How many significant figures will the answer to 3.10 x 4.520 have?

1. How many significant figures will the answer to 3.10 x 4.520 have? You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundredth's place is not recognized as significant when, in fact, it is. 3.10 is the key number which has three significant figures. Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.

2. Another common error is for the student to think that 14 and 14.0 are the same thing. THEY ARE NOT. 14.0 is ten times more precise than 14. The two numbers have the same value, but they convey different meanings about how trustworthy they are. Sometimes student will answer this with five. Most likely you responded with this answer because it says 14.012 on your calculator (the correct answer should be reported as 14.01). This answer would have been correct in your math class because mathematics does not have the significant figure concept.

3. 2.33 x 6.085 x 2.1. How many significant figures in the answer? Answer - two. Which is the key number? Answer - the 2.1. Why? It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Answer = 30

4. (4.52 x 10-4) ÷ (3.980 x 10-6). How many significant figures in the answer? Answer - three. Which is the key number? Answer - the 4.52 x 10-4. Why? It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Notice it is the 4.52 portion that plays the role of determining significant figures; the exponential portion plays no role. Answer = 113.6

5. 4.20x3.52 Which is the key number? Both have 3 significant figures. In this case, the number with smaller value, regardless of the decimal point, is the key number (3.52). The answer is 14.8 (the correct answer should be reported as 14.78 as will be seen shortly.

6. Look at the following multiplication problem: he key number is 35.63 which has 4 significant figures. Therefore, the answer should be 88.55%

7. It is clear that the key number is 891 and the answer should have 3 significant figures. However, in cases where the answer is less than the key number as is our case, the answer retains an extra digit as a subscript. The answer is 546.6

8. When multiple operations are involved, do it in a step by step procedure. The parenthesis above has 97.7 as the key number.

9. The second process has 687 as the key number. Finally as the answer is less than the key number an additional digit was added as a subscript. It is noteworthy to observe that extra digits were retained temporarily in all steps and rounding off to the correct number of significant figures was done in the final answer.

10. Logarithms The digits to the left of the decimal point are not counted since they merely reflect the log 10x and they are not considered significant. The zeros to the right of the decimal point are all significant. Examples Log 2.0x103 = 3.30 (two significant figures in both terms). The blue digit in the answer is not significant.

11. Log 1.18 = 0.072 (three significant figures in both terms, the blue zero in the answer is significant) Antilog of 0.083 = 1.21 (three significant figures in both terms) Log 12.1 = 1.083 (three significant figures in both terms, the blue digit in the answer is not significant)

12. Errors Errors can be classified according to nature of the error into two types, determinate and indeterminate errors. A determinate error (sometimes called a systematic error) is an error which has a direction either positive or negative. An example of such an error is performing a weight measurement on an uncalibrated balance(for instance it always add a fixed amount to the weight).

13. Another important example is measuring volume using a burette that has extra or less volume than indicated on its surface. When using the abovementioned balance or burette, our results will always be higher or lower depending on whether these tools have positive or negative bias.This means that determinate error is unidirectional. Sometimes a determinate error can be significant if the analyst is careless or inexperienced neglecting enough drying times in a gravimetric procedure, using a too high indicator concentration in a volumetric procedure, etc.

14. An indeterminate error is a random error and has no direction where sometimes higher or lower estimates than should be observed are obtained. In most cases, indeterminate errors are encountered by lack of analyst experience and attention. Indeterminate errors are always present but can be minimized to very low levels by good analysts and procedures.