "i" times "i" to the sixth power.

84
00:05:06,667 --> 00:05:11,914
"i" to the sixth power is negative one. "i" times negative one is negative "i".

85
00:05:11,914 --> 00:05:14,839
And if you take "i" to the eighth, once again it'll be one,

86
00:05:14,839 --> 99:59:59,999
"i" to the ninth will be "i" again, and so on and so forth.