Significant digits, Uncertainties, Error Calculations sph3U/3UB 1st Day notes Significant digits, Uncertainties, Error Calculations

Significant digits/figures The concept of significant figures is often used in connection with rounding. A practical calculation that uses any irrational number necessitates rounding the number, and hence the answer, to a finite number of significant figures. The significant digits/figures of a number are those digits that carry meaning contributing to its precision.

Rules for Significant digits/figures All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5). Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2. Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.

Rules for Significant digits/figures Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures.

Rules for Significant digits/figures This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.

Rules for Significant digits/figures The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue, but none that are Universal.

What we will do A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant. (McMaster uses this sometimes.) All labs will use the precision of measuring devices, so there are never any ambiguous cases in labs. On a test: ask if you are unsure. Otherwise, I have to accept a range of answers to various amounts of sig digs.

Scientific Notation Used to represent very large and very small numbers Always written as a value between 1 and 10 with a x10 to a power Sig Digs are maintained (the x10 part is not used for sig digs) Ex) 23456 = 2.3456 x 104 0.0000125 = 1.25 x 10-5

Math Rules When adding or subtracting, round your answers to the least number of decimal places (not sig digs) Ex) 2.01 + 0.007 = 2.017 = 2.02 (2 decimals) When multiplying or dividing, round your answers to the least number of significant figures. 1.50 m/s * 0.50 s = 0.75 m (2 s.d.)

Sci Notn and sig digs ws Hand out

Metric System Metric prefix word Metric prefix symbol Power of ten nano n 10-9 micro µ 10-6 milli m 10-3 centi c 10-2 kilo k 103 mega M 106 giga G 109

Metric system The standard units in Physics are kilograms (kg), seconds (s) and metres (m). The Newton (N) is 1 kgm/s2 and The Joule (J) is 1 kgm2/s2. To use standard units, students must be able to convert units to standard units for effective communication of data in labs and to compare to given values.

Examples 12.0 cm is converted into metres by shifting the decimal place left two spaces. 12.0 cm = 0.120 m 12.0 g is converted into kg by dividing by 1000 or shifting the decimal left three times. 12.0 g = 0.0120 kg If a mass is given as 12.0 mg, then the decimal shifts left 3 times to grams and then 3 more to kg. Once we get to really small/large numbers, scientific notation is needed. 12.0 mg = 0.0000120 kg = 1.20 x 10-5 kg.

Metric practice 12.0 µm 33.45 mm 12.0 µg 12.67 ns 123.4 Gm 7654 Mg Convert to standard units: kg, m, s: 12.0 µm 33.45 mm 12.0 µg 12.67 ns 123.4 Gm 7654 Mg 45.258 Ms 0.000458 km 0.025478 cs

Metric practice 12.0 µm  1.20 x 10-5 m 33.45 mm  0.03345 m Convert to standard units: kg, m, s: 12.0 µm  1.20 x 10-5 m 33.45 mm  0.03345 m 12.0 µg  1.20 x 10-8 kg 12.67 ns  1.267 x 10-8 s 123.4 Gm 1.234 x 1011 m 7654 Mg  7.654 x 106 kg 45.258 Ms  4.5258 x 107 s 0.000458 km  0.458 m 0.025478 cs  2.5478 x 10-4 s

Unit analysis Useful to catch simple algebra mistakes! Carry units all the way through your calculations. Multipliers used when it’s not just a simple decimal shift, when two units are changed at the same time. Ex) Convert 232 km/h into standard units. 232 km (1000 m) (1h) = h km 3600 s 64.4 m/s

ws Unit analysis ws

Uncertainties with labs Uncertainties affect all sciences. Experimental errors and human errors in reading measuring apparatus cause errors in experimental data. A system of rules is required to indicate errors and to plot graphs indicating error. It is important to include errors in your labs and analysis of data problems. Significant digits are one way in which scientists deal with uncertainties.

Uncertainties with labs Sig dig rules are shortcuts to looking at uncertainties. Sig digs are not perfect rules. The error must match the number of decimals of the measurement. (4.55 ± 0.002 is not possible). In experiments, a series of measurements may be done and repeated carefully (precisely) many times but still have differences due to error.

Digital devices Note: Fr digital devices we’re limited to the last decimal, and we can NOT guess beyond that. In this case we use that last decimal as the error. Ex) Mass scale is 12.03 g, so our error must be ± 0.01 g, NOT ± 0.005 g as the device can’t ever give us 3 decimals!

Error types Errors are random uncertainties that may include the observer (momentary lapse) or the environment (temperature, material variations, imperfections.....). Any built in errors with devices are called systematic errors. We usually use half the smallest division to indicate this. Random uncertainties can be reduced by repeating measurements and by using graphs. Errors show the level of confidence we have in a measure.

Error types A measure is written as, for example; 2.08 m ± 0.05 m The ± is the absolute error. This can be converted to a percent of the measure into a relative error: 2.08 ± (0.05/2.08)x100 = 2.08 m ± 2.4% Graphs will be plotted with absolute or relative errors. (Excel handles this easily). See Excel graphing practice (website) for more info on this.

Measures with errors ws Reading rulers, thermometers and triple beam balances…oh my! WS

Errors on Labs All lab reports will require a list of 5 errors. One of these errors must be a list of +/- errors (precision errors) for each device used. (These all count as ONE error). You need to come up with 4 more errors – systematic or random. For each error, you will need to explain how they affect the lab and how you might improve this. Ex: There may be some dirt on the mass pan of the mass scale, which adds extra mass. To fix this, I would clean the mass pan of dust and debris.

Rounding Rules For addition or subtraction, you round the answer to least number of decimal places. Ex) 2.44 + 0.5 = 2.9 For multiplication and division, you round the answer to the least number of sig digs. Ex) 2.40 x 0.50 = 1.2 Ex) 1.20 x 10-3 / 1.0 x 10-5 = 1.2 x 102

Math Ws’s Practice and Hand In sheets

Calculations with Error When adding/subtracting; you add the absolute errors 1) (1.3 ± 0.1) m + (1.1 ± 0.2) m = (2.4 ± 0.3) m 2) (6.6 ± 0.5) m - (1.6 ± 0.5) m = (5.0 ± 1.0) m This method yields a worst case scenario in the errors!! Limitation: a small difference between large numbers give large uncertainties: (400 ± 5) s - (350 ± 5) s = (50 ± 10) s

Calculations with Error When multiplying or dividing; you add the relative errors and express your final answers as absolute errors. 1) (20 m/s ± 2.4%) (4.2 s ± 3.6 %) = 84 m ± 6.0 % = (84 ± 5) m (Note: error is rounded to match decimals of answer calculated (which was rounded by sd)). 2) (5.0 ± 0.5) m / (1.0 ± 0.1) s = 5.0 m/s ± [0.5m/5.0m + 0.1 s/1.0s] (5.0 m/s) = (5.0 ± 1.0) m/s

Error calculation WS

Examples to show rounding rules - Optional 1.234 <--the 4 is "fuzzy" in uncertainty (least significant) x 1.1 <- the 1 is also "fuzzy as it is least significant. 0.1234 <- all these are "fuzzy" as used "fuzzy" 1 to find them. 1.234 1.3574<----- the last 4 digits are "fuzzy" so we round off as 1.4 This is the basis for why we round off to 2 sig digs for that example.

Examples to show rounding rules 1.234 <---------the 4 is fuzzy + 0.011 <---------- the last 1 is fuzzy 1.245 <---------the 5 is the fuzzy digit As the 5 is the last number, no rounding is done. Answer is 1.245 Error worksheet