Course: 7.4 分类真实数字

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum

分类 Real 数字

lesson content

Faith has been planting a square flower bed for many years and has decided that the bedding plants are getting too expensive to fill such an area. This year she is going to plant a circular flower bed within the original square area as shown below:
::信仰多年来一直在种植一个平方花床,并且决定床铺的植物越来越昂贵,无法填满这样的面积。今年,她将在原来的广场内种植一个圆形花床,如下表所示:

As Faith reviews her plan, she has decided to find some exact measurements so she can figure out just how much smaller the new circular flower bed will be compared to the original square flower bed. However, Faith is unsure of what measurements she has to figure out – “I hope I at least come out with some rational numbers!”
::信仰审查她的计划时,她决定寻找一些精确的测量方法,这样她就能知道新的圆形花床比原来的平方花床要小多少。然而,信仰不确定她必须找出什么测量方法 — — “我希望我至少拿出一些合理的数字! ”

In this concept, you will learn to classify real numbers.
::在这个概念中,你会学会对真实数字进行分类。

Real Numbers
::实际数字

All numbers belong to the set of numbers known as the real number system . The real number system consists of every number you have ever dealt with since you were old enough to count. The numbers in the real number system are divided into two main groups. One group is called the rational numbers and the other is called the irrational numbers . The set of irrational numbers consists of all numbers that are not rational. This set of irrational numbers includes those numbers that cannot be written as the ratio of two integers , decimal numbers that are non-terminating and decimals that do not have a repeating pattern of digits. For example, $\begin{align*}pi (\pi), \sqrt{2}, -2.345 876 921\ldots\end{align*}$ are irrational numbers.
::所有数字都属于称为真实数字系统的一组数字。真实数字系统包含您从老到可以计数以来处理过的每一个数字。实际数字系统中的数字分为两大组。一个组称为理性数字, 另一个组称为非理性数字。非理性数字组由所有不合理的数字组成。这个非理性数字组包括无法以两个整数、不终止的十进数和没有重复数字模式的十进数的比例书写的数字。例如, pi, 2, 2- 2.345876921... 是非理性数字。

The set of rational numbers includes natural numbers, whole numbers, integers, numbers that can be expressed as the ratio of two integers, decimal number that terminate and decimal numbers that have a repeating pattern of digits. The natural numbers are the set of positive integers. For example, 1, 2, 3… are all natural numbers. The whole numbers are the natural numbers and zero. For example, 0, 1, 2, 3… are all whole numbers. The integers are the whole numbers and their opposites. For example, -2, -1, 0, 1, 2… are all integers. A rational number is any number that can be expressed in the form $\begin{align*}\frac{a}{b}\end{align*}$ where $\begin{align*}b \neq 0\end{align*}$ . When a rational number is expressed as a decimal then the decimal will terminate (end) or it will have a repeating pattern of digits. For example, $\begin{align*}\frac{1}{2},\sqrt{81},-7.456 \ 545 \ 654\end{align*}$ are all rational numbers.
::理性数字集包括自然数字、整数、整数、整数、以两个整数之比表示的数字、终止数的十进制数和具有重复数字模式的十进制数。自然数字是正整数组。例如,1、2、3...都是自然数。整数是自然数,零。0、1、2、3...就是整数。整数是整数,反数是整数。例如,2-1、2、1、0、1、1、2...都是整数。理性数字是可以用b-b表示的任何数字。当一个理性数字以十进制表示时,小数将终止(结束),或者数字将有一个重复模式。例如,12,81,-7.456,545,654都是理性数字。

When you classify numbers remember that they can often belong to more than one set of numbers. If you think of the number 3, you can call it a natural number, a whole number, an integer and a rational number.
::当对数字进行分类时,记住它们通常可以属于多个一组数字。如果想到数字3,可以称之为自然数字、整数、整数和合理数字。

The following table may help you to better understand the real number system.
::下表可能有助于您更好地了解实际数字系统。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Faith and her circular flower bed. She wants to compare the new circular flower bed to the original square one using exact measurements. First, Faith should compare the perimeter of the square to the circumference of the circle. Then, she should compare the area of the square to the area of the circle.
::早些时候,有人给了你关于Faith和她的圆花床的疑问。她想用精确的测量方法将新的圆花床与原来的广场比较。首先,Faith应该将广场的周界与圆圈的环绕比较。然后,她应该将广场的面积与圆圈的区域比较。

First, determine the diameter of the circle which will be the side length of the square.
::首先,确定圆的直径,即方形的侧长。

The diameter of a circle is two times the length of the radius. The radius of the circle is 3 feet.
::圆的直径是半径的2倍。圆的半径是3英尺。

\begin{aligned} \begin{array}{rcl} d & = & 2 r \\ d & = & 2 (3) \\ d & = & 6 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} d &=& 2r \\ d &=& 2(3) \\ d &=& 6 \end{array}\end{align*}$
::d=2rd=2(3)d=6

The answer is 6.
::答案是6个

The diameter of the circle is 6 feet. Each side of the square is also 6 feet in length.
::圆的直径是6英尺,方形的两侧也是6英尺长。

The answer of 6 is a natural number, a whole number, an integer and a rational number.
::6的回答是自然数字、整数、整数和合理数字。

Next, determine the perimeter of the square. The perimeter of the square is the distance around its outer edges and can be found by adding the lengths of each side or by simply multiplying the side length by four.
::接下来,确定广场的周界。广场的周界是其外部边缘的距离,可以通过增加每侧的长度或简单地将侧长乘以4来找到。

\begin{aligned} \begin{array}{rcl} P_{square} & = & 4 s \\ P_{square} & = & 4 (6) \\ P_{square} & = & 24 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} P_{\text{square}} &=& 4s \\ P_{\text{square}} &=& 4(6) \\ P_{\text{square}} &=& 24 \end{array}\end{align*}$
::Psquare=4sPsquare=4(6)Psquare=24

The answer is 24.
::答案是24个。

The perimeter of the square is 24 feet.
::广场的周界是24英尺

The answer of 24 is a natural number, a whole number, an integer and a rational number.
::24的答案是一个自然数字、一个整数、一个整数和一个合理数字。

Next, determine the circumference of the circle. The perimeter of a circle is known as its circumference and is the distance around the outer edge of the circle. The perimeter of a circle can be found by multiplying the diameter by $\begin{align*}\pi\end{align*}$ .
::下一步, 确定圆的环绕。圆的周围称为环绕, 是圆外边缘的距离。圆的周围可以通过将直径乘以 + 来找到。

\begin{aligned} \begin{array}{rcl} C & = & π d \\ C & = & (3.141592654) (6) \\ C & = & 18.84955592 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} C &=& \pi d \\ C &=& (3.141592654)(6) \\ C &=& 18.84955592 \end{array}\end{align*}$
::CádC= (3141592654)(6) C= 18.84955592

Using the value for $\begin{align*}\pi\end{align*}$ from the calculator (3.141592654) gives the answer 18.84955592.
::使用计算器( 3.141592654) 中的值,回答是 18.84955592。

The circumference of the circle is 18.84955592 feet.
::圆环的周长为18.8455592英尺。

The answer of 18.84955592 is an irrational number. It is a non-terminating, non-repeating decimal.
::18.84955592的回答是一个不合理的数字,是一个不终止的、不重复的十进制数字。

Next, determine the area of the square. The area of the square can be found using the formula :
::下一步,确定方块的区域。可以使用公式找到方块的区域:

$\begin{align*}A=s^2\end{align*}$ where ‘ $\begin{align*}s\end{align*}$ ’ is the side length of the square.
::A=2,其中`s ' 是方形的侧长。

\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & (6)^{2} \\ A & = & 36 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (6)^2 \\ A &=& 36 \end{array}\end{align*}$
::A=S2A=(6)2A=36

The answer is 36.
::答案是36岁

The area of the square flower bed is $\begin{align*}36 \ ft^2\end{align*}$ .
::方花床面积为36平方英尺。

The answer of 36 is a natural number, a whole number, an integer and a rational number.
::36的回答是一个自然数字、一个整数、一个整数和一个合理数字。

Next, determine the area of the circle. The area of the circle can be found using the formula:
::下一步,确定圆的面积。圆的面积可以使用公式找到:

$\begin{align*}A=\pi r^2\end{align*}$ where ‘ $\begin{align*}r\end{align*}$ ’ is the radius of the circle.
::“r”是圆的半径。

\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & (3.141592654) (3)^{2} \\ A & = & (3.141592654) (9) \\ A & = & 28.27433388 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} A &=& \pi r^2 \\ A &=& (3.141592654)(3)^2 \\ A &=& (3.141592654) (9) \\ A &=& 28.27433388 \end{array}\end{align*}$
::r2A=(3141592654)(3)2A=(3141592654)(9)A=28.2743388

The answer is 28.27433388.
::答案是28.27433388。

The area of the circular flower bed is $\begin{align*}28.27433388 \ ft^2\end{align*}$ .
::圆圆花床面积为28.27433388平方英尺。

Using the value for $\begin{align*}\pi\end{align*}$ from the calculator (3.141592654) gives the answer 28.27433388.
::使用计算器( 3.141592654) 中的 __ 值,回答为 28. 27433388。

The answer of 28.27433388 is an irrational number. It is a non-terminating, non-repeating decimal.
::回答28.27433388是一个不合理的数字,是一个不终止的、不重复的十进制数字。

Next, subtract the area of the circular flower bed from the area of the square flower bed. Round the area of the circular flower bed to the nearest tenth and then perform the subtraction .
::接下来,将圆花床面积从方花床面积中减去。将圆花床面积绕到最近的十分之一,然后进行减法。

The difference in the area of the two flower beds is $\begin{align*}36.0 \ ft^2 - 28.3 \ ft^2=7.7 \ ft^2\end{align*}$
::两张花花床面积的差异为36.0平方英尺-2-28.3平方英尺=7.7平方英尺。

The answer of 7.7 is a rational number since it is a terminating decimal.
::7.7的回答是一个合理的数字,因为它是一个终止的小数小数。

Example 2
::例2

Let’s look at classifying some numbers.
::让我们来看看一些数字的分类。

For each of the following numbers displayed in the table, indicate the number set or sets to which they belong.
::对于表中显示的下列数字,请标明其所属的数组或数组。

Classifying Numbers Table
Number	Natural	Whole	Integer	Rational	Irrational
$\begin{align}\frac{22}{7}\end{align}$				$\begin{align}\star\end{align}$
$\begin{align}1.141 \ 141 \ 114 \ldots\end{align}$					$\begin{align}\star\end{align}$
$\begin{align}\frac{17}{4}\end{align}$				$\begin{align}\star\end{align}$
-6			$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$
$\begin{align}\frac{\pi}{2}\end{align}$					$\begin{align}\star\end{align}$
0		$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$
9	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$

$\begin{align*}\frac{22}{7}\end{align*}$ is a rational number because it is written as the ratio of two integers.
::227是一个合理的数字,因为它被写成两个整数之比。

$\begin{align*}1.141 \ 141 \ 114 \ldots\end{align*}$ is an irrational number because it is a non-terminating decimal.
::1.141 141 114... 是一个不合理的数字因为它是一个不终止的小数。

$\begin{align*}\frac{17}{4}\end{align*}$ is a rational number because it is written as the ratio of two integers.
::174是一个合理的数字,因为它是两个整数之比。

-6 is an integer because it is the opposite of the whole number 6. It is a rational number because it can be expressed as the ratio of two integers as $\begin{align*} -\frac{6}{1}\end{align*}$ .
::-6是一个整数,因为它与整个数字6.相反,它是一个合理的数字,因为它可以表示为两个整数之比-61。

$\begin{align*}\frac{\pi}{2}\end{align*}$ is an irrational number because pi is a non-terminating decimal and is an irrational number.
::2是一个不合理的数字,因为 pi 是非终止的十进制小数, 并且是一个不合理的数字。

0 is a whole number, an integer and a rational number. Zero is included in each of these number sets.
::0 是一个整数、一个整数和一个合理数。每个数字组都包含零。

9 is a natural number, a whole number, an integer and a rational number. It is included in each of these sets and can be expressed as the ratio of two integers as $\begin{align*}\frac{9}{1}\end{align*}$ .
::9 是一个自然数字、一个整数、一个整数和一个合理数字。它包含在这些组中的每一组中,可以以两个整数之比表示为91。

The following table may help you to understand the real number system.
::下表可能会帮助您了解实际数字系统。

Example 3
::例3

For each of the following statements state if they are sometimes true, always true, or never true.
::下列每一声明都说明这些声明有时是真实的、总是真实的,还是从来不真实的。

Whole numbers are integers.
::整数是整数。

Integers are whole numbers.
::整数是整数

If a number is an integer, then it is also a rational number.
::如果数字是一个整数,那么它也是一个合理的数字。

The number $\begin{align*}3\pi\end{align*}$ is a rational number.
::数字3是一个合理的数字。

A number can be both rational and irrational
::一个数字可以是理性的,也可以是非理性的

If a number is rational, then it is also an integer.
::如果一个数字是合理的,那么它也是一个整数。

First, read each statement and review what numbers are included in each of the number sets.
::首先,阅读每份说明,并审查每一组数字中包含的数字。

Next, decide your answer based on what you have reviewed.
::接下来,根据你审查过的内容决定你的答复。

Then, justify your answer.
::那么,证明你的回答是有道理的。

a) The numbers that belong to the set of integers are the whole numbers and their opposites.
:a) 属于整数组的数字是整数及其反向数字。

Therefore it is always true that the whole numbers are integers.
::因此,所有数字都是整数,这始终是事实。

b) Integers are only whole numbers if the numbers are positive whole numbers or zero. Integers that are negative do not belong to the set of whole numbers.
:b) 如果整数为正整数或零,整数仅为整数,则整数为整数,负数不属整数组。

Therefore it is sometimes true that integers are whole numbers.
::因此,有时整数确实是整数。

c) The set of rational numbers includes any number that can be expressed as the ratio of two integers.
:c) 一组合理数字包括以两个整数之比表示的任何数字。

Therefore it is always true that if a number is an integer, then it is also a rational number.
::因此,如果一个数字是一个整数,那么它也是合理的数字,这始终是事实。

d) The value of $\begin{align*}\pi \end{align*}$ expressed as a decimal is a non-terminating decimal with no repeated pattern of digits. $\begin{align*}\pi\end{align*}$ belongs to the set of irrational numbers.
:d) 以小数点表示的数值为小数点中未终止的小数点,不重复数字模式。属于非理性数字组。

Therefore it is never true that $\begin{align*}3 \pi\end{align*}$ is a rational number.
::因此, " 3 " 是一个合理的数字从来都不是事实。

e) A number that belongs to the set of rational numbers cannot belong to the set of irrational numbers. Any number that cannot be written as a rational number is an irrational number.
:e) 属于一组合理数字的号码不能属于非理性数字,任何不能以合理数字书写的数字都是非理性数字。

Therefore it is never true that a number can be both rational and irrational.
::因此,一个数字不可能既合理又不合理,这从来都不是事实。

f) If a number is rational, then it is also a natural number or a whole number or an integer.
:f) 如果数字是合理的,那么它也是一个自然数字或整数或整数。

Therefore, it is sometimes true that if a number is rational, then it is an integer.
::因此,有时确实,如果数字是合理的,那么数字就是一个整数。

Example 4
::例4

For each of the following numbers displayed in the table, indicate the number set or sets to which they belong. Justify your answer.
::对于表格中显示的下列数字,请标明其所属的数组或数组。请说明答案的理由。

First, review what numbers are included in each of the number sets.
::首先,审查每一组数字中包含的数字。

Next, decide your answer based on what you have reviewed.
::接下来,根据你审查过的内容决定你的答复。

Then, justify your answer.
::那么,证明你的回答是有道理的。

Remember a number can belong to one or more than one number set.
::记住一个数字可以属于一个或多个数字集。

Classifying Numbers Table
Number	Natural	Whole	Integer	Rational	Irrational	Justification ::理由说明
6.18						Terminating decimal ::终止小数
0.89563...					$\begin{align}\star\end{align}$	Non-terminating decimal
12	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$		Can be expressed as $\begin{align}+12, 12,\frac{12}{1}\end{align}$
-7			$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$		Can be expressed as $\begin{align}-7,- \frac{7}{1}\end{align}$
$\begin{align}-\frac{5}{9}\end{align}$				$\begin{align}\star\end{align}$		fraction
$\begin{align}\sqrt{36}\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$		Value is 6
$\begin{align}-\sqrt{5.5}\end{align}$					$\begin{align}\star\end{align}$	Non-repeating, non-terminating decimal
$\begin{align}\frac{1}{3}\end{align}$				$\begin{align}\star\end{align}$		fraction

Review
::回顾

Classify each of the following numbers as real, whole, integer, rational or irrational. Some numbers will have more than one classification.
::将以下每个数字分类为真实、完整、整数、理性或不合理。有些数字将有一个以上的分类。

1. 3.45

2. -9

3. 1270

4. 1.232323

5. $\begin{align*}\frac{4}{5}\end{align*}$

6. -232 323

7. -98

8. 1.98

9. $\begin{align*}\sqrt{16}\end{align*}$

10. $\begin{align*}\sqrt{2}\end{align*}$

State whether the following statements are true or false.
::说明以下陈述是真实的还是虚假的。

11. An irrational number can also be a real number.
::11. 非理性数字也可以是一个实际数字。

12. An irrational number is a real number and an integer.
::12. 不合理数字是一个实际数字和一个整数。

13. A whole number is also an integer.
::13. 整个数字也是整数。

14. A decimal is considered a real number and a rational number.
::14. 小数点为实际数字和合理数字。

15. A negative decimal can still be considered an integer.
::15. 负十进制仍可被视为整数。

16. An irrational number is a terminating decimal.
::16. 不合理的数字是终止小数。

17. A radical is always an irrational number.
::17. 激进总是不合理的数字。

18. Negative whole numbers are integers and are also rational numbers.
::18. 整个负数是整数,也是合理数字。

19. Pi is an example of an irrational number.
::19. Pi是一个非理性数字的例子。

20. A repeating decimal is also a rational number.
::20. 重复小数点也是合理的数字。

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。

Number	Natural	Whole	Integer	Rational	Irrational
$\begin{align}\frac{22}{7}\end{align}$				$\begin{align}\star\end{align}$
$\begin{align}1.141 \ 141 \ 114 \ldots\end{align}$					$\begin{align}\star\end{align}$
$\begin{align}\frac{17}{4}\end{align}$				$\begin{align}\star\end{align}$
-6			$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$
$\begin{align}\frac{\pi}{2}\end{align}$					$\begin{align}\star\end{align}$
0		$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$
9	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$

Number	Natural	Whole	Integer	Rational	Irrational	Justification ::理由说明
6.18						Terminating decimal ::终止小数
0.89563...					$\begin{align}\star\end{align}$	Non-terminating decimal
12	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$		Can be expressed as $\begin{align}+12, 12,\frac{12}{1}\end{align}$
-7			$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$		Can be expressed as $\begin{align}-7,- \frac{7}{1}\end{align}$
$\begin{align}-\frac{5}{9}\end{align}$				$\begin{align}\star\end{align}$		fraction
$\begin{align}\sqrt{36}\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$	$\begin{align}\star\end{align}$		Value is 6
$\begin{align}-\sqrt{5.5}\end{align}$					$\begin{align}\star\end{align}$	Non-repeating, non-terminating decimal
$\begin{align}\frac{1}{3}\end{align}$				$\begin{align}\star\end{align}$		fraction

Section outline

General

分类 Real 数字

Real Numbers ::实际数字

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)