Course: 9.6 理性数字和不合理数字的属性

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理性数字和不合理数字的属性
Justin accused his little brother of being irrational. Jason was telling a story about gnomes living in the cherry trees in front of their house and coming inside at night to raid the kitchen for flour and milk so they could make cherry pies to feed to the chickens… Jason went on and on forever. Was Justin correct?
::贾斯丁指控他弟弟是非理性的。杰森在讲一个故事,讲的是一个生活在樱桃树中的小矮人,他们住在他们家门口,晚上进来搜查厨房,吃面粉和牛奶,以便他们可以做樱桃派喂养鸡。杰森一直不停地说。贾斯丁是对的吗?

In this concept, you will learn how to recognize rational and irrational numbers.
::在这个概念中,你将学会如何认识合理和不合理的数字。

Recognizing Rational and Irrational Numbers
::确认合理和不合理数字

You may already know that a perfect square is a number whose square root is a whole number. Perfect squares have roots that are rational because whole numbers are rational. Most numbers are not perfect squares and have decimal roots.
::您可能已经知道, 完美的正方块是一个数字, 其平方根是一个整数。完美的正方块的根部是理性的, 因为整个数字是理性的。大多数数字不是完美的正方块, 并且小数根。

Some decimal roots are rational , meaning that they end at a certain place value and/or can be written as fractions. Fractions are rational.
::某些小数点根根是理性的, 意思是它们以某个位置值结束, 并且/ 或者可以写成分数。分数是理性的。

$\begin{aligned} 0.25 = \frac{1}{4} \end{aligned}$

Some roots are negative. Whole or fractional negative numbers are rational.
::有些根源是负的,整体或部分负数是合理的。

$\begin{aligned} \sqrt{81} = \pm 9 \end{aligned}$

Some roots are repeating decimals , as indicated by a line over the repeating digits. Repeating decimals are rational because they can be written as fractions.
::有些根正在重复小数, 重复数字的一行就表明了这一点。重复小数是理性的, 因为它们可以写成分数。

$\begin{aligned} 0.6666666 \dots = 0.6 \bar{6} = \frac{2}{3} \end{aligned}$

More than one place value can be repeated.
::可重复多个位置值。

$\begin{aligned} \frac{21}{55} = 0.38181818 \dots = 0.3 \bar{18} \end{aligned}$

An ellipsis, three dots, following a series of digits indicates that the digits continue. Whether or not the value is terminated at some point down the line is unknown, and your calculator is limited to a certain number of places. Unless you can see the repeated digits, assume the number is irrational, and the digits continue forever.
::省略号为 3 点, 在一系列位数后显示数字继续。无论数值是否在线下某个点终止, 未知该值是否在线下某个点终止, 您的计算器仅限于一定数量的位置。除非您能看到重复的位数, 否则假设数字是不合理的, 数字会永远持续。

When you use the ellipsis to indicate that the number goes on forever, you should not use the approximately equal to symbol, $\begin{aligned} \approx \end{aligned}$ . The $\begin{aligned} \approx \end{aligned}$ indicates that the decimal has been rounded.
::当您使用省略号来表示数字会永远持续时,您不应该使用与符号大致相等的符号, 。表示小数点是四舍五入的。

For example:
::例如:

$\begin{aligned} \sqrt{7} = 2.645751311 \dots \end{aligned}$ or $\begin{aligned} \sqrt{7} \approx 2.646 \end{aligned}$
::7= 2.645751311... 或 72.646

The most famous irrational number pi, or $\begin{aligned} π \end{aligned}$ , is a ratio used in working with circles. $\begin{aligned} Pi = 3.141592653589 \dots \end{aligned}$
::最著名的非理性数字pi, 或 , 是在与圈子合作时使用的比例。 Pi=3.141592653589...

Since pi cannot be expressed as a fraction, and the decimals go on forever, the value used for calculations is usually to the hundredths place.
::由于 pi 不能以分数表示, 而小数点会永远持续下去, 计算所用的数值通常为百位数。

It’s acceptable to say $\begin{aligned} pi = 3.14 \end{aligned}$
::说pi=3.14是可以接受的。

Rational and irrational numbers make up the set of real numbers .
::理性和非理性数字构成真实数字的组合。

Let's look at one more example.
::让我们再举一个例子。

Is the $\begin{aligned} \sqrt{900} \end{aligned}$ rational or irrational?
::900是理性的还是非理性的?

First, determine the root. You already know $\begin{aligned} \sqrt{9} = 3 \end{aligned}$ , so you don’t even need to use your calculator.
::首先,确定根。你已经知道9=3,所以你甚至不需要使用计算器。

$\begin{aligned} \sqrt{900} = 30 \end{aligned}$

Next, 30 is a whole number, and whole numbers are rational.
::接下来,30是一个完整的数字, 整个数字是理性的。

The answer is $\begin{aligned} \sqrt{900} \end{aligned}$ is rational.
::答案是900是理性的。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Justin’s little brother, Jason, who was talking about gnomes making cherry pies.
::Justin的小弟弟Jason(Jason)在谈论Gnomes制作樱桃派,

Jason was not making any sense and going on forever with his story. Justin was correct in saying that Jason was being irrational. But someday, Jason might write a best seller!
::贾斯汀说杰森是非理性的。但有一天,杰森会写出最好的卖家!

Example 2
::例2

Is the root rational or irrational?
::根本是理性的还是非理性的?

$\begin{aligned} \sqrt{50} \end{aligned}$

First, 50 is not a perfect square, so the square root is not a whole number.
::首先,50不是一个完美的平方, 所以平方根不是整数。

Next, you can estimate the root to be between 7 and 8 because 49 and 64 are perfect squares.
::接下来,你可以估计根在7到8之间, 因为49和64是完美的方形。

Then, use a calculator.
::然后,使用计算器。

$\begin{aligned} \sqrt{50} = 7.071067812 \dots \end{aligned}$

You can’t see any repeating decimals, and you can’t tell if the decimals end there or go on forever.
::您看不到任何重复的小数位数, 也无法知道小数位数是在那里结束还是永远继续下去。

The answer is $\begin{aligned} \sqrt{50} \end{aligned}$ is irrational.
::答案是50是非理性的。

Example 3
::例3

Tell whether the solution is rational or irrational.
::说明解决办法是理性的还是非理性的。

$\begin{aligned} (8.7)^{2} \end{aligned}$

First, recognize the root of 8.7 as a decimal. The solution is not a perfect square.
::首先,确认8.7的根值为小数点。解决方案不是一个完美的正方形。

Next, use your calculator.
::下一位,用你的计算器

$\begin{aligned} (8.7)^{2} = 75.69 \end{aligned}$

Then, since the decimals end at the hundredths place, it can be written as a fraction.
::然后,由于小数点结束于第一百个位子,小数点可以写成一个分数。

$\begin{aligned} 75.69 = 75 \frac{69}{100} \end{aligned}$

The answer is 75.69 is a rational number .
::答案是75.69是一个合理的数字。

Example 4
::例4

Determine if the root is rational or irrational.
::确定根是理性的还是非理性的。

$\begin{aligned} \sqrt{414} \end{aligned}$

First, estimate. You know that $\begin{aligned} \sqrt{4} = 2 \end{aligned}$ , so $\begin{aligned} \sqrt{400} = 20 \end{aligned}$ . Your answer should be a bit more than 20.
::首先,估计。你知道4=2, 所以400=20。你的答案应该超过20。

Next, use your calculator.
::下一位,用你的计算器

$\begin{aligned} \sqrt{414} = 20.34698995 \dots \end{aligned}$

Then, recognize that the decimal does not repeat.
::然后,请确认小数点后不再重复。

The answer is $\begin{aligned} \sqrt{414} \end{aligned}$ is irrational.
::答案是414是非理性的。

Example 5
::例5

Is the number -6.9 rational or irrational?
::数字 -6.9是理性的还是非理性的?

First, the number is negative. Negative numbers can be rational.
::首先,数字是负数。负数可以是合理的。

Next, the number contains a decimal. Decimals can be rational.
::接下来,数字包含小数点。小数点可以是理性的。

Then, the decimal is terminated at the tenths place.
::然后,小数点在十分之一处终止。

The answer is that -6.9 is a rational number.
::答案是 -6.9是一个合理的数字。

Review
::回顾

Identify each of the following numbers as rational or irrational. Assume that any decimals with an ellipsis are non-repeating and non-terminating.
::将下列数字中的每一数字识别为合理或不合理。假设任何带有省略号的十进制数字都是不重复和不终止的。

$\begin{aligned} .345 \dots \end{aligned}$

2

-9

-122

$\begin{aligned} 3.456 \dots \end{aligned}$

$\begin{aligned} \sqrt{25} \end{aligned}$

$\begin{aligned} \sqrt{16} \end{aligned}$

$\begin{aligned} \sqrt{12} \end{aligned}$

$\begin{aligned} \sqrt{38} \end{aligned}$

-4.56

$\begin{aligned} π \end{aligned}$

$\begin{aligned} - \frac{4}{5} \end{aligned}$

$\begin{aligned} 9.8712 \dots \end{aligned}$

-19

2,345

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

Section outline

General

理性数字和不合理数字的属性

Recognizing Rational and Irrational Numbers ::确认合理和不合理数字

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Recognizing Rational and Irrational Numbers
::确认合理和不合理数字

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源