十进制介绍
成绩单
Intro to decimals. In this video we will talk about decimals.
Place value and arithmetic with decimals. So let me say at the outset, if this is something relatively easy
for you, if you're very comfortable already with adding, subtracting and multiplying, dividing decimals.
然后你可能不需要观看这个视频。您可能只想简单地跳过，以确保一切都是审查。 但如果你已经知道这个内部，不要花时间观看这个整体视频。 This video is designed for people that are a little shaky with decimals, and want a thorough review of the topic.
So let's begin. First think about positive integers. Just ordinary positive integers. If we have the number 256, what that means is 我们2数百,5数以千万计,6。那些are the place values.
So we could write this number in the form 256 equals 2 times 100 plus 5 times 10 plus 6 times 1. 这在其位置值方面扩展了它。请注意，当我们向左移动数字时，位置值乘以 10, and as we move a digit to the right, the place value is divided by 10. So for example, look at that 8 in the middle.
That 8 is in, is in the 1,000th place. If we move to the left one step, to the 7, the 7 is in the 10,000th place, so when 我们从8到7，十进制移动，地位值乘以10。类似地，如果我们向右移动，从8到3， 该8位于1000个地方，3位于第100位。因此，地点值除以10。
小数只需将相同的模式扩展到小数点的右侧。例如，数字，39.0625，这是什么意思？ 首先。这3位了在距离。 我们有一个9个地方。0s是在数度的地方。
The 6 is in the hundredths place. The 2 is in the thousandth place. 而这五个是十分之一的地方。 我们可以按如下方式写入这个号码。我们可以将其写为3 x 10加9次1加0次1/10 + 6次 1/100th plus 2 times 1/1000th plus 5 times 1/10,000th.
因此，这是在地方值方面的扩展形式。小数点右侧的小数点代表负极 10.如此.1是1/10，那是10到负1。 .01是1/100，10到负数2. .001是10到负3,0001是10到 the negative 4, and so forth, you could extend this pattern.
这有点令人困惑。有一些事情有点令人困惑。 First of all, notice the zero to the let of the decimal point. That's purely decorative. I put that in for clarity, that has no mathematical meaning. All that matters here are the, the numbers to the right of the decimal point.
Now some people get confused. Because how many zeros should these things have? 好的话，如果看一个100到10到10到正面2，那就是一个，然后是两个00s，那很清楚，人们认识到 一个100，所以当我们达到100时。请注意，这只是只有零的东西。
It's 0.01. It only has one zero, and that really confuses people. 但请注意，0.01，这是两个小数点。这非常重要。 如果我们看起来有10的正极力量并计算0s的数量。这个数字零与数字相同 of decimal places in the corresponding negative power of 10.
所以再次说，10,000号，10,000号有四个零。嗯，数字1 10000倾向于负4。 这有四个小数点或小数点右侧的四个地方。还注意到额外的零 任何非零位的右都没有意义。他们绝对没有区别。
So for example I write 3.75 if I tack on extra zeros on right. Those are completely meaningless. They mean absolutely nothing. They may have this dim memory from science class that there was something called significant figures, and sometimes these zeros mattered and sometimes they didn't.
我们不会担心这一点。重要的数字不在测试中。 他们在科学中使用数字的方式不是他们在测试中使用的方式。 所以在测试中，这些零是完全毫无意义的。现在我们可以用小数点看算术， 添加和减去小数非常简单。
We simply line up the decimal points and add or subtract vertically. So suppose we have to add these two decimals. We're gonna line them up vertically so the decimal places are lined up. Once they're lined up then we just add. Similarly if we have to subtract these two numbers we're gonna line them up, the decimal places are lined up together.
Now that they're lined up we're simply going to subtract. And that's all there is to adding and subtracting decimals. 对于乘法，步骤1是将位数的右侧的数字数计算在小数点右侧。 所以假设我们会乘以这两个数字。对于每个人来说，我们必须计算数字数量。
第一个因素有两位小数,second one has three decimal places. 现在我们添加那些，2加3是5.产品将有5位小数。 So now we know how many decimal places the product will have, even before we do the multiplication.
现在，我们只是完全忽略小数点，找到产品，好像它们是两个正整数。 整数。假设我们只有两个正整数，625次48。 What is this product? Well how do we find this without a calculator?
事实证明，我们将在稍后的视频中讨论一下。这被称为加倍和减半的技巧。 事实证明当你乘以两个数字时，当然一直允许的一件事，你可以将其中一个乘以2。 And divide the other by 2. That's one thing that's always allowed.
It turns out that is very advantageous to do this, if one of the numbers is even. And the other number is a multiple of 5. Turns out that is very advantageous to do this. This, so what I'm gonna do, I'm gonna divide 48 by 2, that's gonna become 24. I'm gonna multiply 625 by 2. 625 times 2 is 1250, so I change this to 1250 times 24.
它有一个50，所以我会再次乘以2，所以除了24到2，我得到12。 Double 1250, I get 2500. Well, that also has a 500 in it so it makes sense to do this again. I'm gonna double 2500 and halve 12. Half of 12 is 6 and 2500 doubled is 5,000.
Well, now I've got it into a form where it's very easy to multiple in my head. 5,000 times 6. 那是30,000人。所以这是一个非常方便的技巧要记住。 Now, we know we need give decimal places. We found that out from the original problem.
So we put the five rightmost decimals of this product to the right of the decimal. And notice that this product only has five decimal places. It only uses five places, so all of them are gonna get put to the right of the decimal point. So when we multiply 6.25 times 0.048, we get point, and then we write 3 with all the zeroes.
Well of course, all the zeroes don't matter, we can just write this as 0.3, three-tenths. Notice that to find a power, that's a special case of multiplying. So for example, suppose we have to do 0.03, this is 3/100th, cubed. How do we do this? Well, of course this is 0.03 times 0.03 times 0.03.
Notice what we have here that's 2 plus 2 plus 2 six decimal places the product will have six places to the right of the decimal point. 所以我们知道，即使在我们繁殖之前，现在只考虑数字本身3立方体是27所以最后七个必须借给右边的六个地方。 Of the decimal point it must land in the one million. Place.
So, 0.03 cubed is 0.000027. That seven is six places to the right 小数点，这是27百万分之。现在，小数分裂。 要开始，只需将两个数字的小数移动到右侧，然后继续直到 分母，除数，IA是一个整数。
When we move a decimal one place to the right, say 2.45 to 24.5, we are multiplying the number by 10. 由于我们将分子和分母乘以10，因此商值保持不变。 Essentially what we're doing is we're multiplying by 10 over 10 which is 1. So for example, suppose we start with this or we can start sliding that decimal point to the right.
所以我必须再次绘制一个插槽。此时分子现在是一个 integer, but it's really the denominator that matters. So we have to keep sliding to the right. What this does is adds zeroes at the end of the number in the numerator. And finally we get that number in the denominator down to an integer.
现在我们已经5,600分了7。那很容易做到这只是800。 所以现在让我们看看另一个例子。假设我们划分了这两步。 再次开始向右滑动到右侧幻灯片。我们现在会注意到我们可以继续滑动 正确但通知0.25这是一个季度所以请注意，即取消这将是一个非常简单的方法 是将分子和分母乘以4。
If we multiply by 4, in the denominator, quarter times 4 is 1, so multiply the numerator 到4，因为我们得到了超过1的东西，所以它只是等于.0052，所以52个十万。 In summary, we talked about place values in decimals. And we talked about how to subtract, add and subtract, multiply, and divide decimal numbers.
阅读完整成绩单
然后你可能不需要观看这个视频。您可能只想简单地跳过，以确保一切都是审查。 但如果你已经知道这个内部，不要花时间观看这个整体视频。 This video is designed for people that are a little shaky with decimals, and want a thorough review of the topic.
So let's begin. First think about positive integers. Just ordinary positive integers. If we have the number 256, what that means is 我们2数百,5数以千万计,6。那些are the place values.
So we could write this number in the form 256 equals 2 times 100 plus 5 times 10 plus 6 times 1. 这在其位置值方面扩展了它。请注意，当我们向左移动数字时，位置值乘以 10, and as we move a digit to the right, the place value is divided by 10. So for example, look at that 8 in the middle.
That 8 is in, is in the 1,000th place. If we move to the left one step, to the 7, the 7 is in the 10,000th place, so when 我们从8到7，十进制移动，地位值乘以10。类似地，如果我们向右移动，从8到3， 该8位于1000个地方，3位于第100位。因此，地点值除以10。
小数只需将相同的模式扩展到小数点的右侧。例如，数字，39.0625，这是什么意思？ 首先。这3位了在距离。 我们有一个9个地方。0s是在数度的地方。
The 6 is in the hundredths place. The 2 is in the thousandth place. 而这五个是十分之一的地方。 我们可以按如下方式写入这个号码。我们可以将其写为3 x 10加9次1加0次1/10 + 6次 1/100th plus 2 times 1/1000th plus 5 times 1/10,000th.
因此，这是在地方值方面的扩展形式。小数点右侧的小数点代表负极 10.如此.1是1/10，那是10到负1。 .01是1/100，10到负数2. .001是10到负3,0001是10到 the negative 4, and so forth, you could extend this pattern.
这有点令人困惑。有一些事情有点令人困惑。 First of all, notice the zero to the let of the decimal point. That's purely decorative. I put that in for clarity, that has no mathematical meaning. All that matters here are the, the numbers to the right of the decimal point.
Now some people get confused. Because how many zeros should these things have? 好的话，如果看一个100到10到10到正面2，那就是一个，然后是两个00s，那很清楚，人们认识到 一个100，所以当我们达到100时。请注意，这只是只有零的东西。
It's 0.01. It only has one zero, and that really confuses people. 但请注意，0.01，这是两个小数点。这非常重要。 如果我们看起来有10的正极力量并计算0s的数量。这个数字零与数字相同 of decimal places in the corresponding negative power of 10.
所以再次说，10,000号，10,000号有四个零。嗯，数字1 10000倾向于负4。 这有四个小数点或小数点右侧的四个地方。还注意到额外的零 任何非零位的右都没有意义。他们绝对没有区别。
So for example I write 3.75 if I tack on extra zeros on right. Those are completely meaningless. They mean absolutely nothing. They may have this dim memory from science class that there was something called significant figures, and sometimes these zeros mattered and sometimes they didn't.
我们不会担心这一点。重要的数字不在测试中。 他们在科学中使用数字的方式不是他们在测试中使用的方式。 所以在测试中，这些零是完全毫无意义的。现在我们可以用小数点看算术， 添加和减去小数非常简单。
We simply line up the decimal points and add or subtract vertically. So suppose we have to add these two decimals. We're gonna line them up vertically so the decimal places are lined up. Once they're lined up then we just add. Similarly if we have to subtract these two numbers we're gonna line them up, the decimal places are lined up together.
Now that they're lined up we're simply going to subtract. And that's all there is to adding and subtracting decimals. 对于乘法，步骤1是将位数的右侧的数字数计算在小数点右侧。 所以假设我们会乘以这两个数字。对于每个人来说，我们必须计算数字数量。
第一个因素有两位小数,second one has three decimal places. 现在我们添加那些，2加3是5.产品将有5位小数。 So now we know how many decimal places the product will have, even before we do the multiplication.
现在，我们只是完全忽略小数点，找到产品，好像它们是两个正整数。 整数。假设我们只有两个正整数，625次48。 What is this product? Well how do we find this without a calculator?
事实证明，我们将在稍后的视频中讨论一下。这被称为加倍和减半的技巧。 事实证明当你乘以两个数字时，当然一直允许的一件事，你可以将其中一个乘以2。 And divide the other by 2. That's one thing that's always allowed.
It turns out that is very advantageous to do this, if one of the numbers is even. And the other number is a multiple of 5. Turns out that is very advantageous to do this. This, so what I'm gonna do, I'm gonna divide 48 by 2, that's gonna become 24. I'm gonna multiply 625 by 2. 625 times 2 is 1250, so I change this to 1250 times 24.
它有一个50，所以我会再次乘以2，所以除了24到2，我得到12。 Double 1250, I get 2500. Well, that also has a 500 in it so it makes sense to do this again. I'm gonna double 2500 and halve 12. Half of 12 is 6 and 2500 doubled is 5,000.
Well, now I've got it into a form where it's very easy to multiple in my head. 5,000 times 6. 那是30,000人。所以这是一个非常方便的技巧要记住。 Now, we know we need give decimal places. We found that out from the original problem.
So we put the five rightmost decimals of this product to the right of the decimal. And notice that this product only has five decimal places. It only uses five places, so all of them are gonna get put to the right of the decimal point. So when we multiply 6.25 times 0.048, we get point, and then we write 3 with all the zeroes.
Well of course, all the zeroes don't matter, we can just write this as 0.3, three-tenths. Notice that to find a power, that's a special case of multiplying. So for example, suppose we have to do 0.03, this is 3/100th, cubed. How do we do this? Well, of course this is 0.03 times 0.03 times 0.03.
Notice what we have here that's 2 plus 2 plus 2 six decimal places the product will have six places to the right of the decimal point. 所以我们知道，即使在我们繁殖之前，现在只考虑数字本身3立方体是27所以最后七个必须借给右边的六个地方。 Of the decimal point it must land in the one million. Place.
So, 0.03 cubed is 0.000027. That seven is six places to the right 小数点，这是27百万分之。现在，小数分裂。 要开始，只需将两个数字的小数移动到右侧，然后继续直到 分母，除数，IA是一个整数。
When we move a decimal one place to the right, say 2.45 to 24.5, we are multiplying the number by 10. 由于我们将分子和分母乘以10，因此商值保持不变。 Essentially what we're doing is we're multiplying by 10 over 10 which is 1. So for example, suppose we start with this or we can start sliding that decimal point to the right.
所以我必须再次绘制一个插槽。此时分子现在是一个 integer, but it's really the denominator that matters. So we have to keep sliding to the right. What this does is adds zeroes at the end of the number in the numerator. And finally we get that number in the denominator down to an integer.
现在我们已经5,600分了7。那很容易做到这只是800。 所以现在让我们看看另一个例子。假设我们划分了这两步。 再次开始向右滑动到右侧幻灯片。我们现在会注意到我们可以继续滑动 正确但通知0.25这是一个季度所以请注意，即取消这将是一个非常简单的方法 是将分子和分母乘以4。
If we multiply by 4, in the denominator, quarter times 4 is 1, so multiply the numerator 到4，因为我们得到了超过1的东西，所以它只是等于.0052，所以52个十万。 In summary, we talked about place values in decimals. And we talked about how to subtract, add and subtract, multiply, and divide decimal numbers.
