1.19 摘要:实际数字和代数表达式

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• 选择节摘要:实际数字和代数表达式

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  摘要:实际数字和代数表达式

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  In this chapter, we learned about :
  ::在本章中,我们了解到:

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  Sets
  ::套套套套套套

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  • A set is a group of objects called elements.
    ::一组是一组名称为元素的物体。
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  • The number of elements in a set is called the cardinality of the set and is denoted by |A|.
    ::一组元素的编号称为集的基点,由 A 表示 。
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  • The complement of a set is all of the elements that are not in the set, but are in the universal set. It is denoted by A'.
    ::一组中的补充是非集中的所有元素,而是通用集中的所有元素。它由 A 表示 。
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  • The union of two sets is all of the elements that are in either set. The symbol for union is ∪.
    ::两组组合的结合是两组组合中的所有元素。 工会的符号是 。
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  • The intersection of two sets is all of the elements in both sets. The symbol for intersection is ∩.
    ::两组的交叉点是两组中的所有元素。交叉点的符号是 。
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  • We can use Venn diagrams to represent the relationships between sets.
    ::我们可以使用文恩图表来代表各组之间的关系。

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  Sets of Real Numbers
  ::一组真实数字

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  • The real numbers have the following important subsets: rational numbers, irrational numbers, integers, whole numbers, and natural numbers.
    ::真实数字有以下重要的子集:合理数字、非理性数字、整数、整数和自然数字。
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  • To represent a number on the number line graphically, we plot a point or its coordinate where it is approximately located on the number line.
    ::要用图形方式在数字线上表示数字,我们绘制一个点或其坐标,将其大致放在数字线上。
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  • We can use  $\begin{array}{r}<\end{array}$ , $\begin{array}{r}\le \end{array}$ , $\begin{array}{r}>\end{array}$ , and $\begin{array}{r}\ge \end{array}$  to compare numbers.
    ::我们可以使用 < , \ , > 和 \ 来比较数字 。
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  • To compare numbers, we can use their decimal forms and place value to compare them. 
    ::为了比较数字,我们可以使用他们的小数表格和位置价值来比较它们。
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  • The absolute value of a number is the distance of the number from 0 on the number line.
    ::数字的绝对值是数字线上数字数与 0 的距离。
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  • Intervals on the number line can be bounded, with two endpoints, or unbounded, with either one endpoint or no endpoints.
    ::数字行的中间点可以被捆绑,有两个端点,或者没有绑定,要么一个端点,要么没有端点。
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  • We have several notations for describing intervals, including inequality notation, interval notation, and set-builder notation.
    ::我们有几个标记来描述间隔,包括不平等标记、间隔标记和建筑工的标记。

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  Operations on Real Numbers
  ::实际数字操作

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  • The order that we should perform operations in is parentheses, exponents, multiplication and division from left to right, and lastly, addition and subtraction from left to right.
    ::我们应该执行操作的顺序是括号、引号、乘数和从左到右的分割,最后是从左到右的增减。
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  • There are nine important algebraic properties for the real numbers: commutative properties for addition and multiplication, associative properties for addition and multiplication, identity properties for addition and multiplication, inverse properties for addition and multiplication, and the distributive property.
    ::实际数字有九种重要的代数属性:用于添加和乘法的通量属性;用于添加和乘法的连带属性;用于添加和乘法的识别属性;用于添加和乘法的反向属性;以及分配属性。

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     Addition and Subtraction
  ::加和减

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  • Addition of integers: add the absolute values and keep the common sign if like signs, subtract the absolute values and keep the sign of the larger in absolute value if unlike signs.
    ::增加整数:添加绝对值并保留共同符号,如果与符号相同,则保留共同符号,减去绝对值,如果与符号不同,则保留较大符号的绝对值。
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  • Subtraction of integers: add the opposite.
    ::整数减法:增加反差。
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  • Addition or subtraction of fractions: you need to first express the fractions with a common denominator. Then, you can combine the numerators.
    ::增加或减去分数:您需要先用一个共同分母表示分数。然后,您可以将数数合并。
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  • Addition or subtraction of decimals: stack the numbers so that their place values are aligned and then add or subtract as if the decimal was not there. Remember to bring the decimal point down between the ones and the tenths place.
    ::增加或减去小数点:堆叠数字,使其位置值对齐,然后加或减,仿佛小数点不在那里。记住将小数点移到十点和十点之间。
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  • Addition or subtraction of radicals: combine the numbers outside the radicals only if the radicands are the same.
    ::增加或减减基:只有辐射线相同时,才将激进线以外的数字合并。

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     Multiplication and Division
  ::计算和司

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  • Multiplication and division of integers: if like signs, the result is positive; if unlike signs, the result is negative.
    ::乘数和整数的划分:如果与符号一样,结果为正数;如果与符号不同,结果为负数。
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  • Multiplication of fractions: multiply the numerators and multiply the denominators.
    ::分数的乘法:乘以分子和乘以分母。
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  • Division of fractions: multiply the first fraction by the reciprocal of the second fraction.
    ::分数分数: 第一分数乘以第二分数的对等数。
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  • Multiplication of decimals: multiply the numbers as you would if a decimal point was not present. Then, count the number of decimal places. That is the number of decimal places that should be in your result.
    ::小数点的乘法:如果小数点没有出现,则按小数点的乘法乘以数字。然后,计算小数点位数。这就是结果中应该包含的小数点位数。
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  • Division of decimals: multiply the dividend and the divisor by powers of 10 to make sure the divisor is not a decimal. Then, divide as you would with whole numbers. Remember to bring the decimal point up between the ones and the tenths places.
    ::小数点数的分数:将股息和分数乘以10的权数,以确保分数不是小数点数。然后,以整数进行除法。记住将小数点加到十位数和十位数之间。
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  • Multiplication of radicals: multiply numbers outside the radical with each other and numbers inside the radical with each other.
    ::激进体的乘法: 激进体外的乘数 彼此相伴 和激进体内部的乘数 彼此相伴。
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  • Division of radicals: similar to multiplication in that you can divide numbers outside the radical with each other and numbers inside the radical with each other . If there is a radical in the denominator, you should rationalize the denominator.
    ::激进的分界线:与乘法相似,因为你可以将激进之外的数字和激进内部的数字分隔开来。 如果分母中存在激进,你应该理顺分母。

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     Exponents
  ::指数

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  • The product rule of exponents says that if you multiply two expressions with the same base, you can add the exponents. That is: $\begin{array}{r}{a}^m\times a^n=a^{m+n}\end{array}$  .
    ::引言者的产物规则指出,如果用同一基数乘以两个表达式,您可以添加引言。即: AMxan=am+n。
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  • The quotient rule of exponents says that if you divide two expressions with the same base, you can subtract the exponents. That is: $\begin{array}{r}\frac{a^m}{a^n}=a^{m-n}\end{array}$  .
    ::引言者的商数规则指出,如果用同一基数将两个表达式分开,可以减去引言。这就是:aman=am-n。
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  • The power rule of exponents says that if you raise a base to a power to another power, you can multiply the exponents. That is: $\begin{array}{r}(a^m{)}^n=a^{mn}\end{array}$  .
    ::远征者的权力规则说,如果你把一个基地加到另一个力量上,你就可以使先征者倍增。这就是说am)n=amn。
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  • $\begin{array}{r}{b}^0=1,b\ne 0\end{array}$  
    ::b0=1,b0
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  • $\begin{array}{r}{b}^{-n}=\frac{1}{b^n},b\ne 0\end{array}$  
    ::b_b_b_b_b_b_b_b_b_b__b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b{\b_b_b_b_b_b____b>%0
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  • To write a number in scientific notation, write the significant digits as a number $\begin{array}{r}1\le x<10\end{array}$  and the place value as a power of 10.
    ::要在科学符号中写出数字,请将重要数字写成 1x < 10,将位置值写成10。

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  The Fundamental Theorem of Arithmetic and the Least Common Multiple
  ::人工鉴定和最不常见多重特征的基本理论

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  • Prime numbers are numbers whose only factors are 1 and the number itself.
    ::起始数是数字,其唯一的因素是1和数字本身。
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  • Composite numbers are numbers whose factors include more than 1 and the number itself.
    ::复合数字是指其因素包括1个以上和数字本身的数字。
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  • The Fundamental Theorem of Arithmetic: Every positive integer is either prime or the unique product of prime numbers.
    ::论理学的基本理论:每个正整数要么是质数的质数,要么是质数的独特产物。
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  • To find the least common multiple: Find the prime factorization of all of the numbers. Then, write each prime factor with an exponent that is the largest in any of the prime factorizations, and multiply.
    ::要找到最小常见的倍数 : 查找所有数字的质因数。 然后, 将每个质因数写成一个指数, 该指数是任何质因数中最大的指数, 然后乘以 。

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  Rates and Percents
  ::比率和百分比

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  • To convert between percents and decimals, we can just move the decimal point two units right or left.
    ::要在百分率和小数点之间转换,我们可以将小数点点两单位向右或向左移动。
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  • To convert between percents and fractions, we can create fractions that have a denominator of 100 (per cent).
    ::为了在百分数和分数之间转换,我们可以产生分数,分数分母为100%(%)。
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  • A rate is a ratio that compares two quantities often with different units.
    ::A 率是一种比率,与通常使用不同单位的两个数量相比。
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  • A unit rate is a rate where there is 1 unit of the quantity in the denominator.
    ::单位费率是指分母中数量有一个单位的费率。

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  Algebraic Expressions
  
  ::代数表达式

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  • To evaluate an expression, substitute the  given value for the variable  and perform the indicated operations.
    ::要评价表达式,用给定值替换变量,并进行指定的操作。
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  • To determine if a value is a solution to an equation, evaluate the expressions on each side of the equal sign and determine if the left side of the equation equals the right side of the equation.
    ::要确定一个值是否是方程的解决方案, 评估相等符号两侧的表达式, 并确定方程的左侧是否等于方程的右侧 。
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  • Like terms are terms that have the same variables and each variable is raised to the same power.
    ::类似术语是具有相同变数的术语,每个变数被提升到相同的变数。
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  • To combine like terms, add or subtract the coefficients and keep the variable part of the terms.
    ::将类似术语合并,增减系数,保留术语的可变部分。
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  • To simplify algebraic expressions, perform any operations in parentheses, distribute where possible, and then add and subtract like terms from left to right.
    ::为了简化代数表达式,在括号中进行任何操作,尽可能进行分配,然后从左向右增减类似术语。
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  • To convert a quantity from one set of units to another, use conversion factors to set up a a series of multiplications to cancel out the old units and introduce the new units. 
    ::要将一组单位的数量转换成另一组单位,请使用转换系数设置一系列乘数来取消旧单位并引入新单位。

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  Geometric Formulas
  ::几何公式

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  • Perimeter and circumference measure the distance around the outside of a two-dimensional shape.
    ::周边和环绕测量了二维形状的外部距离。
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  • Area measures the surface of a  two-dimensional shape.
    ::区域测量二维形状的表面。
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  • Surface area measures the area of the surface of a three-dimensional shape.
    ::表面面积测量三维形状表面面积。
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  • Volume measures the space inside a three-dimensional shape.
    ::量度三维形状内的空间。

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  Looking Back, Looking Forward
  ::回顾,展望未来

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  In this chapter, we reviewed number skills and introduced algebraic concepts, like variables, terms, and expressions. We also explored unit conversions and geometric formulas. These skills are used in our day-to-day life from determining our paycheck to recipes for the food we eat, and even to constructing the buildings we live and work in.
  ::在本章中,我们审视了数字技能,引入了代数概念,如变数、术语和表达式。我们还探讨了单位转换和几何公式。这些技能用于日常生活,从确定工资到食用食品的食谱,甚至建造我们居住和工作的建筑。

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  In the next chapter, we will start to attempt the goals of algebra: calculation by completion and balancing. We will consider equations and inequalities with one variable where the variable is raised to the first power. These types of equations are called linear and we consider them in chapter 2.
  ::在下一章,我们将开始尝试代数的目标:通过完成和平衡来计算。我们将考虑方程式和不平等,其中有一个变量,变量被提升到第一权力。这些方程式被称为线性方程式,我们在第二章中考虑它们。

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  Chapter  Review
  ::回顾章次审查

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  The following practice problems review the topics that we explored  in this chapter.  
  ::以下实践问题审视了我们在本章中探讨的专题。