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Properties of Real Numbers
成绩单
算术和分数。实数的属性。
Let me begin by saying this video really starts at square one, it starts with the very basic properties of all of mathematics.
And if you are reviewing math, this is a wonderful place to start. We'll cover everything from the basics.
但是，如果您已经熟悉数学，如果数学是您合理的事情，那么您可能不需要观看所有这些介绍性视频。 What I'd strongly suggest is just skip ahead to the summary, read the summary. If there's something in the summary that makes absolutely no sense, 然后回到视频并观看视频。但不要强迫自己观看视频， if it's stuff that's going to be way to easy for you.
It's very important to know yourself and pace yourself in this process. For folks who really want to learn everything from the beginning onward, let's start at the beginning. When the test says number, it always means a real number. 真实数字是什么意思？嗯，实数是数字线上的任何数字。
This includes round numbers as well as fractions and decimals. So all the numbers pictured here, as well as all the numbers and spaces in between. 数字如三分之一，或pi，或七的平方根，所有这些都是数字。 他们都是真正的数字。当测试说“数字”此数字可能是数字行上的任何数字。
It could be positive, negative, or zero. It could be a whole number, or a fraction, or a decimal. This is one of the way that the test loves to trap people, loves to begin a math problem. X is a number, and then it goes on. And people assume that since x is a number, it can only be one, two, three, four, five, or, in other words, that it has to be a counting number.
People forget about negatives. They forget about fractions and decimals. Very important not to fall into that trap. If we see the word number, you have to be thinking about all these categories all together. We will discuss fractions and decimals in later videos.
Right now, notice that zero is the only number that is neither positive nor negative. 所以我们有这两个非常大的类别，积极和消极。零是数字线上的唯一号码，它不会陷入其中 其中一个类别。一种特殊类型的数字是整数。
整数包括所有正数和负数。如果问题询问整数，则可以是此集合中的任何数字。 它可能是积极或否定。如果问题询问了正整数， then that's ordinary counting numbers, 1, 2, 3, 4, 5, 6, etc. Very important to keep these three sets straight.
Sometimes, questions will ask about numbers, sometimes they'll ask about integers, sometimes they'll ask about positive integers. 那些是三个不同的集合，这是非常重要的不要混淆它们。回到所有真正的数字。 有一些你需要知道的词汇单词。这些涉及四个基本行动。
You may want to pause the video here just to verify to yourself that you know these four terms. The result of addition is called a sum. The result of subtraction is called a difference. 乘法的结果称为产品。划分的结果称为商。
这些是您需要知道的四个词汇单词。对所有实数也有一些基本的算术属性。 要讨论这些，我将使用A，B和C表示实数。这些是您不需要知道的条款，您不需要知道的名称 the properties, all you need to know is the properties themselves. The first one is the commutative property, the ability to switch the order.
So, for example, we can switch the order of addition, a + b = b + a. It doesn't matter which order we do it in, we get the same answer. 同样用乘法,我们交换order around, a * b = b * a. Again we could swap the order around we get the same answer. Addition and multiplication are commutative division and subtraction in general are not commutative.
如果我们在减法周围交换顺序，我们得到了不同的答案。七减4不等于四个减号。 但请记住这一点，如果我们将减法重写为添加负面，那是一个非常复杂的移动，那么它将是换向的。 所以7-4我们可以表达为7 +（-4）。然后我们可以交换订单，（-4）+ 7。
因此，如果我们需要，我们实际上可以执行此次交换。所以所有这些都是你需要知道的属性， you need to what you can do with these numbers. But you don't need to remember the word commutative, 这个词不会在测试中。第二个财产是联合财产，能够分组事物。
So if we're adding three numbers we can group them, we can add b + c first or we could add a + b first. Either way of grouping leads to the same answer. Here we can the three and the five first, or we can add the two and the three first. Either way, we get the same answer. Similarly, with multiplication, we can group them b times c first or 次b第一。
如果我们将3和5乘以或者我们乘以2和3，无论哪种方式，我们都会得到相同的答案。 Addition and multiplication are associative. Subtraction and division, in general, are not associative. 再次，您需要知道数字的作用。您不需要知道这个词，关联。
That's not a word that the test will test you on. The distributive property. This one is a big one. First of all multiplication distributes over addition, and it distributes over subtraction. So we could do the addition and subtraction first and then multiply, or we 可以乘以每件的时间，然后执行添加和减法。
所以，例如，我们可以加上两加三，然后乘以六次，或者我们可以做六次，六次三次，然后单独添加这两种产品。 无论哪种方式，我们都得到了相同的答案。与减法类似，我们可以减去三个和七个首先获得 the four, multiply that by twelve, or we could multiply the 12 * 7 and separately multiply the 12 * 3 and then subtract those two products.
无论哪种方式，我们都得到了相同的答案。还有其他特性涉及特殊数字1和0。 第一个是乘以一个，非常简单地分开。当我们乘以或划分一个时，我们不会改变一个数字。 所以我们可以以象征性地表示这种方式，a * 1 = a，a / 1 = a。如果我们乘以或除以一个，绝对没有变化。
乘以零。零的任何东西等于零。 Again, very simple. So a * 0 = 0 * a = 0. 所以我们完全允许乘以零，我们总是得到零，我们不允许划分为零，这是一个非常重要的观点。
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。 所以，如果a * b = 0，则a = 0，或b = 0 ..并注意单词或 在那里，那是那里的一部分数学设备。这实际上是数学的一部分，那不是装饰。
That creates the logical connection between the two equations. The zero product property is very important in factoring and 在解决二际学中，所以你会遇到它。那些在代数模块视频中。 Dividing a number by itself. When any non-zero number is divided by itself, the quotient has to equal 1.
我们可以将其表达为A / A，或者A / A = 1.再次，非常简单。 So in summary, the test is gonna use the word number and this always means any real number, not just positive integers. 保持数字与正整数直接保持数字非常重要。
那些是三种不同的集合。知道这四个术语，总和，差异，产品和商。 Those are actually vocab words you need to know. And know the properties. 再次，我们不需要知道属性的名称，我们只需要了解属性本身。
Here we have the commutative property, the associative property. Here we have the distributive property, which is a really important one, and then all the properties concerning the special numbers one and zero.
阅读完整成绩单
但是，如果您已经熟悉数学，如果数学是您合理的事情，那么您可能不需要观看所有这些介绍性视频。 What I'd strongly suggest is just skip ahead to the summary, read the summary. If there's something in the summary that makes absolutely no sense, 然后回到视频并观看视频。但不要强迫自己观看视频， if it's stuff that's going to be way to easy for you.
It's very important to know yourself and pace yourself in this process. For folks who really want to learn everything from the beginning onward, let's start at the beginning. When the test says number, it always means a real number. 真实数字是什么意思？嗯，实数是数字线上的任何数字。
This includes round numbers as well as fractions and decimals. So all the numbers pictured here, as well as all the numbers and spaces in between. 数字如三分之一，或pi，或七的平方根，所有这些都是数字。 他们都是真正的数字。当测试说“数字”此数字可能是数字行上的任何数字。
It could be positive, negative, or zero. It could be a whole number, or a fraction, or a decimal. This is one of the way that the test loves to trap people, loves to begin a math problem. X is a number, and then it goes on. And people assume that since x is a number, it can only be one, two, three, four, five, or, in other words, that it has to be a counting number.
People forget about negatives. They forget about fractions and decimals. Very important not to fall into that trap. If we see the word number, you have to be thinking about all these categories all together. We will discuss fractions and decimals in later videos.
Right now, notice that zero is the only number that is neither positive nor negative. 所以我们有这两个非常大的类别，积极和消极。零是数字线上的唯一号码，它不会陷入其中 其中一个类别。一种特殊类型的数字是整数。
整数包括所有正数和负数。如果问题询问整数，则可以是此集合中的任何数字。 它可能是积极或否定。如果问题询问了正整数， then that's ordinary counting numbers, 1, 2, 3, 4, 5, 6, etc. Very important to keep these three sets straight.
Sometimes, questions will ask about numbers, sometimes they'll ask about integers, sometimes they'll ask about positive integers. 那些是三个不同的集合，这是非常重要的不要混淆它们。回到所有真正的数字。 有一些你需要知道的词汇单词。这些涉及四个基本行动。
You may want to pause the video here just to verify to yourself that you know these four terms. The result of addition is called a sum. The result of subtraction is called a difference. 乘法的结果称为产品。划分的结果称为商。
这些是您需要知道的四个词汇单词。对所有实数也有一些基本的算术属性。 要讨论这些，我将使用A，B和C表示实数。这些是您不需要知道的条款，您不需要知道的名称 the properties, all you need to know is the properties themselves. The first one is the commutative property, the ability to switch the order.
So, for example, we can switch the order of addition, a + b = b + a. It doesn't matter which order we do it in, we get the same answer. 同样用乘法,我们交换order around, a * b = b * a. Again we could swap the order around we get the same answer. Addition and multiplication are commutative division and subtraction in general are not commutative.
如果我们在减法周围交换顺序，我们得到了不同的答案。七减4不等于四个减号。 但请记住这一点，如果我们将减法重写为添加负面，那是一个非常复杂的移动，那么它将是换向的。 所以7-4我们可以表达为7 +（-4）。然后我们可以交换订单，（-4）+ 7。
因此，如果我们需要，我们实际上可以执行此次交换。所以所有这些都是你需要知道的属性， you need to what you can do with these numbers. But you don't need to remember the word commutative, 这个词不会在测试中。第二个财产是联合财产，能够分组事物。
So if we're adding three numbers we can group them, we can add b + c first or we could add a + b first. Either way of grouping leads to the same answer. Here we can the three and the five first, or we can add the two and the three first. Either way, we get the same answer. Similarly, with multiplication, we can group them b times c first or 次b第一。
如果我们将3和5乘以或者我们乘以2和3，无论哪种方式，我们都会得到相同的答案。 Addition and multiplication are associative. Subtraction and division, in general, are not associative. 再次，您需要知道数字的作用。您不需要知道这个词，关联。
That's not a word that the test will test you on. The distributive property. This one is a big one. First of all multiplication distributes over addition, and it distributes over subtraction. So we could do the addition and subtraction first and then multiply, or we 可以乘以每件的时间，然后执行添加和减法。
所以，例如，我们可以加上两加三，然后乘以六次，或者我们可以做六次，六次三次，然后单独添加这两种产品。 无论哪种方式，我们都得到了相同的答案。与减法类似，我们可以减去三个和七个首先获得 the four, multiply that by twelve, or we could multiply the 12 * 7 and separately multiply the 12 * 3 and then subtract those two products.
无论哪种方式，我们都得到了相同的答案。还有其他特性涉及特殊数字1和0。 第一个是乘以一个，非常简单地分开。当我们乘以或划分一个时，我们不会改变一个数字。 所以我们可以以象征性地表示这种方式，a * 1 = a，a / 1 = a。如果我们乘以或除以一个，绝对没有变化。
乘以零。零的任何东西等于零。 Again, very simple. So a * 0 = 0 * a = 0. 所以我们完全允许乘以零，我们总是得到零，我们不允许划分为零，这是一个非常重要的观点。
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。 所以，如果a * b = 0，则a = 0，或b = 0 ..并注意单词或 在那里，那是那里的一部分数学设备。这实际上是数学的一部分，那不是装饰。
That creates the logical connection between the two equations. The zero product property is very important in factoring and 在解决二际学中，所以你会遇到它。那些在代数模块视频中。 Dividing a number by itself. When any non-zero number is divided by itself, the quotient has to equal 1.
我们可以将其表达为A / A，或者A / A = 1.再次，非常简单。 So in summary, the test is gonna use the word number and this always means any real number, not just positive integers. 保持数字与正整数直接保持数字非常重要。
那些是三种不同的集合。知道这四个术语，总和，差异，产品和商。 Those are actually vocab words you need to know. And know the properties. 再次，我们不需要知道属性的名称，我们只需要了解属性本身。
Here we have the commutative property, the associative property. Here we have the distributive property, which is a really important one, and then all the properties concerning the special numbers one and zero.
