ROUNDING AND SIGNIFICANT FIGURES
OBJECTIVES Precision VS Accuracy Rounding Numbers Significant Figures Scientific Notation Metric Conversions Factor Label Method Temperature Conversions
Significant Figures When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer. Chapter Two
Significant Figures There are 2 different types of numbers Exact Measured
Exact Exact numbers are infinitely important A. Exact numbers are obtained by: 1. counting 2. definition
EXAMPLES: Exact Numbers Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.
Measured (Inexact) Measured numbers are obtained by 1. using a measuring tool Measured number = they are measured with a measuring device so these numbers have ERROR.
Example: example: any measurement. If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be 215.6 mm (4 significant figures).
Remember Exact numbers you get by counting and definition. Inexact numbers you get by MEASUREMENT
QUESTION Check… do in notes Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10-4 cm. There are 6 hats on the shelf. Gold melts at 1064°C.
SOLUTION 1 yard = 3 feet : EXACT: This is a defined relationship. 2. The diameter of a red blood cell is 6 x 10-4 cm. MEASURED: A measuring tool is used to determine length. 3. There are 6 hats on the shelf. EXACT: The number of hats is obtained by counting. Gold melts at 1064°C. MEASURED: A measuring tool is required.
QUESTION Check A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by
Solution 2. counting 3. definition B. Measured numbers are obtained by A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool
2.6 Rounding Off Numbers 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. Chapter Two
VIDEO Grab your white boards – we are going to make sure you REMEMBER how to round correctly.
Round 286 to the nearest TEN Circle the tens spot – use the one to the right to determine if the ten spot goes up or down… It’s a 6 ….. SO it rounds up to 290
Try this one : Round 387,907 to the nearest hundred thousand: 387,907 Now round it to the nearest hundred…
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. 2.4271 becomes 2.4 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 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater. If a calculation has several steps, it is best to round off at the end. Chapter Two
ROUNDING to SIGNIFICANT FIGURES is a way that scientists keep their numbers in check. It is a way to determine HOW MANY PLACES your answer should be rounded to. Remember last week…. Was it 4.1628 or 4.16 or 4.2 or 4? The idea is to round to the LEAST NUMBER OF SIGNIFICANT DIGITS that were given in the problem.
NOW>>>> which are significant? NOT ALL DIGITS ARE SIGNIFICANT… OF course – there are RULES!
Practice Rule #2 Rounding Make the following into a 3 Sig Fig number Your Final number must be of the same value as the number you started with, 129,000 and not 129 1.5587 .0037421 1367 128,522 1.6683 106 1.56 .00374 1370 129,000 1.67 106
Examples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is <5 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 4965.03 780,582 1999.5 4965 780,600 2000.
RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. Chapter Two
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. Chapter Two
Significant Figures Measured numbers in an answer that matter for reporting. Meant to make writing answers EASY! VIDEO
RULES
ALL NON ZERO’S ARE SIGNIFICANT Rule #1 ALL NON ZERO’S ARE SIGNIFICANT
RULE 2. Zeros in between significant digits are always significant. Thus, 94.072 g has five significant figures.
RULE 2B. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.
RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m.
VIDEO Use the rules! Everytime!
Practice Rules ; Zeros – in your notebook 6 3 5 2 4 All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal 0’s between digits count as well as trailing in decimal form 45.8736 .000239 .00023900 48000. 48000 3.982106 1.00040
How many significant figures are in each of the following? 1) 23.34 4 significant figures 2) 21.003 5 significant figures 3) .0003030 4 significant figures 4) 210 2 significant figures 5) 200 students 1 significant figures 1 significant figures 6) 3000
HOMEWORK : ID SIG FIGS Practice ! Practice! Chapter Two