A positive feature of using scientific notation is that only the significant figures need to appear in the number.
When approximate numbers are added or subtracted, the results should also be represented in terms of the least precise number. Since this is a relatively simple rule to master, note that the answer follows the aforementioned rule of precision. Therefore,
These numbers have two, one, and three decimal places, respectively. The least precise number (least number of decimal places) is 2.8, a value carried only to the tenths position. Therefore, the answer must be calculated to the tenths position only. Thus, the correct answer is 4.7 L (the last 6 and 7 are dropped from the full answer 4.667, and the result is rounded up to provide 4.7 L).
In multiplication and division of approximate numbers, finding the number of significant digits in the appropriate numbers is used to determine how many digits to keep (i.e. where to truncate). One must understand significant digits in order to determine the correct number of digits to keep or remove in multiplication and division problems. As noted earlier in this section, the digits 1 through 9 are considered to be significant. Thus, the numbers 123, 53, 7492, and 5 contain three, two, four, and one significant digit, respectively. The digit zero must be considered separately.
Zeros are significant when they occur between significant digits. In the following examples, all zeros are significant: 10001, 402, 1.1001, and 500.09 (five, three, five, and four significant figures, respectively). Zeroes are not significant when they are used as place holders. When used as a holder, a zero simply identifies where a decimal is located. For example, each of the following numbers has only one significant digit: 1000, 500, 60, 0.09, and 0.0002. In the numbers 1200, 540, and 0.0032, there are two significant digits, and the zeroes are not significant. Once again, when zeros follow a decimal and are preceded by a significant digit, the zeros are significant. In the following examples, all zeroes are significant: 1.00, 15.0, 4.100, 1.90, 10.002, and 10.0400. For 10.002, the zeroes are significant because they fall between two significant digits. For 10.0400, the first two zeroes are significant because they fall between two significant digits; the last two zeroes are significant because they follow a decimal and are preceded by a significant digit. Finally, when approximate numbers are multiplied or divided, the result is expressed as a number having the same number of significant digits as the expression in the problem having the least number of significant digits.
When truncating (removing final, unwanted digits), rounding is normally applied to the last digit to be kept. Thus, if the value of the first digit to be discarded is less than 5, one should retain the last kept digit with no change. If the first digit to be discarded is 5 or greater, increase the last kept digit’s value by one. Assume, for example, only the first two decimal places are to be kept for 25.0847 (the 4 and 7 are to be dropped). The number is then 25.08. Since the first digit to be discarded (4) is less than 5, the 8 is not rounded up. If only the first two decimal places are to be kept for 25.0867 (the 6 and the 7 are to be dropped), it should be rounded to 25.09. Since the first digit to be discarded (6) is 5 or more, the 8 is rounded up to 9.
In some desalination studies, a measurement in gallons or liters may be required. Although a gallon or liter may represent an exact quantity, the measurement instruments that are used are capable of producing approximations only. Using a standard graduated flask in liters as an example, can one determine whether there is exactly 1 liter? Likely not. In fact, one would be pressed to verify that there was a liter within the ± 1/100 of a liter. Therefore, depending upon the instruments used, the precision of a given measurement may vary.
If a measurement is given as 16.0 L, the zero after the decimal indicates that the measurement is precise to within 1/l0 L. Given a measurement of 16.00 L, one has precision to the 1/l00 L. As noted, the digits following the decimal indicate how precise the measurement is. Thus, precision is used to determine where to truncate when approximate numbers are added or subtracted.
3.8 Generic Problem Solving Techniques
Certain methods of logic and techniques of calculation are fundamental to the solution of many problems, and there is a near-infinite number of methods that can be used for technical problem solving. Words such as creativity, ingenuity, originality, etc., appear in all these approaches. What do they all have in common? They provide a systematic, logical approach to solving problems, and what follows is the authors’ definition of a generic approach.
The methodology of solving problems has been discussed by most mathematicians and logicians since the days of Aristotle. Heuristic (“serving to discover”) is the term often given to this study of the methods and rules of solving problems. Nearly always, a stepwise approach to the solution is desirable. The broad steps are:
1 Understand the problem.
2 Devise a plan for the solution.
3 Carry out the plan.
4 Look back and check the problem solution.
Details on each of the four points follows.
3.8.1 Understanding the Problem
The first step in the problem solving approach is to understand the problem. Get acquainted with the problem. Read it and attempt to visualize it in its entirety without concerning yourself with details. Be particularly sure that you understand what is known and what is unknown.
Next, organize the available data. Engineering and science data are often insufficient, redundant, or contradictory. The units may have to be converted. The accuracy of the data must also be verified. Sketch a flow sheet. This step will usually facilitate the “get acquainted” operation. If the problem is a simple one, this sketch may consist of nothing more than a small box with arrows. For chemical processes, the diagrams should indicate the amount and composition of all streams entering or leaving the process.
Finally, introduce suitable and consistent symbols for the unknowns. It is usually helpful to adopt a notation which will serve to identify the unknown. This is particularly important when a problem has many unknowns.
3.8.2 Devising a Plan for the Solution
Once the problem is understood, the next step in the problem solution process is to develop a plan to solve it. Memory plays an important part in even the most creative thinking. It is almost impossible to generate ideas or plans with little to no background knowledge. Although mere memory is not enough to create new ideas, such ideas must be based on experience and acquired knowledge. When devising a plan to solve a problem, here are some things to consider:
1 Have I seen a similar or slightly different problem before?
2 Is there a very general equation or theorem which applies to this type of problem?
3 Can the problem be restated or simplified?
4 If I cannot solve the problem, can I solve a related problem?
5 Can part of the problem be solved? What data are lacking to solve the rest of the problem? Did I use all of the data given?
3.8.3 Carrying Out the Plan
