Significant Digits

 What is the population of Cambridge?  Answer: ____________________ (an estimate)  We got a range of answers but everybody rounded to the closest thousand – Why?  Why didn’t anybody give an exact answer?

134,000 Insignificant Digits These units are not worth counting They constantly change or they can’t be measured accurately Significant Digits We have measured the units in these place values. They can be measured accurately

134,000 Insignificant Digits Significant Digits We have measured the population to 3 significant digits! More significant digits imply greater accuracy.

 Which population sign(s) seems the most reasonable? Note: When teachers correct the number of significant digits – we will often shorten the term to sig dig. Alternately, some teachers use the word significant figures or sig figs

 The amoeba is m in length These place values are too large to use for measurement of an amoeba They are insignificant The last two digits have been measured They are significant

 House Price: $325,000  House price: $205,000  Height of Person: 173 cm  Height of Person: 150 cm  Litres of gasoline at the gas station: 54.7 L  Price of shoes: $35  Price of Shoes: $170  Distance to the sun: 150,000,000 km  Distance to the sun: 147,098,074 km  Radius of calcium atom: m  Radius of calcium atom: 197 pm  Are zeros always insignificant?  Select examples from above where the zeros are significant?  Is there an advantage to using appropriate units.

 Our Discussion tells us that:  Zeros at the ends of large numbers are insignificant  Zeros at the beginning of small numbers after the decimal place are insignificant (ex m)  Zeros in the middle of two significant digits are also significant (since they are measured units/place values) (ex. $205,000)  Zeros after after decimal at the end of a number are significant (since they are measured)

 3 students measure the length of a box. Their results are as follows:  40 cm40.0 cm40.00 cm  Did they get the same results?  Not really, they got the same value but much different levels of accuracy!  The first person used a cheap ruler or meter- stick to get 2 significant digits of accuracy

 3 students measure the length of a box. Their results are as follows:  40 cm40.0 cm40.00 cm  The first person used a cheap ruler or meter-stick to get 2 significant digits of accuracy (I am using logic to suggest that the zero is significant)  The second person used a tape measure with millimeter markings to get 3 significant digits of accuracy.  The third person used a very accurate Vernier measuring tool to get 4 significant digits of accuracy

 We can see that zeros that follow a decimal (and are the last digits of a number) are significant! cm  There would be no need to record the zeros after the decimal unless they were significant.  A GREATER NUMBER OF SIGNIFICANT DIGITS MEANS A GREATER ACCURACY!

 Weight of a rabbit: 1420 g  How many significant digits?  On first inspection, we would say 3 sig dig.  But, maybe the scale measures to the closest gram and we have 4 sig dig.  We can take the ambiguity out by using scientific notation:  If the value is X 10 3, then we know that the fourth digit is significant

1)All nonzero digits are significant.  For example: 457 cm (three significant digits) and 0.25 g (two significant digits) 2)Zeros between nonzero digits are significant.  For example: 1005 kg (four significant digits) and 1.03 cm (three significant digits) 3)Zeros to the left of the first nonzero digit in a number are not significant; they merely indicate the position of the decimal point.  For example: 0.02 g (one significant digit) and cm (two significant digits)

4)When a number ends in zeros that are to the right of the decimal point, they are significant. ◦ For example: g (three significant digits) and 3.0 cm (two significant digits) 5)When a number ends in zero that are not to the right of a decimal digit, the zeros are not necessarily significant.  For example: 130 cm (Is it 2 or 3 significant digits?) 10,300 g (Is it 3,4 or 5 significant digits?)  The way to remove this ambiguity is to use scientific notation.  10,300 g1.03 x 10 4 (three significant digits) x 10 4 (four significant digits)  x 10 4 (five significant digits)

 6)Numbers obtained from counting are not measured. They do not affect the number of significant digits in the answer! Ex.Each section of a bridge weighs 2430 tonnes. The bridge has 24 sections, what is the weight of the bridge? Since the 24 is a counted number, we still use the 3 significant digits in the first number to obtain the number of sig dig in the answer. 24 x 2430 tonnes = tonnes ⇒ tonnes