Course: 2.1 理性数字的属性 | The Way To Learn

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合理数字属性
Properties of Rational Numbers
::合理数字属性

One day, Jason leaves his house and starts walking to school. After three blocks, he stops to tie his shoe and leaves his lunch bag sitting on the path. Two blocks farther on, he realizes his lunch is missing and goes back to get it. After picking up his lunch, he walks six more blocks to arrive at school. How far is the school from Jason’s house? And how far did Jason actually walk to get there?
::有一天,杰森走出家门,开始走去上学。在三个街区后,他停下来系鞋带,把午餐袋留在路上。两个街区后,他发现他的午餐不见了,然后又回去取。在吃午饭后,他走过六个街区就到了学校。学校离杰森家有多远?杰森到底走了多远才到那里去?

Graph and Compare Integers
::图表和比较整数

Integers are the counting numbers (1, 2, 3...), the negative opposites of the counting numbers (-1, -2, -3...), and zero . There are an infinite number of integers and examples are 0, 3, 76, -2, -11, and 995.
::整数是数数(1, 2, 3...),负对数数( 1, 2, 3...) 和零。有无限数量的整数和示例为 0, 3, 76, 2- 2, 11, 995。

Comparing Numbers
::比较数字

Compare the numbers 2 and -5.
::比较数字2和5。

When we plot numbers on a number line , the greatest number is farthest to the right, and the least is farthest to the left.
::当我们在数字线上绘制数字时,最大数字最偏右,最小数字最左。

In the diagram above, we can see that 2 is farther to the right on the number line than -5, so we say that 2 is greater than -5. We use the symbol “>” to mean “greater than”, so we can write 2 > -5.
::在上图中,我们可以看到数字线上的右边距离2比5远,所以我们说2大于5。我们用符号“ >”表示“大于”,这样我们可以写2 > -5。

Classifying Rational Numbers
::分类逻辑数字

When we divide an integer $\begin{aligned} a \end{aligned}$ by another integer $\begin{aligned} b \end{aligned}$ (as long as $\begin{aligned} b \end{aligned}$ is not zero) we get a rational number . It’s called this because it is the ratio of one number to another, and we can write it in fraction form as $\begin{aligned} \frac{a}{b} \end{aligned}$ . (You may recall that the top number in a fraction is called the numerator and the bottom number is called the denominator .)
::当我们将整数a除以另一个整数b(如果b不是零)时,我们就会得到一个合理的数字。这个数字被称为这个数字,因为它是一个数字对另一个数字的比例,我们可以以 ab 的形式以分数形式写成。 (你可能记得,一个分数的顶数被称为数字,底数被称为分母。 )

You can think of a rational number as a fraction of a cake. If you cut the cake into $\begin{aligned} b \end{aligned}$ slices, your share is $\begin{aligned} a \end{aligned}$ of those slices.
::你可以把一个理性数字看作是蛋糕的一小部分。如果你把蛋糕切成B片,你的份额就是这些切片的一部分。

For example, when we see the rational number $\begin{aligned} \frac{1}{2} \end{aligned}$ , we can imagine cutting the cake into two parts. Our share is one of those parts. Visually, the rational number $\begin{aligned} \frac{1}{2} \end{aligned}$ looks like this:
::例如,当我们看到12号理性数字时,我们可以想象把蛋糕切成两部分。我们的份额就是其中一部分。从视觉上看,12号理性数字是这样的:

With the rational number $\begin{aligned} \frac{3}{4} \end{aligned}$ , we cut the cake into four parts and our share is three of those parts. Visually, the rational number $\begin{aligned} \frac{3}{4} \end{aligned}$ looks like this:
::我们把蛋糕切成四个部分我们的份额是其中三个部分。从视觉上看,34号看起来是这样的:

The rational number $\begin{aligned} \frac{9}{10} \end{aligned}$ represents nine slices of a cake that has been cut into ten pieces. Visually, the rational number $\begin{aligned} \frac{9}{10} \end{aligned}$ looks like this:
::理性数字910代表一个蛋糕的九片, 切成十片。从视觉上看,理性数字910是这样的:

Proper fractions are rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one.
::当分子小于分母时,适当分数是合理数字。适当分数代表数字小于1。

Improper fractions are rational numbers where the numerator is greater than or equal to the denominator. An improper fraction can be rewritten as a mixed number – an integer plus a proper fraction. For example, $\begin{aligned} \frac{9}{4} \end{aligned}$ can be written as $\begin{aligned} 2 \frac{1}{4} \end{aligned}$ . An improper fraction represents a number greater than or equal to one.
::错误的分数是当分子大于或等于分母时的合理数字。错误的分数可以重写为混合数 — — 整数加适当分数。比如,94个分数可以写为214。错误的分数代表大于或等于一个数字。

are two fractions that represent the same amount . For example, look at a visual representation of the rational number $\begin{aligned} \frac{2}{4} \end{aligned}$ , and one of the number $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::代表相同数量的两个分数。例如,查看理性数字24的直观表示和12数字之一的直观表示。

You can see that the shaded regions are the same size, so the two fractions are equivalent. We can convert one fraction into the other by reducing the fraction, or writing it in lowest terms. To do this, we write out the prime factors of both the numerator and the denominator and cancel matching factors that appear in both the numerator and denominator.
::您可以看到阴影区域大小相同, 因此两个分数相等。我们可以通过减少分数或以最低值写入, 将一个分数转换成另一个分数。要做到这一点, 我们写出分子和分母的质因数, 并取消在分子和分母中出现的匹配因数。

$\begin{aligned} \frac{2}{4} = \frac{2 \cdot 1}{2 \cdot 2 \cdot 1} = \frac{1}{2 \cdot 1} = \frac{1}{2} \end{aligned}$

Reducing a fraction doesn’t change the value of the fraction—it just simplifies the way we write it. Once we’ve canceled all common factors, the fraction is in its simplest form .
::减少一个分数并不能改变分数的价值 — — 它只是简化了我们写它的方式。一旦我们取消了所有共同因素,这个分数就以最简单的形式出现。

Classifying and Simplifying Numbers
::分类和简化数字

Classify and simplify the following rational numbers
::分类和简化下列合理数字

a) $\begin{aligned} \frac{3}{7} \end{aligned}$
:a) 37

3 and 7 are both prime, so we can't factor them. That means $\begin{aligned} \frac{3}{7} \end{aligned}$ is already in its simplest form. It is also a proper fraction.
::3 和 7 两者都是质数, 所以我们无法计数它们。这意味着 37 已经处于最简单的形式。它也是一个适当的分数。

b) $\begin{aligned} \frac{9}{3} \end{aligned}$
:b) 93

$\begin{aligned} \frac{9}{3} \end{aligned}$ is an improper fraction because $\begin{aligned} 9 > 3 \end{aligned}$ . To simplify it, we factor the numerator and denominator and cancel: $\begin{aligned} \frac{3 \cdot 3}{3 \cdot 1} = \frac{3}{1} = 3 \end{aligned}$ .
::93是一个不适当的分数, 因为 9> 3。为了简化它, 我们乘以分子和分母, 取消: 3331=31=3 。

Order Rational Numbers
::顺序有理数字

Ordering rational numbers is simply a matter of arranging them by increasing value—least first and greatest last.
::合理数字的排序仅仅是通过增加价值来安排它们的问题,最起码的,最起码的,最起码的,是最后的价值。

Ordering Fractions
::顺序分数

1. Put the following fractions in order from least to greatest : $\begin{aligned} \frac{1}{2}, \frac{3}{4}, \frac{2}{3} \end{aligned}$
::1. 将下列分数排列为从最小到最大:12,34,23

$\begin{aligned} \frac{1}{2} < \frac{2}{3} < \frac{3}{4} \end{aligned}$

Simple fractions are easy to order—we just know, for example, that one-half is greater than one quarter, and that two thirds is bigger than one-half. But how do we compare more complex fractions?
::简单分数容易排序 — — 我们只是知道,比如说,一半大于四分之一,三分之二大于一半。但我们如何比较更复杂的分数呢?

2. Which is greater, $\begin{aligned} \frac{3}{7} \end{aligned}$ or $\begin{aligned} \frac{4}{9} \end{aligned}$ ?
::2. 哪一个更大,37or49?

In order to determine this, we need to rewrite the fractions so we can compare them more easily. If we rewrite each as an equivalent fraction so that both have the same denominators, then we can compare them directly. To do this, we need to find the lowest common denominator (LCD), or the least common multiple of the two denominators.
::为了确定这一点, 我们需要重写分数, 以便更容易地比较它们。如果我们重写每个分数, 以便两者都有相同的分数, 那么我们就可以直接比较它们。要做到这一点, 我们需要找到最小的公分数, 或者两个分数中最小的公分数。

The lowest common multiple of 7 and 9 is 63. Our fraction will be represented by a shape divided into 63 sections. This time we will use a rectangle cut into 9 by $\begin{aligned} 7 = 63 \end{aligned}$ pieces.
::最小常见的乘数为 7 和 9 是 63. 我们的分数将以63个区块的形状表示。这次我们用矩形切成 9 乘 7 = 63 个区块。

7 divides into 63 nine times, so $\begin{aligned} \frac{3}{7} = \frac{9 \cdot 3}{9 \cdot 7} = \frac{27}{63} \end{aligned}$ .
::7乘以63,9次,37=9-39-7=2763

We can multiply the numerator and the denominator both by 9 because that’s really just the opposite of reducing the fraction. To get back from $\begin{aligned} \frac{a}{b} \end{aligned}$ to $\begin{aligned} \frac{3}{7} \end{aligned}$ , we’d just cancel out the 9’s. Or, to put that in more formal terms:
::我们可以将分子数和分母乘以9, 因为这与减少分数完全相反。如果要从AB返回到37,我们就会取消9。或者,用更正式的术语说:

The fractions $\begin{aligned} \frac{a}{b} \end{aligned}$ and $\begin{aligned} \frac{c \cdot a}{c \cdot b} \end{aligned}$ are equivalent as long as $\begin{aligned} c \neq 0 \end{aligned}$ .
::ab和cacb的分数与c0相同。

Therefore, $\begin{aligned} \frac{27}{63} \end{aligned}$ is an equivalent fraction to $\begin{aligned} \frac{3}{7} \end{aligned}$ . Here it is shown visually:
::因此,2763是相当于37的分数。

9 divides into 63 seven times, so $\begin{aligned} \frac{4}{9} = \frac{7 \cdot 4}{7 \cdot 9} = \frac{28}{63} \end{aligned}$ .
::9乘以637次,49=7479=2863

$\begin{aligned} \frac{28}{63} \end{aligned}$ is an equivalent fraction to $\begin{aligned} \frac{4}{9} \end{aligned}$ . Here it is shown visually:
::2863是49的等值分数。

By writing the fractions with a common denominator of 63, we can easily compare them. If we take the 28 shaded boxes out of 63 (from our image of $\begin{aligned} \frac{4}{9} \end{aligned}$ above) and arrange them in rows instead of columns, we can see that they take up more space than the 27 boxes from our image of $\begin{aligned} \frac{3}{7} \end{aligned}$ :
::通过以63的共分母来写分数,我们可以很容易地比较它们。如果我们在63个分母中(从我们上面的49个图像中)取出28个阴影盒,然后把它们排成行而不是列,我们可以看到它们占用的空间比我们37个的27个框还要大:

Since $\begin{aligned} \frac{28}{63} \end{aligned}$ is greater than $\begin{aligned} \frac{27}{63} \end{aligned}$ , $\begin{aligned} \frac{4}{9} \end{aligned}$ is greater than $\begin{aligned} \frac{3}{7} \end{aligned}$ .
::由于2863大于2763,49大于37。

Graph and Order Rational Numbers
::图表和顺序理性数字

To plot non-integer rational numbers (fractions) on the number line, we can convert them to mixed numbers (graphing is one of the few occasions in algebra when it’s better to use mixed numbers than improper fractions), or we can convert them to decimal form.
::为了在数字线上绘制非整数合理数字(折数),我们可以将其转换为混合数字(在代数中,如果使用混合数字比不适当的分数更好,则在代数中只是少数次数字之一),或者我们可以将其转换为小数格式。

Plotting Numbers on a Number Line
::数字行的绘图数字

Plot the following .
::绘制以下地图。

a) $\begin{aligned} \frac{2}{3} \end{aligned}$
:a) 23

If we divide the number line into sub-intervals based on the denominator of the fraction, we can look at the fraction’s numerator to determine how many of these sub-intervals we need to include.
$\begin{aligned} \frac{2}{3} \end{aligned}$ falls between 0 and 1. Because the denominator is 3, we divide the interval between 0 and 1 into three smaller units. Because the numerator is 2, we count two units over from 0.
::如果我们根据分母的分母将数字行分为分数分数分数,我们可以查看分数的分子,以确定需要包括多少次数。 23个分数介于0:1之间。 23个分数介于0:1之间, 因为分数为3, 我们将分数间隔介于0和1之间, 分成三个小单位。因为分子分数为2, 我们从0点算出两个单位。

b) $\begin{aligned} - \frac{3}{7} \end{aligned}$
:b)-37

$\begin{aligned} - \frac{3}{7} \end{aligned}$ falls between 0 and -1. We divide the interval into seven units, and move left from zero by three of those units.
::-37在0到-1之间。我们把间隔分成7个单位, 由零移动3个单位。

Examples
::实例

Example 1
::例1

Classify and simplify the rational number $\begin{aligned} \frac{50}{60} \end{aligned}$ .
::分类和简化第5060号合理数字。

$\begin{aligned} \frac{50}{60} \end{aligned}$ is a proper fraction, and we can simplify it as follows: $\begin{aligned} \frac{50}{60} = \frac{5 \cdot 5 \cdot 2}{5 \cdot 3 \cdot 2 \cdot 2} = \frac{5}{3 \cdot 2} = \frac{5}{6} \end{aligned}$
::5060是一个适当的分数,我们可以简化如下: 5060=5552532=532=56。

Example 2
::例2

Plot the rational number $\begin{aligned} \frac{17}{5} \end{aligned}$ on the number line.
::在数字线上绘制合理数字175。

$\begin{aligned} \frac{17}{5} \end{aligned}$ as a mixed number is $\begin{aligned} 3 \frac{2}{5} \end{aligned}$ and falls between 3 and 4. We divide the interval into five units, and move over two units.
::175个混合数为325个,在3到4之间。我们把间隔分成5个单位, 移动两个单位。

Another way to graph this fraction would be as a decimal. $\begin{aligned} 3 \frac{2}{5} \end{aligned}$ is equal to 3.4, so instead of dividing the interval between 3 and 4 into 5 units, we could divide it into 10 units (each representing a distance of 0.1) and then count over 4 units. We would end up at the same place on the number line either way.
::将这个分数图解为小数的另一种方式是小数。 325等于3.4, 因此,与其将3和4之间的间隔分为5个单位, 不如将其分成10个单位( 每一个单位代表0. 1 的距离) , 然后计数超过 4 个单位。我们最终会在同一地点, 以两种方式在数字线上。

Review
::回顾

Order the numbers $\begin{aligned} 2, - \frac{5}{2}, \frac{5}{2} \end{aligned}$ from least to greatest.
::命令2 -52,52,52 从最少到最多。

::命令2 -52,52,52 从最少到最多。

The tick-marks on the number line represent evenly spaced integers. Find the values of $\begin{aligned} a, b, c, d, and e : \end{aligned}$
::数字行上的刻号表示平均间距整数。查找 a、b、c、d 和 e 的值:

::数字行上的刻号表示平均间距整数。查找 a、b、c、d 和 e 的值:

In 3-5, determine what fraction of the whole each shaded region represents.
::在3-5中,确定每个阴影区域整体的分数。

For 6-10, place the following sets of rational numbers in order, from least to greatest.
::6-10时,将以下各组合理数字排列为从最少到最多。

$\begin{aligned} \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \end{aligned}$

$\begin{aligned} \frac{1}{10}, \frac{1}{2}, \frac{2}{5}, \frac{1}{4}, \frac{7}{20} \end{aligned}$

$\begin{aligned} \frac{39}{60}, \frac{49}{80}, \frac{59}{100} \end{aligned}$

$\begin{aligned} \frac{7}{11}, \frac{8}{13}, \frac{12}{19} \end{aligned}$

$\begin{aligned} \frac{9}{5}, \frac{22}{15}, \frac{4}{3} \end{aligned}$

For 11-15, find the simplest form of the following rational numbers.
::11-15 找到以下最简单的合理数字形式。

$\begin{aligned} \frac{22}{44} \end{aligned}$

$\begin{aligned} \frac{9}{27} \end{aligned}$

$\begin{aligned} \frac{12}{18} \end{aligned}$

$\begin{aligned} \frac{315}{420} \end{aligned}$

$\begin{aligned} \frac{244}{168} \end{aligned}$

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。

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