节: 使用 Real 数字和 Real 数字属性操作 | 1.6 使用真实数字和真实数字属性操作

章节大纲

When calculating a bill, it is important to add the purchases and calculate the tax in the right order . In this section, we discuss the correct order to perform operations in and real number properties.
::计算帐单时,必须增加购买量,并按正确的顺序计算税额。在本节中,我们讨论在房地产中进行操作和实际数目的正确顺序。

lesson content

Order of Operations
::行动命令令

The is a set of guidelines that allow us to perform computations the same way every time. The order is as follows:
::这是一套指导准则,让我们每次都能以同样的方式进行计算。顺序如下:

Order of Operations
::行动命令令

" data-term="Parentheses" role="term" tabindex="0"> Parentheses Do anything in parentheses first. ( Absolute value bars are on the same level as parentheses.)
::括号首先在括号中做任何事情。 (绝对值栏与括号相同。 )

Next, all powers or exponents need to be evaluated since exponents are repeated multiplication .
::其次,需要评估所有权力或推手,因为推手是重复乘法。

Multiplication/ Division Multiplication and division must be performed at the same time, from left to right, because they are inverse operations. These go before addition and subtraction because they are repeated addition and repeated subtraction, respectively.
::乘法/司的乘法和除法必须同时从左向右进行,因为它们是逆向作业,在增加和减法之前,因为它们是重复增加和重复减法。

Addition/Subtraction Addition and subtraction are also calculated at the same time , from left to right.
::同时从左到右计算增减。

Example 1
::例1

Compute $\begin{aligned} 2^{2} + 6 \cdot 3 - (5 - 1) \end{aligned}$ .
::计算22+63-(5-1))。

Solution: We follow the order of operations starting with parentheses.
::解决办法:我们遵循从括号开始的操作顺序。

Parentheses $\begin{aligned} \to 2^{2} + 6 \cdot 3 - 4 \end{aligned}$
::括号 22+63-4

Exponents $\begin{aligned} \to 4 + 6 \cdot 3 - 4 \end{aligned}$
::4+63-4的指数

Multiplication $\begin{aligned} \to 4 + 18 - 4 \end{aligned}$
::乘数 _4+18-4

Add/Subtract $\begin{aligned} \to 18 \end{aligned}$
::添加/减号_________________________________________________________________________________________________________________

Example 2
::例2

Compute $\begin{aligned} \frac{9 - 4 \div 2 + 13}{2^{2} \cdot 3 - 7} \end{aligned}$ .
::计算 9 - 4 @% 2+1322_ 3- 7。

Solution: Think of everything in the numerator in its own set of parentheses. Do the same with the denominator. The problem can be rewritten as $\begin{aligned} (9 - 4 \div 2 + 13) \div (2^{2} \cdot 3 - 7) \end{aligned}$ . When there are multiple operations inside parentheses, use the order of operations within each.
::解决方案 : 在自己的括号中 , 以自己的括号来思考分子中的一切。对分母也这样做。问题可以重写为 (9 - 4 2+13) (22 - 3 - 7 ) 。当括号内有多个操作时, 请使用各自内部的操作顺序。

\begin{aligned} (9 - 4 \div 2 + 13) \div (2^{2} \cdot 3 - 7) \\ (9 - 4 \div 2 + 13) \div (4 \cdot 3 - 7) & exponents \\ (9 - 2 + 13) \div (12 - 7) & multiplication and division \\ 20 \div 5 & addition and subtraction \\ 4 \end{aligned}

9 - 42+13) (223-7) 9 - 42+13) (43-7) 病原体(9 - 2+13) (12-7) 乘法和除法 205 增减 4

Parentheses can also be written within another set of parentheses. This is called embedding parentheses. When embedded parentheses are in a problem, you may see brackets , [ ], in addition to parentheses.
::括号也可以写在另一组括号内。这称为嵌入括号。如果嵌入括号有问题,除了括号外,您还可以看到括号[ ]。

Example 3

::例3

Use the order of operations to compute $\begin{aligned} 8 + [4^{2} - 6 \div (5 + 1)] \end{aligned}$ .
::使用操作顺序来计算 8+[42-6](5+1)]。

Solution: This is an example of embedded parentheses. Start by simplifying the parentheses that are inside the brackets. Then, simplify what is inside the brackets according to the order of operations.
::解决方案 : 这是嵌入括号的一个例子。首先简化括号中的括号。然后根据操作顺序简化括号中的内容。

\begin{aligned} 8 + [4^{2} - 6 \div (5 + 1)] \\ 8 + [4^{2} - 6 \div 6] & parentheses \\ 8 + [16 - 6 \div 6] & exponents \\ 8 + [16 - 1] & division \\ 8 + 15 & subtraction inside brackets \\ 23 \end{aligned}

::8+8+[42-6(5+1)]8+[42-66]、8+[42-66]、8+[16-66]、8+[16-66]、8+[16-1]、8+[16-1]、8+15括号内减法23

Even though the brackets are on the right, we do the subtraction inside the brackets first, before the addition on the left.
::尽管括号在右边,但我们先在括号内减法,然后在左边添加。

by Mathispower4u demonstrates how to simplify expressions using the order of operations.
::由 Mathispower4u 演示如何使用操作顺序简化表达式。

Example 4

::例4

Abigail buys 2 shirts that each cost $12. She also buys a pair of jeans that were originally $50, but are discounted by 15%. She also has a coupon for $10 off her purchase. How much was her purchase?
::Abigail购买了两件每件12美元的衬衫。她还购买了一条原50美元的牛仔裤,但折扣了15%。她还购买了10美元的优惠券。她购买了多少?

Solution: First, we need to write an expression to represent this situation.
::解决方案:首先,我们需要写一个表达方式来代表这种情况。

cost of two shirts: $\begin{aligned} 2 \times 12 \end{aligned}$
::两件衬衫的费用: 2x12

cost of jeans with the discount: $\begin{aligned} 50 - (50 \times 0.15) \end{aligned}$
::贴贴牛仔牛仔裤成本:50-(50x0.15)

The total cost of the purchase is the sum of the cost of the items minus the $10 coupon. That gives us,
::购买总成本是物品成本减去10美元优惠券的总和。

\begin{aligned} [(2 \times 12) + (50 - (50 \times 0.15))] - 10 \end{aligned}

We want to begin with our innermost parentheses. In this problem, there are two expressions in innermost sets of parentheses.
::我们希望从我们最深层的括号开始。在这个问题中, 最深层的括号中有两个表达式。

\begin{aligned} [(2 \times 12) + (50 - (50 \times 0.15))] - 10 \\ = [24 + (50 - 7.5)] - 10 \end{aligned}

We still have another set of parentheses in our problem, so we should evaluate that next.
::我们的问题还有另一套括号,因此,我们应该评估下一个问题。

\begin{aligned} [24 + (50 - 7.5)] - 10 \\ = [24 + 42.5] - 10 \end{aligned}

Next, we want to add the numbers in the parentheses.
::接下来,我们要在括号中加上数字。

\begin{aligned} [24 + 42.5] - 10 \\ = 66.5 - 10 \\ = 56.5 \end{aligned}

Our final answer is $56.50. It is important to add the dollar sign to indicate the unit for this problem.
::我们的最后一个答案是56.50美元,必须加上美元符号,以表明这一问题的单位。

Real Number Properties
::真实数字属性

The properties of r eal numbers enable us to solve mathematical problems. Notice that these properties hold for addition and multiplication. Subtraction and division can be defined in terms of addition and multiplication, respectively, by using these properties. (Since these properties hold, we can describe the set of real numbers as a field ).
::实际数字的属性可以让我们解决数学问题。请注意, 这些属性属于附加和乘法。减法和除法可以分别用这些属性来定义增法和乘法。 (自这些属性持有以来, 我们可以将这些实际数字描述为字段 ) 。

Property	Example
Commutative	$\begin{aligned} a + b = b + a, a b = b a \end{aligned}$
Associative	$\begin{aligned} a + (b + c) = (a + b) + c, a (b c) = (a b) c \end{aligned}$
Identity	$\begin{aligned} a + 0 = a, a \cdot 1 = a \end{aligned}$
Inverse	$\begin{aligned} a + (- a) = 0, a \cdot \frac{1}{a} = 1 \end{aligned}$
Distributive	$\begin{aligned} a (b + c) = a b + a c \end{aligned}$

From the identity property , we can say that 0 is the additive identity and 1 is the multiplicative identity . Similarly, from the inverse property, $\begin{aligned} - a \end{aligned}$ is the additive inverse of $\begin{aligned} a \end{aligned}$ and $\begin{aligned} \frac{1}{a} \end{aligned}$ is the multiplicative inverse of $\begin{aligned} a \end{aligned}$ because when you add or multiply a number and its inverse, you get the identity as a result.
::从身份属性中,我们可以说0是添加身份,1是倍增效应身份。同样,从反向属性中,a是a和a的复数反差,a是a和a的复数反差,因为当您增加或乘以一个数字及其反数时,你就会因此获得身份。

Example 5
::例5

Identify the property used in the equations below.
::标明以下方程式中使用的财产。

a. $\begin{aligned} 2 (4 x - 3) = 8 x - 6 \end{aligned}$
::a. (2(4x-3)=8x-6)

b. $\begin{aligned} 5 \cdot \frac{1}{5} = 1 \end{aligned}$
::b. 515=1

c. $\begin{aligned} 6 \cdot (7 \cdot 8) = (6 \cdot 7) \cdot 8 \end{aligned}$
::c. 6____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

d. $\begin{aligned} 6 \cdot 7 = 7 \cdot 6 \end{aligned}$
::d. 6+7=7+6

Solution:
::解决方案 :

a. This is an example of the .
::a. 这是一个例子。

b. This is an example of the inverse property.
::b. 这是反向财产的例子。

c. This is an example of the associative property .
::c. 这是连带财产的一个例子。

d. This is an example of the commutative property.
::d. 这是通货财产的例子。

by Mathispower4u demonstrates how to use the distributive property.
::Mathispower4u 演示如何使用分配财产。

WARNING
::警告

When you have

\begin{aligned} - (a + b) \end{aligned}

, you are distributing -1 to each each one of the terms, not distributing the negative sign:

\begin{aligned} - (a + b) = - 1 \cdot a + - 1 \cdot b \end{aligned}

Summary
::摘要

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。

There are nine important 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9个重要属性: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。

Review
::回顾

Determine which property is used in each problem below.
::确定在下面每个问题中使用哪些财产。

1. $\begin{aligned} 8 + 5 = 5 + 8 \end{aligned}$

2. $\begin{aligned} 7 (3 - 2) = 21 - 14 \end{aligned}$

3. $\begin{aligned} \frac{2}{3} \cdot \frac{3}{2} = 1 \end{aligned}$

4. $\begin{aligned} 4 \cdot (5 \cdot 2) = (4 \cdot 5) \cdot 2 \end{aligned}$

5. $\begin{aligned} - \frac{1}{4} + 0 = - \frac{1}{4} \end{aligned}$

6. $\begin{aligned} - 6 + 6 = 0 \end{aligned}$

Compute the following using the order of operations.
::使用操作顺序计算以下内容。

7. $\begin{aligned} 12 \div 4 + 3^{3} \cdot 2 - 10 \end{aligned}$

8. $\begin{aligned} 8 \div 4 + (15 \div 3 - 2^{2}) \cdot 6 \end{aligned}$

9. $\begin{aligned} \frac{10 - 4 \div 2}{7 \cdot 2 + 2} \end{aligned}$

10. $\begin{aligned} \frac{1 + 20 - 16 \div 4^{2}}{(5 - 3)^{2} + 12 \div 2} \end{aligned}$

11. $\begin{aligned} [3 + (4 + 7 \cdot 3) \div 5]^{2} - 47 \end{aligned}$

12. $\begin{aligned} \frac{7 \cdot 4 - 4}{\frac{6}{2} + 5} \cdot 4^{2} - 18 \end{aligned}$

13. $\begin{aligned} 6^{2} - [9 + (7 - 5)^{3}] + 49 \div 7 \end{aligned}$

14. $\begin{aligned} \frac{27 \div 3^{2} + (6 - 2^{2})}{(32 \div 8 + 1) \cdot 3} \end{aligned}$

15. $\begin{aligned} 6 + 5 \cdot 2 - 9 \div 3 + 4 \end{aligned}$

Explore More
::探索更多

1. Simplify $\begin{aligned} 6 (4 - 9 + 5) \end{aligned}$ using:
::1. 使用下列方法简化6(4-9+5):

the distributive property
::分配款 -- -- 分配款

the order of operations
::操作顺序

Do you get the same answer? Why do you think that is?
::你得到同样的答案吗?

2. $\begin{aligned} 6 + 5 \cdot 2 - 9 \div 3 + 4 \end{aligned}$

a. Using the expression above, insert parentheses to make the expression equal 1. You may need to use more than one set of parentheses.
::a. 使用上面的表达式,插入括号,使表达式等于1。可能需要使用不止一套括号。

b. Using the expression above, insert parentheses to make the expression equal 23. You may need to use more than one set of parentheses.
::b. 使用上面的表达式,插入括号,使表达式等于23。您可能需要使用不止一套括号。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

章节大纲

Order of Operations ::行动命令令

Order of Operations ::行动命令令

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Real Number Properties ::真实数字属性

Example 5 ::例5

WARNING ::警告

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Order of Operations
::行动命令令

Order of Operations
::行动命令令

Example 1
::例1

Example 2
::例2

Example 3

::例3

Example 4

::例4

Real Number Properties
::真实数字属性

Example 5
::例5

WARNING
::警告

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX