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复杂数字介绍
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复杂数字简介。我将首先说，一个迷人数学家的财产是关闭的，
是否关闭了特定数量。现在这是一个你不必知道这一行为的词。
例如，实数，编号线上的数字在此外关闭。
他们正在补充。换句话说，如果我们添加任何两个数字，我们得到另一个实数。 We can pick any two numbers on the number line, no matter what ones we pick. When we add, we'll land somewhere else on the number line. 我们永远不会通过添加两个数字的过程离开数字线，这是一个封闭的系统。
Similarly, real numbers are closed under subtraction and multiplication. We could subtract any two numbers, 乘以任何两个数字，结果仍将在数字线上的某个位置。我们还可以分开我们不划分零的规定。 Now that's not a problem for closure, if we just have an exception of a single number.
不能除以零，但除此之外，我们可以划分任意两个数字。我们仍然在数字线上。 所以换句话说，在加法，减法，乘法，划分方面，实数是封闭系统。 They have closure. Real numbers are also closed under exponentiation 只要所有的指数都是整数。
So we can pick any number on the number line, raise it to any integer power. Now again, the exception is with zero we can't raise zero to the power of negative 整数Cuz这就是拍摄它的互惠。但除了那个单例之外， 实数在指数下关闭。我们可以选择任何实数，将其提升到整数电源， 我们得到另一个真实数字。
It's still a closed system. We're not leaving the number line. 虽然有根源，但我们缺乏关闭。平方根或不存在的数量答案 任何甚至是负数的根源。所以我们找不到消极的平方根，我们找不到第四根或 阴性的第六根。
All of those leave the number line, all of those do not have solutions that are on the number line. Among other things, this means that a very simple algebraic equation such as x squared plus 4 equls 0 have no solution. There is absolutely no number on the number line that satisfies that particular equation.
Now that's kinda crazy if you think about it. Such an easy equation. How could such an easy equation have no solution? But nothing on the number line satisfies that particular equation. So this kind of thing bothers mathematicians. Mathematicians in the 16th century realized we could solve a great many 仅仅通过允许负面的平方根来关闭问题。
So this sounds odd because you probably have always heard you can't take the square root of a negative. It is perfectly true that the square root of negative does not exist anywhere on the real number line. 所以现在我们正在做的是我们要离开真正的数字线。我们正在扩展我们作为数字的重要概念。
所以我们首先定义这个奇数i，这是负数1.的平方根1.所以数字行上不存在这个数字。 它取消了数字线。我们在等式x平方中定义这个等于负1， 将有两个解决方案积极的我和负面i。类似于等式的两个解决方案，例如x平方等方程 正4。
这将使溶液阳性2和负数2.积极根和负根。 这么多的方式，积极的我和消极的我。这个我和它的倍数有不幸的名字想象。 So this is the word that everyone uses to refer to these. So you see, earlier mathematicians such as Mr.
Rene Descartes，他是发明了XY图形计划的人。他是一个很棒的数学家。 他使用这个术语，因为他不相信这些数字真的被视为数字线上的数字的相等。 He really thought the numbers on the number line, those were really real numbers.
那些是真正的坚实真爱数字。和这些其他事情，有点像欺骗使用它们。 所以他使用了虚构的词。我们现在，明白这些数字与任何其他数字一样合法。 事实证明，我们用这些数字测量现实世界中的东西。我们实际测量电力。
And an electrical current with an imaginary amplitude could kill you. It could have very real results in your life. 所以换句话说，我们用这个的现实世界中有真实的东西。所以想象中的名字是一个错误的人。 It's an earlier misunderstanding. Unfortunately, this sobriquet this nickname has stuck 即使它不准确。
So I just want to emphasize even though I'll be calling these imaginary, cuz that's what people call them. I want you to appreciate they're not in fact imaginary. They're just as bona fide and legitimate as the numbers on the number line. This number i makes it possible to find the square root of any negative number. So for example, suppose we have to find the square root of negative 9.
嗯，我们可以表达负数9作为缺点的产物1.分开平方根。 当然9的平方根，是3.负1的平方根是我，所以我们得到3i。 类似地，任何负面的平方根将是绝对值的平方根。
This number i also makes it possible to solve algebraic equations that previously had no solution on the real number line. 例如，x平方等于负25.在满足该等式的数字线上没有数字， 但我们可以使用虚数来解决它。事实上，解决方案是正5i或负5i。
考虑这种不可行的二次方程。所以这是一个等式。 There's no way to factor it. If we graph this, 我们发现它是一个完全高于X轴的抛物线。它甚至从未与X轴相交。
所以换句话说，数字上所有无穷大数字的数字线上没有点 线路，没有一个满足该特定方程的单个。所以我们知道答案不是一个实数。 But we could still solve it, so what we're gonna do is we're gonna subtract 4 from both sides and 当我们这样做时，我们会得到左边的差异模式的平方。
所以这是我们在代数课上谈到的模式，如果这对你不熟悉，那就值得回去 watching those videos in the algebra module about square of a sum, square of a difference, those are important patterns to know. 事实证明，左边的表达式可以对x减去3个平方，因此我们得到的是x等于3个平方等于负4。
Take a square root of both sides. We get x minus 3 equals plus or minus 2i. 向两侧添加3，我们得到x等于3加或减去2i。和这些数字，3加2I和 3减去2I，那些是该等式的解决方案。所以它没有真正的解决方案，但如果我们涉及虚数，它有一个解决方案。
真实数字加上一个假想数的这样的和差异称为复数。 一般复数号是形式A PLUS BI。如果它有助于您可视化此， 你能想到的与躺在复数plane. So the horizontal axis, that's the ordinary real number line.
这是我们已经知道的数字线。我们始终知道并喜欢它。 垂直轴是另一个数字线。这是虚构的轴，所以我们看到我们有我，2i，3i，4i。 如果我们走下了那个轴，我们会得到负面i，负面的2i，负3i。这是虚构的轴。
And then a complex number such as 3 plus 2i. Well we can imagine that as a point in the plane, drawn here. 所以我们在真正的轴上超过3，我们上升到虚构的轴上，这是我们绘制的地方。 So every complex number would rely on a different point in that plane. I wanna emphasize the ACT does not test this.
您不需要了解复杂的飞机。为了做任何事情，这个行为要问， 我只是分享这个，因为对于一些人特别是视觉思想家来说，它有点帮助你想象，复杂的数字在哪里生活？ I've been saying that they don't live on the real number line. Well if they're not on the number line, where do they live?
Now, we can see the real number line is just one part of the larger complex plane, and these numbers live in other places on the plane off the real number line. 请注意，当我们解决上述两种代数方程时，我们得到了两种解决方案，其中虚部的零件相反或减去标志。 例如，我们有3加2I和3减去2i。这不是巧合。
具有相同真实部件和相反符号的虚部的两个复杂数字称为复杂缀合物。 A plus bi and a minus bi are complex conjugates of each other. We will see some uses of complex conjugates in the next video on operations with complex numbers. So right now, just hold onto that thought.
我们将在下一个视频中讨论它更多。最后，在这个视频中，我会谈谈我的权力。 该法令希望学生了解有关我的权力的模式。所以，当然，我第一个是我。 我按定义平方为负1.我差点呢？
好吧，我的立方体，这将是我的平方时间我。所以这是一个负面的1次，所以这将是消极的我。 And finally, what about i to the fourth? Well i to the 4th, one way to think of it is i cubed times i, so that would be negative i times positive i. Well the i times i is negative 1, so 我们的负数为1，这将为我们提供积极的1。
Another way to think about that, we could also express i to the 4th as i squared times i squared. 因此，这将是负1倍的阴性1，这是正面的1.所以我们这样做的方式，我到第四个是负1。 So first of all, it's very important to know that pattern i to the 1st is i, i to the 2nd is negative 1, i to the 3rd is negative i, 我到第四个是积极的1.
这是基本模式。因为模式返回正面1， 意味着接下来的四个权力将遵循相同的模式。所以我到第5次到第四次1， 我到第6次是我到第4次到第四次，等等。因此，在第4到第4个意味着模式会重复。
它会再次重复一次。你只是像壁纸一样重复这个重复。 这将达到无限。所以这对欣赏来说非常重要，这是模式。 We know that i to the power of any multiple of four has to equal positive 1. This allows us to evaluate i to the power of any large number.
So for example, the ACT might ask us what is i to the power of 91? So it seems like a ridiculously large number, how are we gonna calculate this? Well turns out, we could just write it this way. I to the 91 equals i to the 88, times i to the 3, because 88 plus 3 is 91. 我到88,88被4所以4，所以我到88必须是积极的1.所以它只是我的二手的积极1次，你可能会记得我的立方是消极的我， 所以我们只是得到消极的我。
如果您记得这个简单的模式，您可以评估I的任何力量。总之，我们谈到了想象中的号码， which is the square root of negative 1. We also talked about why the word imaginary's not the best word for this, but it's the word that people use. We discussed how to use i to write the square root of any negative number and 注意到我如何用来解决以前无法解决的代数方程。
所以没有实际数字解决方案的代数方程，现在我们可以在复杂的飞机中找到解决方案。 We introduced the idea of complex conjugates and we discussed the patterns of powers of i.
阅读完整成绩单
他们正在补充。换句话说，如果我们添加任何两个数字，我们得到另一个实数。 We can pick any two numbers on the number line, no matter what ones we pick. When we add, we'll land somewhere else on the number line. 我们永远不会通过添加两个数字的过程离开数字线，这是一个封闭的系统。
Similarly, real numbers are closed under subtraction and multiplication. We could subtract any two numbers, 乘以任何两个数字，结果仍将在数字线上的某个位置。我们还可以分开我们不划分零的规定。 Now that's not a problem for closure, if we just have an exception of a single number.
不能除以零，但除此之外，我们可以划分任意两个数字。我们仍然在数字线上。 所以换句话说，在加法，减法，乘法，划分方面，实数是封闭系统。 They have closure. Real numbers are also closed under exponentiation 只要所有的指数都是整数。
So we can pick any number on the number line, raise it to any integer power. Now again, the exception is with zero we can't raise zero to the power of negative 整数Cuz这就是拍摄它的互惠。但除了那个单例之外， 实数在指数下关闭。我们可以选择任何实数，将其提升到整数电源， 我们得到另一个真实数字。
It's still a closed system. We're not leaving the number line. 虽然有根源，但我们缺乏关闭。平方根或不存在的数量答案 任何甚至是负数的根源。所以我们找不到消极的平方根，我们找不到第四根或 阴性的第六根。
All of those leave the number line, all of those do not have solutions that are on the number line. Among other things, this means that a very simple algebraic equation such as x squared plus 4 equls 0 have no solution. There is absolutely no number on the number line that satisfies that particular equation.
Now that's kinda crazy if you think about it. Such an easy equation. How could such an easy equation have no solution? But nothing on the number line satisfies that particular equation. So this kind of thing bothers mathematicians. Mathematicians in the 16th century realized we could solve a great many 仅仅通过允许负面的平方根来关闭问题。
So this sounds odd because you probably have always heard you can't take the square root of a negative. It is perfectly true that the square root of negative does not exist anywhere on the real number line. 所以现在我们正在做的是我们要离开真正的数字线。我们正在扩展我们作为数字的重要概念。
所以我们首先定义这个奇数i，这是负数1.的平方根1.所以数字行上不存在这个数字。 它取消了数字线。我们在等式x平方中定义这个等于负1， 将有两个解决方案积极的我和负面i。类似于等式的两个解决方案，例如x平方等方程 正4。
这将使溶液阳性2和负数2.积极根和负根。 这么多的方式，积极的我和消极的我。这个我和它的倍数有不幸的名字想象。 So this is the word that everyone uses to refer to these. So you see, earlier mathematicians such as Mr.
Rene Descartes，他是发明了XY图形计划的人。他是一个很棒的数学家。 他使用这个术语，因为他不相信这些数字真的被视为数字线上的数字的相等。 He really thought the numbers on the number line, those were really real numbers.
那些是真正的坚实真爱数字。和这些其他事情，有点像欺骗使用它们。 所以他使用了虚构的词。我们现在，明白这些数字与任何其他数字一样合法。 事实证明，我们用这些数字测量现实世界中的东西。我们实际测量电力。
And an electrical current with an imaginary amplitude could kill you. It could have very real results in your life. 所以换句话说，我们用这个的现实世界中有真实的东西。所以想象中的名字是一个错误的人。 It's an earlier misunderstanding. Unfortunately, this sobriquet this nickname has stuck 即使它不准确。
So I just want to emphasize even though I'll be calling these imaginary, cuz that's what people call them. I want you to appreciate they're not in fact imaginary. They're just as bona fide and legitimate as the numbers on the number line. This number i makes it possible to find the square root of any negative number. So for example, suppose we have to find the square root of negative 9.
嗯，我们可以表达负数9作为缺点的产物1.分开平方根。 当然9的平方根，是3.负1的平方根是我，所以我们得到3i。 类似地，任何负面的平方根将是绝对值的平方根。
This number i also makes it possible to solve algebraic equations that previously had no solution on the real number line. 例如，x平方等于负25.在满足该等式的数字线上没有数字， 但我们可以使用虚数来解决它。事实上，解决方案是正5i或负5i。
考虑这种不可行的二次方程。所以这是一个等式。 There's no way to factor it. If we graph this, 我们发现它是一个完全高于X轴的抛物线。它甚至从未与X轴相交。
所以换句话说，数字上所有无穷大数字的数字线上没有点 线路，没有一个满足该特定方程的单个。所以我们知道答案不是一个实数。 But we could still solve it, so what we're gonna do is we're gonna subtract 4 from both sides and 当我们这样做时，我们会得到左边的差异模式的平方。
所以这是我们在代数课上谈到的模式，如果这对你不熟悉，那就值得回去 watching those videos in the algebra module about square of a sum, square of a difference, those are important patterns to know. 事实证明，左边的表达式可以对x减去3个平方，因此我们得到的是x等于3个平方等于负4。
Take a square root of both sides. We get x minus 3 equals plus or minus 2i. 向两侧添加3，我们得到x等于3加或减去2i。和这些数字，3加2I和 3减去2I，那些是该等式的解决方案。所以它没有真正的解决方案，但如果我们涉及虚数，它有一个解决方案。
真实数字加上一个假想数的这样的和差异称为复数。 一般复数号是形式A PLUS BI。如果它有助于您可视化此， 你能想到的与躺在复数plane. So the horizontal axis, that's the ordinary real number line.
这是我们已经知道的数字线。我们始终知道并喜欢它。 垂直轴是另一个数字线。这是虚构的轴，所以我们看到我们有我，2i，3i，4i。 如果我们走下了那个轴，我们会得到负面i，负面的2i，负3i。这是虚构的轴。
And then a complex number such as 3 plus 2i. Well we can imagine that as a point in the plane, drawn here. 所以我们在真正的轴上超过3，我们上升到虚构的轴上，这是我们绘制的地方。 So every complex number would rely on a different point in that plane. I wanna emphasize the ACT does not test this.
您不需要了解复杂的飞机。为了做任何事情，这个行为要问， 我只是分享这个，因为对于一些人特别是视觉思想家来说，它有点帮助你想象，复杂的数字在哪里生活？ I've been saying that they don't live on the real number line. Well if they're not on the number line, where do they live?
Now, we can see the real number line is just one part of the larger complex plane, and these numbers live in other places on the plane off the real number line. 请注意，当我们解决上述两种代数方程时，我们得到了两种解决方案，其中虚部的零件相反或减去标志。 例如，我们有3加2I和3减去2i。这不是巧合。
具有相同真实部件和相反符号的虚部的两个复杂数字称为复杂缀合物。 A plus bi and a minus bi are complex conjugates of each other. We will see some uses of complex conjugates in the next video on operations with complex numbers. So right now, just hold onto that thought.
我们将在下一个视频中讨论它更多。最后，在这个视频中，我会谈谈我的权力。 该法令希望学生了解有关我的权力的模式。所以，当然，我第一个是我。 我按定义平方为负1.我差点呢？
好吧，我的立方体，这将是我的平方时间我。所以这是一个负面的1次，所以这将是消极的我。 And finally, what about i to the fourth? Well i to the 4th, one way to think of it is i cubed times i, so that would be negative i times positive i. Well the i times i is negative 1, so 我们的负数为1，这将为我们提供积极的1。
Another way to think about that, we could also express i to the 4th as i squared times i squared. 因此，这将是负1倍的阴性1，这是正面的1.所以我们这样做的方式，我到第四个是负1。 So first of all, it's very important to know that pattern i to the 1st is i, i to the 2nd is negative 1, i to the 3rd is negative i, 我到第四个是积极的1.
这是基本模式。因为模式返回正面1， 意味着接下来的四个权力将遵循相同的模式。所以我到第5次到第四次1， 我到第6次是我到第4次到第四次，等等。因此，在第4到第4个意味着模式会重复。
它会再次重复一次。你只是像壁纸一样重复这个重复。 这将达到无限。所以这对欣赏来说非常重要，这是模式。 We know that i to the power of any multiple of four has to equal positive 1. This allows us to evaluate i to the power of any large number.
So for example, the ACT might ask us what is i to the power of 91? So it seems like a ridiculously large number, how are we gonna calculate this? Well turns out, we could just write it this way. I to the 91 equals i to the 88, times i to the 3, because 88 plus 3 is 91. 我到88,88被4所以4，所以我到88必须是积极的1.所以它只是我的二手的积极1次，你可能会记得我的立方是消极的我， 所以我们只是得到消极的我。
如果您记得这个简单的模式，您可以评估I的任何力量。总之，我们谈到了想象中的号码， which is the square root of negative 1. We also talked about why the word imaginary's not the best word for this, but it's the word that people use. We discussed how to use i to write the square root of any negative number and 注意到我如何用来解决以前无法解决的代数方程。
所以没有实际数字解决方案的代数方程，现在我们可以在复杂的飞机中找到解决方案。 We introduced the idea of complex conjugates and we discussed the patterns of powers of i.
