课程： 2.5 合理数字的乘法 | The Way To Learn

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选择节合理数字乘法

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合理数字乘法
Multiplication of Rational Numbers
::合理数字乘法

Whenever we multiply a number by negative one, the sign of the number changes. In more mathematical terms, multiplying by negative one maps a number onto its opposite. The number line below shows two examples: $\begin{aligned} 3 \cdot - 1 = - 3 \end{aligned}$ and $\begin{aligned} - 1 \cdot - 1 = 1 \end{aligned}$ .
::当我们乘以一个数字为负数乘以一个数字时,数字的符号就会改变。用数学术语来说,如果以负数乘以一个数字,数字就会反向。下面的数字线显示两个例子: 313和- 11=1。

When we multiply a number by negative one, the absolute value of the new number is the same as the absolute value of the old number, since both numbers are the same distance from zero .
::当我们乘以一个数字乘以一个负数时,新数字的绝对值与旧数字的绝对值相同,因为这两个数字与零的距离相同。

The product of a number “ $\begin{aligned} x \end{aligned}$ ” and negative one is $\begin{aligned} - x \end{aligned}$ . This does not mean that $\begin{aligned} - x \end{aligned}$ is necessarily less than zero! If $\begin{aligned} x \end{aligned}$ itself is negative, then $\begin{aligned} - x \end{aligned}$ will be positive because a negative times a negative (negative one) is a positive.
::数字“x”和负一的产值为 -x。这并不意味着x一定低于零。如果x本身为负值,那么-x将是正值,因为负值为负值(负值为负值)是正值。

When you multiply an expression by negative one, remember to multiply the entire expression by negative one.
::当将表达式乘以负值时,请记住将整个表达式乘以负值。

Multiplying Rational Numbers
::乘数合理数字

Multiply the following by negative one.
::用负乘法乘以下列数。

a) 79.5
:a) 79.5

-79.5

b) $\begin{aligned} π \end{aligned}$
:b) ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} - π \end{aligned}$

c) $\begin{aligned} (x + 1) \end{aligned}$
:c) (x+1)

$\begin{aligned} - (x + 1) or - x - 1 \end{aligned}$
:x+1)或-(x-1)

d) $\begin{aligned} | x | \end{aligned}$
:d) ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} - | x | \end{aligned}$
::{\fn华文楷体\fs16\1cHE0E0E0}#x##############################################################################################################################################################################################################################################################################################################################################################################################################################################################################################################

Note that in the last case the negative sign outside the absolute value symbol applies after the absolute value. Multiplying the argument of an absolute value equation (the term inside the absolute value symbol) does not change the absolute value. $\begin{aligned} | x | \end{aligned}$ is always positive. $\begin{aligned} | - x | \end{aligned}$ is always positive. $\begin{aligned} - | x | \end{aligned}$ is always negative.
::请注意, 在最后一例中, 绝对值符号外的负符号应用到绝对值之后。乘以绝对值方程( 绝对值符号内的术语) 的参数不会改变绝对值。 x 总是正的。 x 总是正的。 x 总是负的。 x 总是负的。 @x 总是负的。

Whenever you are working with expressions, you can check your answers by substituting in numbers for the variables. For example, you could check part $\begin{aligned} d \end{aligned}$ of Example 1 by letting $\begin{aligned} x = - 3 \end{aligned}$ . Then you’d see that $\begin{aligned} | - 3 | \neq - | 3 | \end{aligned}$ , because $\begin{aligned} | - 3 | = 3 \end{aligned}$ and $\begin{aligned} - | 3 | = - 3 \end{aligned}$ .
::使用表达式时,您可以通过替换变量的数值来检查您的答案。例如,您可以通过让 x*%3 来检查例1中的 d 部分。然后您可以看到 3\\\ 3\ 3\ , 因为 @ 3\ 3 3 和 3\ 3 3 3 。

Careful, though—plugging in numbers can tell you if your answer is wrong, but it won’t always tell you for sure if your answer is right!
::数字插插可以告诉你答案是否正确, 但它不会总是告诉你答案是否正确!

Multiplying Fractions
::乘数分数

1. Simplify $\begin{aligned} \frac{1}{3} \cdot \frac{2}{5} \end{aligned}$ .
::1. 简化第1325号。

One way to solve this is to think of money. For example, we know that one third of sixty dollars is written as $\begin{aligned} \frac{1}{3} \cdot $ 60 \end{aligned}$ . We can read the above problem as one-third of two-fifths . Here is a visual picture of the fractions one-third and two-fifths .
::解决这一难题的一个办法是考虑金钱。例如,我们知道60美元的三分之一是13-60美元。我们可以将上述问题看成五分之二的三分之一。这里是三分之一和五分之二的图象。

If we divide our rectangle into thirds one way and fifths the other way, here’s what we get:
::如果我们把矩形一分为三, 五分成五,

Here is the intersection of the two shaded regions. The whole has been divided into five pieces width-wise and three pieces height-wise. We get two pieces out of a total of fifteen pieces.
::这是两个阴影区域的交叉点。整个区域被分为五块宽度和三块高度。我们从总共十五块中分了两块。

$\begin{aligned} \frac{1}{3} \cdot \frac{2}{5} = \frac{2}{15} \end{aligned}$

Notice that $\begin{aligned} 1 \cdot 2 = 2 \end{aligned}$ and $\begin{aligned} 3 \cdot 5 = 15 \end{aligned}$ . This turns out to be true in general: when you multiply rational numbers, the numerators multiply together and the denominators multiply together. Or, to put it more formally:
::注意 12=2 和 35=15 。这在一般情况下是真实的: 当您乘以理性数字时, 分子会一起乘, 分母会一起乘。或者, 要更正式地说:

When multiplying fractions: $\begin{aligned} \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d} \end{aligned}$
::当乘以分数时: abcd=abbd

This rule doesn’t just hold for the product of two fractions, but for any number of fractions.
::这条规则不仅维持两个分数的产物, 也维持任何一个分数的产物。

2. Evaluate and simplify $\begin{aligned} \frac{12}{25} \cdot \frac{35}{42} \end{aligned}$ .
::2. 评价和简化1225-3542。

We can see that 12 and 42 are both multiples of six, 25 and 35 are both multiples of five, and 35 and 42 are both multiples of 7. That means we can write the whole product as $\begin{aligned} \frac{6 \cdot 2}{5 \cdot 5} \cdot \frac{5 \cdot 7}{6 \cdot 7} = \frac{6 \cdot 2 \cdot 5 \cdot 7}{5 \cdot 5 \cdot 6 \cdot 7} \end{aligned}$ . Then we can cancel out the 5, the 6, and the 7, leaving $\begin{aligned} \frac{2}{5} \end{aligned}$ .
::我们可以看到,12和42的倍数是6,25和35的倍数是5,35和42的倍数是7的倍数是7,这意味着我们可以把整个产品写成6-25-5-5-5-5-5-76-7=6-2-5-5-5-5-5-5-5-5-5-6-17。然后我们可以取消5,6和7,离开25。

Identify and Apply Properties of Multiplication
::识别和应用乘数属性

The four mathematical properties which involve multiplication are the Commutative, Associative, Multiplicative Identity and Distributive Properties .
::涉及乘法的四种数学属性是交流、联合、多复制特性和分配属性。

Commutative property : When two numbers are multiplied together, the product is the same regardless of the order in which they are written.
::公用财产:当两个数字相乘时,产品是相同的,不管其写法的顺序如何。

Example: $\begin{aligned} 4 \cdot 2 = 2 \cdot 4 \end{aligned}$
::示例:42=24

We can see a geometrical interpretation of The Commutative Property of Multiplication to the right. The Area of the shape $\begin{aligned} (l e n g t h \times w i d t h) \end{aligned}$ is the same no matter which way we draw it.
::我们可以看到对权利乘法的通量属性的几何解释。形状区域(长xxwidth)是相同的,不管我们如何绘制。

Associative Property : When three or more numbers are multiplied, the product is the same regardless of their grouping.
::共有财产:当三个或三个以上数字乘以乘以时,不论产品组别如何,产品都是相同的。

Example: $\begin{aligned} 2 \cdot (3 \cdot 4) = (2 \cdot 3) \cdot 4 \end{aligned}$
::示例:2______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Multiplicative Identity Property : The product of one and any number is that number.
::多重身份财产:一个号码和任何一个号码的产物就是这个号码。

Example: $\begin{aligned} 5 \cdot 1 = 5 \end{aligned}$
::示例:51=5

Distributive Property : The product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For expressions $\begin{aligned} a, b, \end{aligned}$ and $\begin{aligned} c \end{aligned}$ , $\begin{aligned} a (b + c) = a b + a c \end{aligned}$ .
::分配财产:一个表达式和一个总和的产物等于表达式产品和每个术语的总和。对于a、b和c, a(b+c)=ab+ac, a(b+c)=ab+ac。

Example: $\begin{aligned} 4 (6 + 3) = 4 \cdot 6 + 4 \cdot 3 \end{aligned}$
::示例:4(6+3)=46+43

Real-World Application
::真实世界应用

A gardener is planting vegetables for the coming growing season. He wishes to plant potatoes and has a choice of a single $\begin{aligned} 8 \times 7 \end{aligned}$ meter plot, or two smaller plots of $\begin{aligned} 3 \times 7 \end{aligned}$ and $\begin{aligned} 5 \times 7 \end{aligned}$ meters. Which option gives him the largest area for his potatoes?
::园丁在即将到来的种植季节种植蔬菜,他想种土豆,并选择一个8×7米的地块,或两块3×7和5×7米的小地块。

In the first option, the gardener has a total area of $\begin{aligned} (8 \times 7) \end{aligned}$ or 56 square meters.
::在第一种办法中,园丁总面积为8x7或56平方米。

In the second option, the gardener has $\begin{aligned} (3 \times 7) \end{aligned}$ or 21 square meters, plus $\begin{aligned} (5 \times 7) \end{aligned}$ or 35 square meters. $\begin{aligned} 21 + 35 = 56 \end{aligned}$ , so the area is the same as in the first option.
::在第二种办法中,园丁有(3×7)或21平方米,加上(5×7)或35平方米。 21+35=56,因此面积与第一种办法相同。

Examples
::实例

Multiply the following :
::乘以下列数 :

Example 1
::例1

$\begin{aligned} \frac{2}{5} \cdot \frac{5}{9} \end{aligned}$

With this problem, we can cancel the fives: $\begin{aligned} \frac{2}{5} \cdot \frac{5}{9} = \frac{2 \cdot 5}{5 \cdot 9} = \frac{2}{9} \end{aligned}$
::面对这个问题,我们可以取消5: 2559=2559=29。

Example 2
::例2

$\begin{aligned} \frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5} \end{aligned}$

With this problem, we can multiply all the numerators and all the denominators:
::面对这个问题,我们可以乘以所有的计数器和分母:

$\begin{aligned} \frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5} = \frac{1 \cdot 2 \cdot 2}{3 \cdot 7 \cdot 5} = \frac{4}{105} \end{aligned}$

Example 3
::例3

$\begin{aligned} \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \end{aligned}$

With this problem, we multiply all the numerators and all the denominators, and then we can cancel most of them. The 2’s, 3’s, and 4’s all cancel out, leaving $\begin{aligned} \frac{1}{5} \end{aligned}$ .
::面对这个问题,我们乘以所有的分子和分母,然后我们就可以取消其中的大部分。 2 、 3 和 4 全部取消,剩下 15 个。

With multiplication of fractions, we can simplify before or after we multiply. The next example uses factors to help simplify before we multiply.
::乘以分数的乘法, 我们可以在乘以之前或之后简化。下一个示例使用系数来帮助在乘以之前简化。

Review
::回顾

In 1-4, multiply the following expressions by negative one.
::在 1-4 中,将以下表达式乘以负1。

25

-105

$\begin{aligned} x^{2} \end{aligned}$
::x2x2

$\begin{aligned} (3 + x) \end{aligned}$
:3+x)

In 5-10, multiply the following rational numbers. Write your answer in the simplest form .
::5 - 10, 乘以以下合理数字。以最简单的形式写下您的答复。

$\begin{aligned} \frac{5}{12} \times \frac{9}{10} \end{aligned}$

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

$\begin{aligned} \frac{3}{4} \times \frac{1}{3} \end{aligned}$

$\begin{aligned} \frac{15}{11} \times \frac{9}{7} \end{aligned}$

$\begin{aligned} \frac{1}{13} \times \frac{1}{11} \end{aligned}$

$\begin{aligned} \frac{12}{15} \times \frac{35}{13} \times \frac{10}{2} \times \frac{26}{36} \end{aligned}$

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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