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Significant Figures Powerpoint FEB 2024 (2).pdf
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- 2. Significant Digits or“Figures” • How to recognize significant figures when: – Taking a measurement – Reading a measurement – Performing a calculation
- 3. Uncertainty in Measurement Adigit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.
- 4. Why Is thereUncertainty? Measurements are performed with instruments. No instrument can read to an infinite number of decimal places Which of these balances has the greatest uncertainty in measurement?
- 5. Accuracy and Precisionin Measurements Accuracy: how close a measurement is to the accepted value. Precision: how close a series of measurements are to one another or how far out a measurement is taken. A measurement can have high precision, but not be as accurate as a less precise one.
- 6. Precision and Accuracy Accuracy refers to the agreement of a particular value with the true value. Precision refers to the degree of agreement among several measurements made in the same manner. Neither accurate nor precise Precise but not accurate Precise and accurate
- 7. Types of error Random Error (indeterminate Error) – measurement has an equal probability of being high or low. Systematic Error ( Determinate Error) – Occurs in the same direction each time (high or low), often resulting from poor technique or incorrect calibration.
- 8. Random Errors –Many experiments, when repeated, will yield slightly different results each time. Random errors are produced by unknown and unpredictable variations in the experimental situation. ➢Ex – when shooting bullets, it is unlikely that they have the same exit speed due to variations in the cartridge, fluctuations in gunpowder explosion etc… ➢Ex – when bouncing a ball, it is unlikely to rebound to the same height each time bounce is repeated. This may be due to orientation of ball in air, orientation of ball as it hits surface, uneven surface etc… ➢Small random errors means experiment has high precision.
- 9. 3.Systematic Errors– areerrors associated with a particular instrument or experimental technique. Systematic errors are not reduced by repeating measurements. ➢ -is measuring device perfectly accurate ( calibration)? ➢ - is device zeroed ? ➢ -is the experiment being performed in a wind or presence of air currents? ➢ -is there a bias on behalf of the experimenter? ( right handed/left handed) ➢ -is odometer on car accurate if tires have been changed ? ➢ most electrical equipment is assumed to have a 5% systematic error ➢ Small systematic errors means experiment has high accuracy
- 10. Percent error % error= |experimental value – theoretical value| x 100% theoretical value
- 11. Significant Figures Scientistuse significant figures to determine how precise a measurement is Significant digits in a measurement include all of the known digits plus one estimated digit
- 12. Significant Figures areused to indicate the precision of a measured number or to express the precision of a calculation with measured numbers. In any measurement the digit farthest to the right is considered to be estimated. 0 1 2 1.3 2.0
- 13. Using two differentrulers, I measured the width of my hand to be 4.5 centimeters and 4.54 centimeters. Explain the difference between these two measurements. Question For Thought
- 14. The first measurementimplies that my hand is somewhere between 4.5 and 4.9 cm long. There is a uncertainty in this number because we have to estimate. The second measurement implies that my hand is between 4.5 and 4.6 cm long. This measurement is more certain due to its greater precision.
- 15. Significant figures arenecessary to reduce uncertainty in our measurements. Significant figures indicate the precision of the measured value!! 4.5 cm 2 significant figures Uncertain 4.54 cm 3 significant figures More certain due to greater precision
- 16. Significant Figures Scientistuse significant figures to determine how precise a measurement is Significant digits in a measurement include all of the known digits plus one estimated digit So when reading an instrument… Read instrument to the last digit that you know Estimate or “eyeball” the final digit
- 17. For example… Lookat the ruler below Each line is 0.1cm You can read that the arrow is on 13.3 cm However, using significant figures, you must estimate the next digit That would give you 13.30 cm
- 18. Let’s try thisone Look at the ruler below What can you read before you estimate? 12.8 cm Now estimate the next digit… 12.85 cm
- 19. The same rulesapply with all instruments The same rules apply Read to the last digit that you know Estimate the final digit
- 20. Let’s try graduatedcylinders Look at the graduated cylinder below What can you read with confidence? 56 ml Now estimate the last digit 56.0 ml
- 21. One more graduatedcylinder Look at the cylinder below… What is the measurement? 53.5 ml
- 22. Rules for Significantfigures Rule #1 All non zero digits are ALWAYS significant How many significant digits are in the following numbers? •274 •25.632 •8.987 •3 Significant Figures •5 Significant Digits •4 Significant Figures
- 23. Rule #2 Allzeros between significant digits are ALWAYS significant How many significant digits are in the following numbers? 504 60002 9.077 3 Significant Figures 5 Significant Digits 4 Significant Figures
- 24. Rule #3 AllFINAL zeros to the right of the decimal ARE significant How many significant digits are in the following numbers? 32.0 19.000 105.0020 3 Significant Figures 5 Significant Digits 7 Significant Figures
- 25. Rule #4 Allzeros that act as place holders are NOT significant Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal
- 26. For example 0.0002 6.02 x1023 100.000 150000 800 1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits 1 Significant Digit How many significant digits are in the following numbers?
- 27. Rule #5 Allcounting numbers and constants have an infinite number of significant digits For example: 1 hour = 60 minutes 12 inches = 1 foot 24 hours = 1 day
- 28. How many significantdigits are in the following numbers? 0.0073 100.020 2500 7.90 x 10-3 670.0 0.00001 18.84 2 Significant Digits 6 Significant Digits 2 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit 4 Significant Digits
- 29. Numbers with nodecimal are ambiguous... Does 5000 ml mean exactly 5000? Maybe.... Maybe Not! So 5000, 500, 50, and 5 are all assumed to have 1 significant figure If a writer means exactly 5000, he/she must write 5000. or 5.000 x 103
- 30. Rules Rounding SignificantDigits Rule #1 If the digit to the immediate right of the last significant digit is less that 5, do not round up the last significant digit. For example, let’s say you have the number 43.82 and you want 3 significant digits The last number that you want is the 8 – 43.82 The number to the right of the 8 is a 2 Therefore, you would not round up & the number would be 43.8
- 31. Rounding Rule #2 If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure Let’s say you have the number 234.87 and you want 4 significant digits 234.87 – The last number you want is the 8 and the number to the right is a 7 Therefore, you would round up & get 234.9
- 32. Rounding Rule #3 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up 78.657 (you want 3 significant digits) The number you want is the 6 The 6 is followed by a 5 and the 5 is followed by a non zero number Therefore, you round up 78.7
- 33. Rounding Rule #4 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even. 2.5350 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even 2.54
- 34. Say you havethis number 2.5250 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even and it already is 2.52
- 35. Let’s try theseexamples… 200.99(want 3 SF) 18.22 (want 2 SF) 135.50 (want 3 SF) 0.00299 (want 1 SF) 98.59 (want 2 SF) 201 18 136 0.003 99
- 36. Calculations with SigFigs Adding or subtracting: answer can have no more places after the decimal than the LEAST of the measured numbers. Count # decimal places held (nearest .1? .01? .001?) Answer can be no more accurate than the LEAST accurate number that was used to calculate it.
- 37. For Example: 5.50 grams +8.6 grams -------- 14.1 grams 52.09 ml - 49.7 ml ------------- 2.39 ml --> 2.4 ml
- 38. Calculations with SigFigs • Multiplying or dividing: round result to least # of sig figs present in the factors – Answer can’t have more significant figures than the least reliable measurement. • COUNT significant figures in the factors
- 39. 56.78 cm x2.45cm = 139.111 cm2 Round to 3 sig figs = 139cm2 75.8cm x 9.6cm = ?
- 40. Now let’s dosome math..... (round answers to correct sig figs!) 5.0033 g + 1.55 g answer: 6.55 g Did you need to count sig figs? NO!
- 41. Try this one.... 4.80ml - .0015 ml answer: 4.80 ml (one might say .0015 is insignificant COMPARED TO 4.80)
- 42. Now try these... 5.0033g / 5.0 ml answer: 1.0 g/ml Did you have to count sig figs? YES!
- 43. Rules for significantfigures in mathematical operations Multiplication and Division – number of significant figures you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements. Ex. 6.38 x 2.0 = 12.76 -→ 13 (2 significant figures)
- 44. Rules for significantfigures in mathematical operations Addition and Subtraction: The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. 6.8 + 11.934= 18.734 → 18.7 ( 3 sig figs)
- 45. Here’s a tougherone..... 3.0 C/s x 60 s/min x 60 min/hr = answer: 10800 C/hr rounds to 11000 C/hr Note: Standard conversion factors never limit sig. figures- instruments and equipment do.
- 46. Scientific Notation Scientificnotation is used to express very large or very small numbers I consists of a number between 1 & 10 followed by x 10 to an exponent The exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal
- 47. Large Numbers Ifthe number you start with is greater than 1, the exponent will be positive Write the number 39923 in scientific notation First move the decimal until 1 number is in front – 3.9923 Now at x 10 – 3.9923 x 10 Now count the number of decimal places that you moved (4) Since the number you started with was greater than 1, the exponent will be positive 3.9923 x 10 4
- 48. Small Numbers Ifthe number you start with is less than 1, the exponent will be negative Write the number 0.0052 in scientific notation First move the decimal until 1 number is in front – 5.2 Now at x 10 – 5.2 x 10 Now count the number of decimal places that you moved (3) Since the number you started with was less than 1, the exponent will be negative 5.2 x 10 -3
- 49. Scientific Notation Examples 99.343 4000.1 0.000375 0.0234 94577.1 9.9343x 101 4.0001 x 103 3.75 x 10-4 2.34 x 10-2 9.45771 x 104 Place the following numbers in scientific notation:
- 50. Going from ScientificNotation to Ordinary Notation You start with the number and move the decimal the same number of spaces as the exponent. If the exponent is positive, the number will be greater than 1 If the exponent is negative, the number will be less than 1
- 51. Going to OrdinaryNotation Examples 3 x 106 6.26x 109 5 x 10-4 8.45 x 10-7 2.25 x 103 3000000 6260000000 0.0005 0.000000845 2250 Place the following numbers in ordinary notation: