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Mobile App Chapter 2: Problem 51
Round each number to two significant figures. (a) \(2.34\) (b) \(2.35\) (c) \(2.349\) (d) \(2.359\)
Short Answer
Expert verified
(a) 2.3, (b) 2.4, (c) 2.3, (d) 2.4
Step by step solution
01
Identify the Significant Figures
To round a number to two significant figures, locate the second digit from the left that is not zero. This will be the last digit in our rounded number.
02
Determine the Rounding Digit
Look at the digit immediately to the right of the second significant figure. If this digit is 5 or more, round up the second significant figure by one. If it is less than 5, keep the second significant figure unchanged.
03
Round (a) 2.34
The first two significant figures of 2.34 are 2 and 3. The digit to the right of the second significant figure is 4, which is less than 5. So, we keep the second figure unchanged and (a) 2.34 rounded to two significant figures is 2.3.
04
Round (b) 2.35
The first two significant figures of 2.35 are 2 and 3. The digit to the right is 5, therefore we round up the second significant figure by one. Hence, (b) 2.35 rounded to two significant figures is 2.4.
05
Round (c) 2.349
The first two significant figures of 2.349 are 2 and 3. The third digit is 4, which is less than 5, so we do not round up. Thus, (c) 2.349 rounded to two significant figures is 2.3.
06
Round (d) 2.359
The first two significant figures of 2.359 are 2 and 3. The third digit is 5, so we round the second significant figure up by one. Therefore, (d) 2.359 rounded to two significant figures is 2.4.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rounding Numbers
Rounding numbers is a crucial technique to simplify figures when precision is not necessary or the amount of data is too overwhelming for practical use. It involves adjusting the value of a number to a certain degree of accuracy, which is usually dictated by the significant figures or decimal places that one is aiming for.
In the process of rounding, the focus is on the digit that determines if the number will be rounded up or down. For instance, in the context of our exercise, numbers like 2.34 and 2.35 are rounded to the nearest significant figure, which is the second digit from the left. Here, the critical concept is the next digit (third from the left), which influences whether the second digit stays the same or increases by one. Remember that if this third digit is five or more, you round up; if less than five, you do not round up.
In the process of rounding, the focus is on the digit that determines if the number will be rounded up or down. For instance, in the context of our exercise, numbers like 2.34 and 2.35 are rounded to the nearest significant figure, which is the second digit from the left. Here, the critical concept is the next digit (third from the left), which influences whether the second digit stays the same or increases by one. Remember that if this third digit is five or more, you round up; if less than five, you do not round up.
Chemical Calculations
In chemical calculations, the proper use of significant figures is paramount as it maintains the precision and accuracy of measurements. These calculations often involve complex equations and the use of constants that already have a defined number of significant figures. Thus, maintaining the consistency of significant figures throughout your calculations ensures that the final answer is reliable and reflects the precision of the chemicals and instruments used.
For example, when calculating molar concentrations or reactant-product yields, the number of significant figures might dictate the validity of the experiment’s results. In case the calculations produce more figures, the tendency is to round off to match the figures provided by the measurements' precision.
For example, when calculating molar concentrations or reactant-product yields, the number of significant figures might dictate the validity of the experiment’s results. In case the calculations produce more figures, the tendency is to round off to match the figures provided by the measurements' precision.
Digits and Decimal Places
Understanding the difference between digits and decimal places is also essential when working with significant figures. Digits refer to any numerical symbol from 0 to 9 that makes up a number, whereas decimal places are the number of digits located to the right of the decimal point.
Decimal places are particularly important when considering the accuracy of a measurement or calculation. The exercise in question emphasizes the importance of decimal places in the rounding process. As we noted, the number behind our second significant figure determines if we maintain or increase the figure before it. This orientation towards the decimal places helps ensure consistency in rounding and is a foundational concept in mathematics and science.
Decimal places are particularly important when considering the accuracy of a measurement or calculation. The exercise in question emphasizes the importance of decimal places in the rounding process. As we noted, the number behind our second significant figure determines if we maintain or increase the figure before it. This orientation towards the decimal places helps ensure consistency in rounding and is a foundational concept in mathematics and science.
