25. DETERMINING SIGNIFICANT FIGURES The objectives are to: The objectives are to: -Know what significant figures are. -Know which numbers are significant. -Distinguish between significant and not- significant zeros. -Be aware that the number of figures in the scientific notation also indicate the precision. -Be able to express the result of a calculation in the right amount of significant figures. -Be able to express the result of a measurement in the right amount of significant figures. -Know the four additional principles concerning significant figures -Know what significant figures are. -Know which numbers are significant. -Distinguish between significant and not- significant zeros. -Be aware that the number of figures in the scientific notation also indicate the precision. -Be able to express the result of a calculation in the right amount of significant figures. -Be able to express the result of a measurement in the right amount of significant figures. -Know the four additional principles concerning significant figures -Know what significant figures are. -Know which numbers are significant. -Distinguish between significant and not- significant zeros. -Be aware that the number of figures in the scientific notation also indicate the precision. -Be able to express the result of a calculation in the right amount of significant figures. -Be able to express the result of a measurement in the right amount of significant figures. -Know the four additional principles concerning significant figures -Know what significant figures are. -Know which numbers are significant. -Distinguish between significant and not- significant zeros. -Be aware that the number of figures in the scientific notation also indicate the precision. -Be able to express the result of a calculation in the right amount of significant figures. -Be able to express the result of a measurement in the right amount of significant figures. -Know the four additional principles concerning significant figures

26. WHAT ARE SIGNIFICANT FIGURES? During any calculation, addition, subtraction, multiplication, or division, your answer could be expressed with too few or too many significant figures. These numeric values may imply a precision that might not exist in the experiment being evaluated. If you round off incorrectly, your answer will have an incorrect number of significant figures and will lose precision. Therefore, we will need to develop rules for determining the correct number of significant figures in a number and apply these rules to calculations. During any calculation, addition, subtraction, multiplication, or division, your answer could be expressed with too few or too many significant figures. These numeric values may imply a precision that might not exist in the experiment being evaluated. If you round off incorrectly, your answer will have an incorrect number of significant figures and will lose precision. Therefore, we will need to develop rules for determining the correct number of significant figures in a number and apply these rules to calculations. EXAMPLE EXAMPLE 1.024 x 1.2 = 1.2288 Too many numerals Too precise1.024 x 1.2 = 1 Too few numerals Not precise enough 1.024 x 1.2 = 1.2288 Too many numerals Too precise1.024 x 1.2 = 1 Too few numerals Not precise enough

27. WHICH NUMBERS ARE SIGNIFICANT? First we need to learn how to evaluate the number of significant figures a given number contains. This is necessary for calculations of any type: addition, subtraction, multiplication and division. In this table the significant figures are underlined. 5 significant figures4 significant figures3 significant figures 678.90 345.6 3.456 0.03456 23.4 2.34 0.234 Note that the zeros in 0.234 and 0.03456 are NOT significant figures.

28. ZEROS In large numbers Large numbers containing zeroes, such as 45,600 grams, pose a special problem. As the number is written, we cannot tell whether the two zeroes indicate the precision of the measurement or whether the zeroes merely locate the decimal point. If the zeroes indicate the precision, they are significant and the implied uncertainty is + 1. This means that the measurement lies between 45,599 and 45,601. If, however, the zeroes merely locate the decimal point and are not significant, the implied uncertainty is + 100. Then we know the measurement lies between 45,500 and 45,700. Thus a number written in this form might be ambiguous.

29. EXAMPLE 45,600 grams SIGNIFICANT 45,600 grams + 1 gram \ / 45,60145,599 5 significant figures INSIGNIFICANT 45,600 grams + 100 grams \/ 45,70045,500 3 significant figures In small numbers Numbers less than one may also contain zeros that are not significant. Nobody will disagree that the zero in 0.12 is insignificant and merely marking the decimal point. However, all subsequent zeros are also insignificant, until the leftmost digit of the actual number. In 0.00123 the number of significant digits is three.

30. SCIENTIFIC NOTATION Precision as to the significance of a value may be avoided by expressing the number in standard exponential notation. Scientific notation Normal notation 4.5600 x 104 4.56 x 104 45,600 4,5600 1.23 x 10-3 0.00123 0.00123

31. If the number 45,600.0 contains 5 significant figures, the number would be expressed as 4.5600 x 104. This notation implies 5 significant figures. If only 3 numbers are significant, the number would be expressed as 4.56 x 104. If the number 45,600.0 contains 5 significant figures, the number would be expressed as 4.5600 x 104. This notation implies 5 significant figures. If only 3 numbers are significant, the number would be expressed as 4.56 x 104. The number 0.00123 can be expressed as 1.23 x 10-3. The zeroes are merely marking the decimal point; therefore. 1.23 x 10-3 contains 3 significant figures. The number 0.00123 can be expressed as 1.23 x 10-3. The zeroes are merely marking the decimal point; therefore. 1.23 x 10-3 contains 3 significant figures.

32. CALCULATIONS Adding and subtracting You have learned how to determine how many significant figures are in a number. Now you will learn how many significant figures should be expressed in the result of a calculation. When adding or subtracting quantities, the rule is to determine which number in the calculation has the least number of digits to the right of the decimal point. Your result will have that same number of digits to the right of the decimal point.

33. EXAMPLE 26.46 - 4.123 22.337 26.46 + 4.123 30.583 22.34 30.58

34. EXAMPLE 2.634 - -0.02 - 2.614 2.634 + 0.02 2.654 2.61 2.61 2.65 2.65

35. Multiplying and dividing The multiplication and division of numbers use a different rule for determining the number of significant figures. When multiplying or dividing, determine which number entering the calculation has the smallest total number of significant figures regardless of the decimal points position. Your result will have that same number of significant figures.

36. Example least number of significant figures 2.61 / 1.2 = 2.175 2.61 x 1.2 = 3.132 Round to: 2.2 Round to: 3.1