Significant Digits or “Figures” How to recognize significant figures when: Taking a measurement Reading a measurement Performing a calculation

Accuracy and Precision in Measurements Accuracy: how close a measurement is to the accepted value. Precision: how close a series of measurements are to one another or how far out a measurement is taken. A measurement can have high precision, but not be as accurate as a less precise one.

Significant Figures are used to indicate the precision of a measured number or to express the precision of a calculation with measured numbers. In any measurement the digit farthest to the right is considered to be estimated. 1 2 2.0 1.3

Question For Thought Using two different rulers, I measured the width of my hand to be 4.5 centimeters and 4.54 centimeters. Explain the difference between these two measurements.

The first measurement implies that my hand is somewhere between 4 The first measurement implies that my hand is somewhere between 4.5 and 4.9 cm long. There is a uncertainty in this number because we have to estimate. The second measurement implies that my hand is between 4.5 and 4.6 cm long. This measurement is more certain due to its greater precision.

More certain due to greater precision 4.5 cm 2 significant figures Uncertain 4.54 cm 3 significant figures More certain due to greater precision Significant figures are necessary to reduce uncertainty in our measurements. Significant figures indicate the precision of the measured value!!

Significant Figures Scientist use significant figures to determine how precise a measurement is Significant digits in a measurement include all of the known digits plus one estimated digit So when reading an instrument… Read instrument to the last digit that you know Estimate or “eyeball” the final digit

For example… Look at the ruler below Each line is 0.1cm You can read that the arrow is on 13.3 cm However, using significant figures, you must estimate the next digit That would give you 13.30 cm

Let’s try this one Look at the ruler below What can you read before you estimate? 12.8 cm Now estimate the next digit… 12.85 cm

Let’s try graduated cylinders Look at the graduated cylinder What can you read with confidence? = 56 ml Now estimate the last digit = 56.0 ml What is this measurement? _____________

Recognizing # Sig Figs in a Number All non zero digits are ALWAYS significant How many significant digits are in the following numbers? ______Significant Figures ______Significant Digits 274 25.632 8.987

All zeros between significant digits are ALWAYS significant How many significant digits are in the following numbers? 504 60002 9.077 ________ Significant Figures ________ Significant Digits

All FINAL zeros to the right of the decimal ARE significant How many significant digits are in the following numbers? ______Significant Figures ______Significant Digits 32.0 19.000 105.0020

_____Significant Digit _____Significant Digits All zeros that act as place holders are NOT significant Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal How many significant digits are in the following numbers? _____Significant Digit _____Significant Digits 0.0002 6.02 x 1023 100.000 150000 800

Numbers with no decimal are ambiguous... Does 5000 ml mean exactly 5000? Maybe.... Maybe Not! So 5000, 500, 50, and 5 are all assumed to have 1 significant figure If a writer means exactly 5000, he/she must write 5000. or 5.000 x 103

All counting numbers and constants have an infinite number of significant digits For example: 1 hour = 60 minutes 12 inches = 1 foot 24 hours = 1 day

How many significant digits are in the following numbers? _______________ 0.0073 100.020 2500 7.90 x 10-3 670.0 0.00001 18.84

Here is a one sentence rule for counting sig figs: All digits ARE significant except Zeros preceding a decimal fraction (ex: 0.0045) and Zeros at the end of a number containing NO decimal point (ex: 45,000)

Calculations with Sig Figs Adding or subtracting: answer can have no more places after the decimal than the LEAST of the measured numbers. Count # decimal places held (nearest .1? .01? .001?) Answer can be no more accurate than the LEAST accurate number that was used to calculate it.

5.50 grams For Example: + 8.6 grams - 49.7 ml -------- ------------- 2.39 ml --> 2.4 ml

Calculations with Sig Figs Multiplying or dividing: round result to least # of sig figs present in the factors Answer can’t have more significant figures than the least reliable measurement. COUNT significant figures in the factors

56.78 cm x 2.45cm = 139.111 cm2 Round to 3 sig figs = 139cm2 75.8cm x 9.6cm = ?

Now let’s do some math..... (round answers to correct sig figs!) 5.0033 g + 1.55 g answer: 6.55 g Did you need to count sig figs? NO!

Try this one.... 4.80 ml - .0015 ml answer: 4.80 ml (one might say .0015 is insignificant COMPARED TO 4.80)

Now try these... 5.0033 g / 5.0 ml answer: 1.0 g/ml Did you have to count sig figs? YES!

Here’s a tougher one..... 3.0 C/s x 60 s/min x 60 min/hr = answer: 10800 C/hr rounds to 11000 C/hr Note: Standard conversion factors never limit sig. figures- instruments and equipment do.

Scientific Notation Scientific notation is used to express very large or very small numbers It consists of a number between 1 & 10 followed by x 10 to an exponent Exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal

Large Numbers If the number you start with is greater than 1, the exponent will be positive Write the number 39923 in scientific notation First move the decimal until 1 number is in front 3.9923 Now add x 10 Now count the number of decimal places you moved (4) Since the number you started with was greater than 1, the exponent will be positive 3.9923 x 10 4

Small Numbers If the number you start with is less than 1, the exponent will be negative Write the number 0.0052 in scientific notation First move the decimal until 1 number is in front = 5.2 Now add x 10 Now count the number of decimal places moved (3) Since the number you started with was less than 1, the exponent will be negative 5.2 x 10 -3

Scientific Notation Examples Place the following numbers in scientific notation Note: all sig figs from the original number must be present! 99.343 4000.1 0.000375 0.0234 94577.1 ______________

Going to Ordinary Notation Examples Place the following numbers in ordinary notation: ________________ 3 x 106 6.26x 109 5 x 10-4 8.45 x 10-7 2.25 x 103