Introduction to Significant Figures & Scientific Notation

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

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

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

The same rules apply with all instruments Read to the last digit that you know Estimate the final digit

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

One more graduated cylinder Look at the cylinder below… What is the measurement? 53.5 ml

Rules for Significant figures Rule #1 All non zero digits are ALWAYS significant How many significant digits are in the following numbers? 3 Significant Figures 5 Significant Digits 4 Significant Figures 274 25.632 8.987

Rule #2 All zeros between significant digits are ALWAYS significant How many significant digits are in the following numbers? 3 Significant Figures 5 Significant Digits 4 Significant Figures 504 60002 9.077

Rule #3 All FINAL zeros to the right of the decimal ARE significant How many significant digits are in the following numbers? 32.0 19.000 105.0020 3 Significant Figures 5 Significant Digits 7 Significant Figures

Rule #4 Trailing zeros – at the end of numbers but to the right of the decimal point 2.00 g - has 3 sig. digits (what this means is that the measuring instrument can measure exactly to two decimal places. 100 m has 1 sig. digit Zeros are significant if a number contains decimals

For example 1 Significant Digit 0.0002 3 Significant Digits How many significant digits are in the following numbers? 1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits 0.0002 6.02 x 1023 100.000 150000 800

Exact Numbers Are numbers that are not obtained by measuring Referred to as counting numbers EX : 12 apples, 100 people

Exact Numbers Also arise by definition 1” = 2.54 cm or 12 in. = 1 foot Are referred to as conversion factors that allow for the expression of a value using two different units

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 2 Significant Digits 6 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit

Rules Rounding Significant Digits Rule #1 If the digit to the immediate right of the last significant digit is less that 5, do not round up the last significant digit. For example, let’s say you have the number 43.82 and you want 3 significant digits The last number that you want is the 8 – 43.82 The number to the right of the 8 is a 2 Therefore, you would not round up & the number would be 43.8

Rounding Rule #2 If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure Let’s say you have the number 234.87 and you want 4 significant digits 234.87 – The last number you want is the 8 and the number to the right is a 7 Therefore, you would round up & get 234.9

Rounding Rule #3 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up 78.657 (you want 3 significant digits) The number you want is the 6 The 6 is followed by a 5 and the 5 is followed by a non zero number Therefore, you round up 78.7

Rounding Rule #4 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even. 2.5350 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even 2.54

Say you have this number 2.5250 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even and it already is 2.52

Let’s try these examples… 200.99 (want 3 SF) 18.22 (want 2 SF) 135.50 (want 3 SF) 0.00299 (want 1 SF) 98.59 (want 2 SF) 201 18 136 0.003 99

Science Problems w/Sig Figs When combining measurements with different degrees of accuracy and precision, the accuracy of the final answer can be no greater than the least accurate measurement.

Adding and Subtracting Sig. Figures This principle can be translated into a simple rule for addition and subtraction: When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

Significant Figures Adding and subtracting sig figs - your answer must be limited to the value with the greatest uncertainty.

Line up decimals and Add 150.0 g H2O (using significant figures) 0.507 g salt 150.5 g solution 150.5 g solution 150.0 is the least precise so the answer will have no more than one place to the right of the decimal.

Example Answer will have the same number of decimal places as the least precise measurement used. 12.11 cm 18.0 cm 1.013 cm 31.132 cm 9.62 cm 71.875 cm Correct answer would be 71.9 cm – the last sig fig is “8”, so you will round using only the first number to the right of the last significant digit which is “7”.

Significant Figures Multiplication and division of sig figs - your answer must be limited to the measurement with the least number of sig figs. 5.15 X 2.3 11.845 3 sig figs 2 sig figs only allowed 2 sig figs so 11.845 is rounded to 12 5 sig fig 2 sig figs

Multiplication and Division Answer will be rounded to the same number of significant figures as the component with the fewest number of significant figures. 4.56 cm x 1.4 cm = 6.38 cm2 = 6.4 cm2

28.0 inches 2.54 cm 1 inch Computed measurement is 71.12 cm Answer is 71.1 cm x = 71.12 cm

When both addition/subtraction and multiplication/division appear in the same problem In addition/subtraction the number of significant digits is limited by the value of greatest uncertainty. In multiplication/division, the number of significant digits is limited by the value with the fewest significant digits. Since the rules are different for each type of operation, when they both occur in the same problem, complete the first operation and establish the correct number of significant digits. Then proceed with the second and set the final answer according to the correct number of significant digits based on that operation

(1.245 + 6.34 + 8.179)/7.5 Add 1.245 + 6.34 + 8.179 = Then divide by 7.5 =

Scientific Notation Scientific notation is used to express very large or very small numbers I consists of a number between 1 & 10 followed by x 10 to an exponent The 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 at x 10 – 3.9923 x 10 Now count the number of decimal places that 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 at x 10 – 5.2 x 10 Now count the number of decimal places that you 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: 99.343 4000.1 0.000375 0.0234 94577.1 9.9343 x 101 4.0001 x 103 3.75 x 10-4 2.34 x 10-2 9.45771 x 104

Going from Scientific Notation to Ordinary Notation You start with the number and move the decimal the same number of spaces as the exponent. If the exponent is positive, the number will be greater than 1 If the exponent is negative, the number will be less than 1

Going to Ordinary Notation Examples Place the following numbers in ordinary notation: 3000000 6260000000 0.0005 0.000000845 2250 3 x 106 6.26x 109 5 x 10-4 8.45 x 10-7 2.25 x 103