 Uncertainty: not surely known; doubtful; varying  Significant: full of meaning; important

Uncertainty in Measurements  Measurements made (by ruler, graduated cylinder, etc.) require estimation  imagine distance Above is a ruler, where the labeled numbers (i.e. “1” and “2”) are considered “certain numbers.” gif Above is a graduated cylinder, where the labeled numbers (i.e. “40” and “50”) are “certain.” a.gif

Uncertainty in Measurements continued…  Certain numbers: same regardless of who made them; estimated  Uncertain numbers: estimated values; depend on device  NOTE!: measurements always have some level of uncertainty

Example  Why is there always uncertainty in a measurement? In a measuring device, there are only a limited amount of “tick marks” that indicate the certain numbers. There can always be a measurement between two indicated tick marks on a measuring device, and another measurement within that. Since the measurement can never be exact, you would need to estimate. This estimation makes the measurement uncertain. In a measuring device, there are only a limited amount of “tick marks” that indicate the certain numbers. There can always be a measurement between two indicated tick marks on a measuring device, and another measurement within that. Since the measurement can never be exact, you would need to estimate. This estimation makes the measurement uncertain.

Rules for Significant Numbers  Significant figures (sig figs): numbers recorded (all certain numbers and one estimated uncertain number)  Nonzero integers ALWAYS count; include {…-3,- 2,-1,1,2,3…}  NOTE!: integers are whole numbers and NOT decimals/fraction  Exact numbers: determined by counting/definition (ex: 5 pencils); unlimited sig figs

Flow Chart on Sig Figs

Types of Zeros Below is a chart of zeros and examples of each Below is a chart of zeros and examples of each Zeros Captive zeros: between nonzero digits; ALWAYS count as significant numbers captive zeros Leading zeros: precede nonzero digits; NEVER count as significant numbers leading zeros Trailing Zeros: follow nonzero digits at the end; counts as significant number if they follow a period; does not count without the period Count does not count

Example  How many sig figs are present in “100?” Why do the zeros do/do not count? There is only one sig fig in this value, because the zeros are considered “trailing.” In the case of trailing zeros, they can be considered significant or not. They are only counted as significant if and only if they follow a decimal point. In this case, however, there is no decimal point; therefore, they do not count. There is only one sig fig in this value, because the zeros are considered “trailing.” In the case of trailing zeros, they can be considered significant or not. They are only counted as significant if and only if they follow a decimal point. In this case, however, there is no decimal point; therefore, they do not count.

Rules for Rounding Off  Less than 5  preceding number stays the same  ≥ 5  preceding number increases by one  NOTE!: carry extra digits to the final result and round  When rounding use only the first number to the right of the last sig fig

Example  Round to the nearest hundredth Since “4” is less than 5, the “6” is left as is Since “4” is less than 5, the “6” is left as is.

Rules for Multiplying/Dividing Significant Numbers  Number of sig figs act as the “limiting factor”  Number with smallest number of sig figs acts as “limiting factor” 1.08 x x sig figs 2 sig figs 3 sig figs 2 sig figs

Example  1.4 x Find the answer and give the limiting factor The limiting factor is “1.4”, because compared to “5.78,” which has 3 sig figs, it has only 2 sig figs.  limits to 2 sig figs in answer 8.1. The limiting factor is “1.4”, because compared to “5.78,” which has 3 sig figs, it has only 2 sig figs.  limits to 2 sig figs in answer

Rules for Adding/Subtracting Significant Numbers  Decimal places counted as “limiting factor”  Number with the least decimal places is “limiting factor” 1.08 x x places 1 place 2 places 1 place

Example HHHHow many sig figs should the answer of – – 2.1 contain? What is the limiting factor? Solve The limiting factor is 2.1, because it has only one decimal place.

Quiz 1. In the picture below, estimate the measurement of the purple line in inches. Are the estimated values certain/uncertain?

Quiz continued… 2. When counting 30 apples, how many sig figs are there? Why? 3. Round to the nearest thousandth. 4. What is meant by the term limiting factor when multiplying/dividing sig figs? 5. How many sig figs should the answer of contain? Give the limiting factor and solve.

Answers 1. The measurement of the line is about 1.55 inches. The estimated part is uncertain, because there is no mark on the ruler that physically displays a measurement beyond the hundredth place. Therefore, the estimated part is uncertain, since you are only “imagining” tick marks between the 5 th and 6 th tick mark.

Answers continued… 2. There are unlimited significant figures, because this measurement is done by counting Since the number after “4” is “5”, which is ≥ 5, the “4” is increased by one. 4. When multiplying/dividing, look at the number with the least number of sig figs. This term limits the answer to that amount of sig figs ; the “5.1” limits the answer to 2 sig figs.

Reference  content/uploads/2008/04/ruler_0_10.jpg content/uploads/2008/04/ruler_0_10.jpg content/uploads/2008/04/ruler_0_10.jpg  ol/TX/G08S06/20graphicaa.gif ol/TX/G08S06/20graphicaa.gif ol/TX/G08S06/20graphicaa.gif  s1/sigfigs/sigfig/node152.html s1/sigfigs/sigfig/node152.html s1/sigfigs/sigfig/node152.html