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Notes on Significant Figures (“Sig Figs”) Mathematical tool used by scientists when expressing measured data and calculations involving measured data Why am I just now finding out about this? • Sig figs are important “tools” used by ALL scientists, but are generally not emphasized below Chemistry on the high school level (why trouble their heads and Bio is not as quantitative as Chemistry or Physics) • Chemistry and Physics (the “Physical Sciences”) involve many measurements – among them time, displacement, velocity, moles, grams, liters, molarities, voltage, frequency, radius, watts, etc!!! • Chemists and Physicists therefore must present data in a manner that all scientists will be able to: – A) Trust the data – B) Understand the level of sophistication of the equipment that was used to take the measurments Do Math Teachers use this system? • Generally speaking, NO. The reason for that is simple… • Scientists use MEASURED VALUES (taken from some instrument) and most math problems involve “counted” numbers. (ie: How many students are in the room? Not: How tall is that student?) • How tall must have a unit associated with it! • Sig Figs help communicate the level of sophistication of the measurement device •THAT’S WHAT SIG FIGS DO!!! Example • Let’s say Ron takes a piece of yarn and ties a knot the length of the yarn each meter for 10 meters. • Let’s say Jill has a meter stick. • They both have to measure the length of the lab counters in the classroom. • Each one MUST have a different answer – Ron’s will be less “precise” but each will be “correct” if they “follow the rules”. What is this measurement? I’m measuring the same thing. Shouldn’t the recorded value be the same? Rule 1 – All non-zero digits are significant Measurement Number of Sig Figs 0.456m 3 1200g 2 95.7cm 3 0.88117m 5 Rule 2 – Zeros between two non-zero digits are significant (the “sandwich” rule) Measurement Number of Sig Figs 0.405g 3 9001m 4 20.03ml 4 10.8007kg 6 Rule 3 – Zeros to the right of a decimal are significant if they are also to the right of a non-zero digit Measurement Number of Sig Figs 4.00m 3 79.0030g 6 0.007cm 65000L 1 2 Let’s Explore that rule: • Look at the difference in meaning between: • 4 miles – this person might be using a map with a ruler that has a 10 mile scale • 4.0 miles – this person might be reading a map with a ruler that has a 1 mile scale • 4.00 miles – this person might be reading a map with a ruler that has a 0.1 mile scale Rule 4 – Lone zeros are never significant Measurement Number of Sig Figs 0.456m 3 0.15g 2 0.230cm 3 0.88117m 5 Rule 5 – Zeros to the left of an understood decimal are not significant unless a line is used (or a decimal is properly placed ) Measurement Number of Sig Figs 40m 1 40.m 2 40.0m 3 93,000,000 2 How is a line used? Measurement Number of Sig Figs 1 2 3 4 Let’s Get Some Practice First: • 35.1 kg 3 • 45,000 cm • 80.2 mg • 0.003 mm 2 3 1 0.40 m 200.0 L 2 4 2.500 km 4.50 x 103 g 4 3 Going Further… • Well, at this point, any time a person on the street asks you how many sig figs there are in a number, now you’ll know. • Seriously, though…none of this is important unless you are taking empirical data. • Empirical data describes the simplest type of data – such as the length of an index card in cm or the mass of a kidney bean in grams, • As chemists and physicists, this is what we’re all about! Using Sig Figs in Calculations • Just stating that a pane of glass measures 9.92 cm on a side communicates something about the device used to measure that length. • However, often times we need to know the area covered or the volume of the object. • Most often, your labs will involve calculating some values. There are some simple rules for manipulating this data. Rule for Multiplying and Dividing with Significant Figures • The final answer must have no more sig figs than the measurement with the least number of sig figs • Let’s say all measurements are in cm: 890 • 42.4 x 21 = 890.4 (calculator) • 100 x 24.887 = 2488.7 (calculator) 2000 • 29.9 x 0.005 = 0.1495 (calculator) 0.1 • 87.9 / 0.40 = 2197.5 (calculator) 2200 • 35.000 / 7.00 = 5 (calculator) 5.00 Rule for Adding and Subtracting with Significant Figures • The final answer must have no more decimal places than the measurement with the least number of decimal places • Let’s say all measurements are in cm: 63 • 42.4 + 21 = 63.4 (calculator) • 0.10 + 24.887 = 24.987 (calculator) 24.99 • 30.0 - 0.005 = 29.995 (calculator) 30.0 88.30 • 87.90 + 0.40 = 88.3 (calculator) • 35.000 - 7.00 = 28 (calculator) 28.00