The Rules of the Game

Over hundreds of years ago, physicists and other scientists developed a traditional way of expressing their observations.  International System of Units (SI) – metric system.

 The amount of information that the measuring instrument can provide.  The precision of a measuring tool depends on how readable the tool is.  Ruler example ◦ 14.5 m is all you can read off of a cm ruler ◦ To say that the line was would give the impression that a very precise tool had been used not a ruler.

 When measuring with a mechanical instrument (ruler, triple beam balance etc), record all the digits that are marked on the instrument’s scale and estimate (and only one) more digit.  When measuring with an electronic instrument, record all the digits on the readout. Consider the last digit to be approximate.  Round calculated answers only once, at the end of the calculation, so that the number of significant digits reflects the precision of the original measurements.

 Every experimental measurement has a degree of uncertainty.  The volume, V, at right is certain in the 10’s place, 10mL<V<20mL  The 1’s digit is also certain, 17mL<V<18mL  A best guess is needed for the tenths place. 5

 We can see the markings between cm  We can’t see the markings between the.6-.7  We must guess between.6 &.7  We record 1.67 cm as our measurement  The last digit an 7 was our guess...stop there 6

What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) cm

cm ?

 Do you see why Measured Numbers have error…you have to make that Guess!  All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate.  To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

1) All non- zero digits are significant Ex. (34 cm sig fig) 2) Zeros in zero sandwiches are significant Ex. ( sig fig) 3) In order for zeros at the end of a measurement to be significant, there must be a decimal point. Ex. ( sig fig but sig fig) 4) Zeros to the left of the first non-zero are NOT significant. They are just place holders. Ex. ( sig fig)

Measurement# of Sig Figs 1) ) 300 3) ) ) ) 5.00

 When we are using measurements in different calculations in Chemistry and Physics and even Biology, we need to account for the level of precision. To do so, scientists use significant figures.  The last sig fig in a measurement is always the doubtful digit. But that is not always clear. For example  12340cm or cm or cm

 If Jenn measures a line to be 12.0 cm, what number is she doubtful about and how many sig. figs. are there?  If Darren measures a mass to be 1300 g, what number is he doubtful about and how many sig. figs. are there?

 Scientists use scientific notation as a method to show the proper amount of sig. figs.  300 km can be written as 3 x 10 2 km. (10 2 is 100 – so 3 x 100 =300)  300 km (with 2 sig. figs) can be written as 3.0 x 10 2 km. Notice we clearly have 2 sig. figs.

 When multiplying by 10 +x we can move the decimal x times to make the number bigger. Ex.4 x 10 5 N = N Ex.3.00 x 10 4 kg = kg (3 sig. figs – shown in sci. notation)

 When multiplying by 10 -x, we can move the decimal x times to make the number smaller.  6.01 x m = m (3 sig. figs)  3.10 x N = N (3 sig. figs.)

 Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified.  How do you decide how many digits to keep?  Simple rules exist to tell you how.

 Once you decide how many digits to retain, the rules for rounding off numbers are straightforward:  RULE 1. If the first digit you remove is 4 or less, drop it and all following digits becomes 2.6 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less.

 RULE 2. If the first digit removed is greater than 5, round up by adding 1 to the last digit kept is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater.  RULE 3. If the first digit removed is equal to 5 and the digit before it is an even number, drop it and all following digits. If the digit before it is an odd number, round up by adding 1 to the last digit kept.

 * If there are any numbers written after the five, then you must round up.  Example: is 1.24 when rounding to 3 sig figs  Example: is 1.25 because there is a number after the 5.

RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

 RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.

Make the following into a 3 Sig Fig number ,  ,  10 6 Your Final number must be of the same value as the number you started with, 129,000 and not 129. It is because there is a 5 with numbers after it.

For example you want a 4 Sig Fig number , Number after 5, so rounds up 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig ,

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= = = – = Look for the last important digit

Multiplication and division  1.54 =  =  =  10 6  =     = 

 When deciding the number of significant digits kept for calculations involving a series of operations, round for each set of rules (multiplication/division and addition/subtraction). Ex: 3.56g + 1.3g= 4.86g = 4.9g/1mL 3.3mL -2mL 1.3 mL = 5g/mL