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Significant Figures and UncertaintyUncertaintyAll measurements are approximations—no measuring device can give perfect measurements without experimental uncertainty. Ex 1. We measure a mass at 13.2g and the analog scale measures in 0.1 increments. Therefore we have to measure to the nearest 0.1g and are therefore somewhat uncertain about the last digit. The last digit could be between 13.2g and 13.3g but the devise just isn’t accurate enoughEx. 2.What is a "significant figure"?The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures. Rules for deciding the number of significant figures in a measured quantity:(1) All nonzero digits are significant:1.234 g has 4 significant figures,1.2 g has 2 significant figures. (2) Zeroes between nonzero digits are significant:1002 kg has 4 significant figures,3.07 mL has 3 significant figures. (3) Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: 0.001 oC has only 1 significant figure,0.012 g has 2 significant figures. (4) Trailing zeroes that are also to the right of a decimal point in a number are significant: 0.0230 mL has 3 significant figures,0.20 g has 2 significant figures. (5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 miles may be 2 or 3 significant figures,50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as:5.06 × 104 calories (3 significant figures)By writing a number in scientific notation, the number of significant figures is clearly indicated by the number of numerical figures in the 'digit' term as shown by these examples.All digits in scientific notation are significantTo write numbers in scientific notationScientific Notation (also called Standard Form in Britain) is a special way of writing numbers: Like this:???Or this:?It makes it easy to use big and small values. OK, How Does it Work?Example 1: 700Why is 700 written as 7 × 102 in Scientific Notation ?700 = 7 × 100 and 100 = 102 so 700 = 7 × 102 ?Both 700 and 7 × 102 have the same value, just shown in different ways.?Example 2: 4,900,000,000 ? ? 1,000,000,000 = 109 , so 4,900,000,000 = 4.9 × 109 in Scientific Notation The number is written in two parts:Just the digits (with the decimal point placed after the first digit), followed by × 10 to a power that puts the decimal point where it should be (i.e. it shows how many places to move the decimal point). Example 3: 5326.6In this example, 5326.6 is written as 5.3266 × 103, because 5326.6 = 5.3266 × 1000 = 5.3266 × 103PRACTICE:How many sig figs?1.54568 10 1020 501.2 501.20 Convert to scientific nototation:1.54568 100 150.0 501.2 501.20 Rules for mathematical operationsIn carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate measurement involved in the calculation. (1) In addition and subtraction, the result is rounded off so that it has the same number of decimal places as the measurement having the fewest decimal places (or digits to the right). For example, 100 (assume 3 significant figures) + 23.643 (5 significant figures)?=?123.643,rounded to 124 (3 significant figures and no decimal places). (2) In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example, 3.0 (2 significant figures ) × 12.60 (4 significant figures)?=?37.8000rounded to 38 (2 significant figures).Rules for rounding off numbers *** Examples are to be rounded to 2 significant figures(1) If the digit to be dropped is greater than 5, the last retained digit is increased by one. For example, 12.6 is rounded to 13. (2) If the digit to be dropped is less than 5, the last remaining digit is left as it is. For example, 12.4 is rounded to 12. (3) If the digit to be dropped equals 5, but the number to the left is odd, then we increase the number to the left by 1. For example, 13.5 is rounded to 14(4) If the digit to be dropped is 5, but the number to the left is even, then we keep the number to the left the same. For example,12.5 is rounded to 12The rationale for this rule is to avoid bias in rounding: half of the time we round up, half the time we round down.Sample problems on significant figures1. ???37.76 + 3.907 + 226.4?=?2. ???319.15 - 32.614?=?3. ???125 - 0.23 + 4.109?=?4. ???2.02 × 2.5?=?5. ???600.0 / 5.2302?=?6. ???(5.5)3?=?7. ???0.556 × (40 - 32.5)?=?8. ???What is the average of 0.1707, 0.1713, 0.1720, 0.1704, and 0.1715? 9. ???3.00 x 105 - 1.5 x 102?=?(Give the exact numerical result, and then express that result to the correct number of significant figures).10. 1.251 -1.2510 = Rearranging FormulasTo transpose or rearrange a formula you may?add or subtract the same quantity to or from both sides?multiply or divide both sides by the same quantityGeneral rule- whatever function is on one side, to isolate, do the opposite function on the other sideExample 1:Rearrange the formula to isolate for xY = x+ 8 in order to makeSolutionTo make x the subject we must remove the 8 from the right. So, we subtract 8 from the right, but we remember that we must do the same to the left. So ify = x +8, subtracting 8 yieldsy?8=x+8?8y?8= x or x = y-8Example 2:Rearrange the formula y=3x to make x the subject.SolutionThe reason why x does not appear on its own is that it is multiplied by 3. If we divide 3x by 3 we obtainy3=3x3So, we can obtain x on its own by dividing both sides of the formula by 3.y3=x or x=y3Example 1: Isolate for mass D = M/VExample 2: Isolate for volumev = u + atConversions:Remember- King Henry Died Unusually Drinking Chocolate milkcenter11874500Practice:convert 56.2km to metres162.0cm to metres130km/h to m/sTimeThere are 60 sec in 1 minThere are 60 min in 1hourThere are 24 hours in 1 dayTo convert your units for time first write your given unit then apply your conversion factor23.4h to seconds103 seconds to min10045 s to days ................
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