Transcript
算术和分数. Properties of Real Numbers.
让我首先说这个视频真的在广场上开始，它从所有数学的基本属性开始。
如果您正在查看数学，这是一个令人兴奋的地方。我们将涵盖基础知识的所有内容。
But, if you're already familiar with math, if math is something you're reasonably good at, then you probably don't need to watch all these introductory videos. 我强烈建议的只是跳过摘要，阅读摘要。如果摘要中有一些东西，这绝对没有意义， then go back to the video and watch the video. But don't force yourself to watch the video, 如果它是让你轻松的方式。
在这个过程中了解自己并节奏自己非常重要。对于真正想要从开始前开始学习一切的人， 让我们从一开始就开始。当测试说号码时，它总是表示实数。 What's meant by a real number? Well, a real number is any number on the number line.
这包括圆形数字以及分数和小数。所以这里描绘的所有数字以及之间的所有数字和空格。 Numbers like one-third, or pi, or the square root of seven, all of these are numbers. They're all real numbers. When the test says "number" this number could be any number on the number line.
它可能是积极的，消极的或零。它可以是整数，或分数或小数。 这是测试爱人的人的方式之一，喜欢开始数学问题。 x是一个数字，然后它继续。人们认为，由于x是一个数字，它只能是一个，两个，三， 四，五，或者换句话说，它必须是计数号码。
人们忘记了否定。他们忘了分数和小数。 非常重要，不要陷入陷阱。如果我们看到单词号码， 你必须思考所有这些类别都在一起。我们将在以后的视频中讨论分数和小数。
现在，请注意，零是唯一既积极也不为负的数字。 So we have these two very large categories, positive and negative. Zero is the only number on the number line that doesn't fall into either one of those categories. One special type of number is the integer.
Integers include all positive and negative whole numbers. If the question asks about integers, it could be any number in this set. It could be the positives or the negatives. If the question asks about positive integers, 然后这是普通计数数字，1,2,3,4,5,6等非常重要的是保持这三个套装。
有时，问题会询问数字，有时他们会询问整数，有时他们会询问积极整数。 Those are three different sets, it's very important not to confuse them. Back to all the real numbers. There are a few vocab words you need to know. These concern the four fundamental operations.
您可能希望在此处暂停视频只是为了验证自己知道这四个术语。 加法结果称为总和。减法的结果称为差异。 The result of multiplication is called a product. The result of division is called the quotient.
These are four vocab words that you need to know. There are also some fundamental arithmetic properties common to all real numbers. To discuss these, I will use A, B and C to represent real numbers. These are terms you don't need to know, you don't need to know the names of 属性，您需要知道的只是属性本身。第一个是换向财产，能够切换订单。
因此，例如，我们可以切换添加顺序，a + b = b + a。我们做到了哪个顺序并不重要，我们得到了相同的答案。 与乘法相似，我们会绕过围绕顺序，a * b = b * a。我们可以再次换掉我们的顺序我们得到相同的答案。 加法和乘法是换向分割，减法一般不是换向。
If we swap the order around the subtraction, we get different answers. Seven minus four does not equal four minus seven. But keep this in mind, if we rewrite the subtraction as the addition of a negative, that's a very sophisticated move, then it would be commutative. So 7- 4 we can express that as 7 + (-4). And then we could swap the order, (-4) + 7.
So we could actually perform that swap if we needed. So all these are properties you need to know, 您需要对这些数字可以做的。但是你不需要记住换向的这个词， that word will not be on the test. The second property is the associative property, the ability to group things.
因此，如果我们添加三个数字，我们可以将B + C归档，或者我们可以先添加A + B. 分组的任何方式都会导致相同的答案。在这里，我们可以这三个和五个首先，或者我们可以添加两个和三个。 无论哪种方式，我们都会得到相同的答案。同样，通过乘法，我们可以首先将它们分组 a times b first.
If we multiply the 3 and the 5 first or we multiply the 2 and the 3 first, either way, we get the same answer. 加法和乘法是关联的。一般而言，减法和划分并不关联。 Again, you need to know what the numbers do. You don't need to know this word, associative.
这不是测试将测试您的一句话。分配财产。 这个是一个很大的人。首先乘法分发添加， 它分配减法。所以我们可以首先做加法和减法，然后乘以或我们 could multiply times each piece and then perform the addition and the subtraction.
So for example, we can add two plus three first and then multiply, or we can do six times two, six times three first and then add those two products separately. Either way we get the same answer. Similarly with subtraction we could subtract the three and the seven first get 四个，乘以十二，或者我们可以将12 * 7乘以，并单独乘以12 * 3，然后减去这两种产品。
Either way we get the same answer. There are other properties that concern the special numbers 1 and 0. The first one is multiplying and dividing by one, very simple. We don't change a number when we multiply or divide by one. So we can represent that symbolically this way, a * 1 = a, a/1 = a. Absolutely no change if we multiply or divide by one.
Multiplying by zero. Anything times zero equals zero. 再次，非常简单。所以a * 0 = 0 * a = 0。 So we're perfectly allowed to multiply by zero and we always get zero, we're not allowed to divide by zero, that's a very important point.
零产品属性。如果两个数字的乘积为零，则其中一个因素必须为零。 So if a * b = 0, then a = 0, or b = 0.. And notice that the word or there, that's an essential piece of mathematical equipment right there. That's actually part of the math, that's not decoration.
这在两个方程之间创建了逻辑连接。零产品属性在保理和中非常重要 in solving quadratics, so you'll run into it there. Those are in the algebra module videos. 分开一个数字。当任何非零数自身除以时，商必须等于1。
We can express this as a/a, or a/a = 1. Again, very straightforward. 所以总之，测试将使用单词号码，这始终意味着任何实数，而不仅仅是正整数。 It's very important to keep numbers verses integers versus positive integers straight.
Those are three different sets. Know those four terms, sum, difference, product, and quotient. 那些实际上是你需要知道的词汇词。并了解属性。 Again, we don't need to know the names of the properties, we just need to know the properties themselves.
在这里，我们拥有协会财产的换向财产。在这里，我们有分配的属性，这是一个非常重要的财产，而且 然后所有有关特殊编号和零的属性。
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But, if you're already familiar with math, if math is something you're reasonably good at, then you probably don't need to watch all these introductory videos. 我强烈建议的只是跳过摘要，阅读摘要。如果摘要中有一些东西，这绝对没有意义， then go back to the video and watch the video. But don't force yourself to watch the video, 如果它是让你轻松的方式。
在这个过程中了解自己并节奏自己非常重要。对于真正想要从开始前开始学习一切的人， 让我们从一开始就开始。当测试说号码时，它总是表示实数。 What's meant by a real number? Well, a real number is any number on the number line.
这包括圆形数字以及分数和小数。所以这里描绘的所有数字以及之间的所有数字和空格。 Numbers like one-third, or pi, or the square root of seven, all of these are numbers. They're all real numbers. When the test says "number" this number could be any number on the number line.
它可能是积极的，消极的或零。它可以是整数，或分数或小数。 这是测试爱人的人的方式之一，喜欢开始数学问题。 x是一个数字，然后它继续。人们认为，由于x是一个数字，它只能是一个，两个，三， 四，五，或者换句话说，它必须是计数号码。
人们忘记了否定。他们忘了分数和小数。 非常重要，不要陷入陷阱。如果我们看到单词号码， 你必须思考所有这些类别都在一起。我们将在以后的视频中讨论分数和小数。
现在，请注意，零是唯一既积极也不为负的数字。 So we have these two very large categories, positive and negative. Zero is the only number on the number line that doesn't fall into either one of those categories. One special type of number is the integer.
Integers include all positive and negative whole numbers. If the question asks about integers, it could be any number in this set. It could be the positives or the negatives. If the question asks about positive integers, 然后这是普通计数数字，1,2,3,4,5,6等非常重要的是保持这三个套装。
有时，问题会询问数字，有时他们会询问整数，有时他们会询问积极整数。 Those are three different sets, it's very important not to confuse them. Back to all the real numbers. There are a few vocab words you need to know. These concern the four fundamental operations.
您可能希望在此处暂停视频只是为了验证自己知道这四个术语。 加法结果称为总和。减法的结果称为差异。 The result of multiplication is called a product. The result of division is called the quotient.
These are four vocab words that you need to know. There are also some fundamental arithmetic properties common to all real numbers. To discuss these, I will use A, B and C to represent real numbers. These are terms you don't need to know, you don't need to know the names of 属性，您需要知道的只是属性本身。第一个是换向财产，能够切换订单。
因此，例如，我们可以切换添加顺序，a + b = b + a。我们做到了哪个顺序并不重要，我们得到了相同的答案。 与乘法相似，我们会绕过围绕顺序，a * b = b * a。我们可以再次换掉我们的顺序我们得到相同的答案。 加法和乘法是换向分割，减法一般不是换向。
If we swap the order around the subtraction, we get different answers. Seven minus four does not equal four minus seven. But keep this in mind, if we rewrite the subtraction as the addition of a negative, that's a very sophisticated move, then it would be commutative. So 7- 4 we can express that as 7 + (-4). And then we could swap the order, (-4) + 7.
So we could actually perform that swap if we needed. So all these are properties you need to know, 您需要对这些数字可以做的。但是你不需要记住换向的这个词， that word will not be on the test. The second property is the associative property, the ability to group things.
因此，如果我们添加三个数字，我们可以将B + C归档，或者我们可以先添加A + B. 分组的任何方式都会导致相同的答案。在这里，我们可以这三个和五个首先，或者我们可以添加两个和三个。 无论哪种方式，我们都会得到相同的答案。同样，通过乘法，我们可以首先将它们分组 a times b first.
If we multiply the 3 and the 5 first or we multiply the 2 and the 3 first, either way, we get the same answer. 加法和乘法是关联的。一般而言，减法和划分并不关联。 Again, you need to know what the numbers do. You don't need to know this word, associative.
这不是测试将测试您的一句话。分配财产。 这个是一个很大的人。首先乘法分发添加， 它分配减法。所以我们可以首先做加法和减法，然后乘以或我们 could multiply times each piece and then perform the addition and the subtraction.
So for example, we can add two plus three first and then multiply, or we can do six times two, six times three first and then add those two products separately. Either way we get the same answer. Similarly with subtraction we could subtract the three and the seven first get 四个，乘以十二，或者我们可以将12 * 7乘以，并单独乘以12 * 3，然后减去这两种产品。
Either way we get the same answer. There are other properties that concern the special numbers 1 and 0. The first one is multiplying and dividing by one, very simple. We don't change a number when we multiply or divide by one. So we can represent that symbolically this way, a * 1 = a, a/1 = a. Absolutely no change if we multiply or divide by one.
Multiplying by zero. Anything times zero equals zero. 再次，非常简单。所以a * 0 = 0 * a = 0。 So we're perfectly allowed to multiply by zero and we always get zero, we're not allowed to divide by zero, that's a very important point.
零产品属性。如果两个数字的乘积为零，则其中一个因素必须为零。 So if a * b = 0, then a = 0, or b = 0.. And notice that the word or there, that's an essential piece of mathematical equipment right there. That's actually part of the math, that's not decoration.
这在两个方程之间创建了逻辑连接。零产品属性在保理和中非常重要 in solving quadratics, so you'll run into it there. Those are in the algebra module videos. 分开一个数字。当任何非零数自身除以时，商必须等于1。
We can express this as a/a, or a/a = 1. Again, very straightforward. 所以总之，测试将使用单词号码，这始终意味着任何实数，而不仅仅是正整数。 It's very important to keep numbers verses integers versus positive integers straight.
Those are three different sets. Know those four terms, sum, difference, product, and quotient. 那些实际上是你需要知道的词汇词。并了解属性。 Again, we don't need to know the names of the properties, we just need to know the properties themselves.
在这里，我们拥有协会财产的换向财产。在这里，我们有分配的属性，这是一个非常重要的财产，而且 然后所有有关特殊编号和零的属性。
