章节大纲

  • A calculator, pencil, and graph paper on a desk, indicating a homework setting.

    Jerod has a homework problem that involves finding the area of a rectangle. He knows that the area of a rectangle equals its length times its width. The rectangle in question has a length of 6.9 m and a width of 6.5 m, so he multiplies the two numbers on his calculator. The answer he gets is 44.85 m 2 , which he records on his homework. To his surprise, his teacher marks this answer wrong. The reason? The answer has too many significant figures.  
    ::Jerod的作业问题涉及寻找矩形区域。 他知道矩形区域的长度等于其宽度的倍数。 有关矩形的长度为6.9米,宽度为6.5米, 因此他将计算器上的两个数字乘以倍数。 他得到的答案是44.85平方米, 他记录在家庭作业上。 令他惊讶的是, 他的老师记错了答案。 原因? 答案有太多重要的数字 。

    What Are Significant Figures?
    ::什么是重要数字?

    In any measurement, the number of significant figures is the number of digits thought to be correct by the person doing the measuring. It includes all digits that can be read directly from the measuring device plus one estimated digit.
    ::在任何测量中,重要数字的数目是进行测量的人认为正确的数字数,包括可从测量设备直接读取的所有数字数加上一个估计数字数。

    Look at the sketch of a beaker below. How much blue does the beaker contain? The top of the liquid falls between the mark for 40 mL and 50 mL, but it’s closer to 40 mL. A reasonable estimate is 43 mL. In this measurement, the first digit (4) is known for certain and the second digit (3) is an estimate, so the measurement has two significant figures.
    ::看看下面的烧杯的草图。烧杯包含多少蓝色?液体的顶部在40毫升到50毫升之间,但接近40毫升。合理的估计是43毫升。在这个测量中,第一个数字(4)是肯定的,第二个数字(3)是估计,因此测量有两个重要数字。

    A beaker with blue liquid indicating a measurement of approximately 43 mL.

    Now look at the graduated cylinder sketched below. How much  gray liquid does it contain? First, it’s important to note that you should read the amount of liquid at the bottom of its curved surface. This falls about half way between the mark for 36 mL and the mark for 37 mL, so a reasonable estimate would be 36.5 mL.
    ::现在看看下面绘制的毕业气瓶。 它包含多少灰色液体? 首先, 需要注意的是, 您应该读读其弯曲表面底部的液体数量。 这大约在36毫升的标记和37毫升的标记之间下降了一半, 因此合理的估计是36.5毫升。

    A graduated cylinder showing liquid measurement between 36 mL and 37 mL.

    Q: How many significant figures does this measurement have?
    ::问题:这一计量有多少重要数字?

    A: There are three significant figures in this measurement. You know that the first two digits (3 and 6) are accurate. The third digit (5) is an estimate.
    ::A:这一计量有三个重要数字。你知道前两个数字(3和6)是准确的。第三个数字(5)是估计数。

    Rules for Counting Significant Figures
    ::计算重要数字的规则

    The examples above show that it’s easy to count the number of significant figures when you are making a measurement. But what if someone else has made the measurement? How do you know which digits are known for certain and which are estimated? How can you tell how many significant figures there are in the measurement? There are several rules for counting significant figures:
    ::上面的例子表明,在进行测量时,计算重要数字的数量很容易。 但如果有人做了测量呢? 你怎么知道哪些数字是已知的,哪些是估计的?你如何知道测量中有多少重要数字?计算中有多少重要数字?计算中有几个规则:

    • Leading zeros are never significant. For example, in the number 006.1, only the 6 and 1 are significant.
      ::主要零位数从来就不重要,例如,在006.1中,只有6和1位数是显著的。
    • Zeros within a number between nonzero digits are always significant. For example, in the number 106.1, the zero is significant, so this number has four significant figures.
      ::非零位数之间的数数内零数总是显著的,例如,在106.1中,零是显著的,因此这一数字有四个重要数字。
    • Zeros that show only where the decimal point falls are not significant. For example, the number 470,000 has just two significant figures (4 and 7). The zeros just show that the 4 represents hundreds of thousands and the 7 represents tens of thousands. Therefore, these zeros are not significant.
      ::零点只显示小数点下降位置的零点并不重要。 比如,470,000个数字只有两个重要数字(4和7 ) 。 零点只是显示4个代表数十万,7个代表数万。 因此,这些零并不重要。
    • Trailing zeros that aren't needed to show where the decimal point falls are significant. For example, 4.00 has three significant figures.
      ::不需要显示小数点下降位置的轨道零值。 例如, 4. 00 有 3 个重要数字 。

    Q: How many significant figures are there in each of these numbers: 20,080, 2.080, and 2000?
    ::问:每个数字中有多少重要数字:20 080、2.080和2000?

    A: Both 20,080 and 2.080 contain four significant figures, but 2000 has just one significant figure.
    ::A:20 080和2.080都包含四个重要数字,但2000年只有一个重要数字。

    Determining Significant Figures in Calculations
    ::确定计算中的重要数字

    When measurements are used in a calculation, the answer cannot have more significant figures than the measurement with the fewest significant figures. This explains why the homework answer above is wrong. It has more significant figures than the measurement with the fewest significant figures. As another example, assume that you want to calculate the volume of the block of wood shown below.
    ::当在计算中使用测量数据时,答案不能有比用数量最少的重要数字衡量的数据更重要的数字。 这解释了为什么上面的作业回答是错误的。 它有比用数量最少的重要数字衡量更重要的数字。 另一个例子是,假设你想计算下面显示的木块块的体积。

    A brown block of wood, representing the object for volume calculation in significant figures.

    The volume of the block is represented by the formula:
    ::区块的体积由公式表示:

    Volume = length × width × height

    Therefore, you would do the following calculation:
    ::因此,你将做以下计算:

    Volume = 1.2 cm × 1.0 cm × 1 cm = 1.2 cm 3

    Q: Does this answer have the correct number of significant figures?
    ::问题:这一答案是否含有重要数字的正确数字?

    A: No, it has too many significant figures. The correct answer is 1 cm 3 . That’s because the height of the block has just one significant figure. Therefore, the answer can have only one significant figure.
    ::A:不,它有太多重要数字。正确的答案是1厘米3。 这是因为街区的高度只有一个重要数字。 因此,答案只有一个重要数字。

    Rules for Rounding
    ::四舍五入规则

    To get the correct answer in the volume calculation above, rounding was necessary. Rounding is done when one or more ending digits are dropped to get the correct number of significant figures. In this example, the answer was rounded down to a lower number (from 1.2 to 1). Sometimes the answer is rounded up to a higher number. How do you know which way to round? Follow these simple rules:
    ::要获得上述量计算中的正确答案, 舍入是必要的。 当一个或一个以上末位数字被丢弃以获得重要数字的正确数字时, 要进行四舍五入。 在此示例中, 答案被四舍五入到一个较低的数字( 从 1.2 到 1 ) 。 有时答案被四舍五入到一个更高的数字 。 您如何知道如何四舍五入 ? 遵循这些简单的规则 :

    • If the digit to be rounded (dropped) is less than 5, then round down. For example, when rounding 2.344 to three significant figures, round down to 2.34.
      ::如果要四舍五入(删除)的数字小于5,那么四舍五入。例如,如果将2.344到3个重要数字四舍五入,则四舍五入到2.34。
    • If the digit to be rounded is greater than 5, then round up. For example, when rounding 2.346 to three significant figures, round up to 2.35.
      ::如果四舍五入的位数大于5, 则四舍五入。 例如, 当将2.346到3个重要数字四舍五入时, 将组合到2.35 。
    • If the digit to be rounded is 5, round up if the digit before 5 is odd, and round down if digit before 5 is even. For example, when rounding 2.345 to three significant figures, round down to 2.34. This rule may seem arbitrary, but in a series of many calculations, any rounding errors should cancel each other out.
      ::如果四舍五入的位数是 5 , 如果 5 之前的位数是奇数, 则四舍五入, 如果 5 之前的位数是偶数, 则四舍五入 。 例如, 当将2.345 到 3 个重要数字四舍五入时, 则四舍五入到2.334 。 这一规则可能看起来是武断的, 但在许多计算中, 任何四舍五入的错误都应该相互抵消 。

    Summary
    ::摘要

    • In any measurement, the number of significant figures is the number of digits thought to be correct by the person doing the measuring. It includes all digits that can be read directly from the measuring device plus one estimated digit.
      ::在任何测量中,重要数字的数目是进行测量的人认为正确的数字数,包括可从测量设备直接读取的所有数字数加上一个估计数字数。
    • To determine the number of significant figures in a measurement that someone else has made, follow the rules for counting significant figures.
      ::为确定他人计量中的重要数字数量,遵循计算重要数字的规则。
    • When measurements are used in a calculation, the answer cannot have more significant figures than the measurement with the fewest significant figures.
      ::当在计算时使用测量数据时,答案不能有比用数量最少的测量数据更重要的数字。
    • Rounding is done when one or more ending digits are dropped to get the correct number of significant figures. Simple rules state when to round up and when to round down.
      ::当一个或一个以上末位数字被丢弃以获得重要数字的正确数字时,就会进行四舍五入。 简单规则规定了何时进行四舍五入和何时进行四舍五入。

    Review
    ::回顾

    1. How do you determine the number of significant figures when you make a measurement?
      ::在进行测量时,如何确定重要数字的数量?
    2. Measure the width of a sheet of standard-sized (8.5 in x 11.0 in) loose-leaf notebook paper. Make the measurement in centimeters and express the answer with the correct number of significant figures.
      ::测量标准尺寸( 8.5 英寸 x 11. 0 英寸) 松叶笔记本纸的宽度。 以厘米计度, 用重要数字的正确数字表示答案 。
    3. How many significant figures do each of these measurements have?
      1. 0.04
      2. 500
      3. 1.50

      ::每个测量结果有多少重要数字? 0.04 500 1.50
    4. In this calculation, how many significant figures should there be in the answer? 1.0234 + 1.1 + 0.0056
      ::在这一计算中,答案中应有多少重要数字? 1.0234+1.1+0.0056
    5. Round each of these numbers to three significant figures:
      1. 1258
      2. 3274
      3. 6845

      ::每一轮上述数字的轮数达到3个重要数字:1258 3274 6845

    Resources
    ::资源