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Significant Figures:
Significant Figures are the minimum number of figures (digits) required to report a measurement
without loss of accuracy or precision. It is important to use significant figures when recording a
measurement so that it does not appear to be more accurate than a given measuring tool is
capable of determining. The most significant digit is the non-zero digit farthest left. For
example, the 4 in the number 0.004567. If the number includes a decimal, the least significant
digit is the digit farthest to the right. For example, in the number 12.390, the 0 is the least
significant digit. If the number includes no decimal, the least significant digit is the digit to the
far right. For example, in 33500, the 5 is the least significant digit.
---- TYPES OF NUMBERS -------------------------------------------------
INEXACT Numbers - these can be any measurement
Numbers in measurements that have actually been measured (according to the
scale/tool used) plus one estimated (within a reasonable range) number. FOR
EXAMPLE: A graduated cylinder is marked in intervals of 1 milliliter. A
measurement from this tool would be reported to the nearest tenth of a milliliter.
In the example shown below, the proper measurement would be 16.5 ml
20
15
TO DETERMINE SIGNIFICANT FIGURES FOR INEXACT NUMBERS
• NONZEROS- any nonzero number is significant – 567 has three, 2.8 has two.
• ZEROSa. Leading Zeros - are never significant. - 0.0025 has 2 significant figures
b. Captive Zeros - are always significant. - 2.005 has 4 significant figures
c. Trailing Zeros - are only significant if there is a decimal point.
200 has 1 significant figure, 200. has 3 significant figures; and 200.00 has 5
10
EXACT Numbers - DO NOT affect the number of significant figures
• Numbers used for counting (There are 16 students in a class)
• ALL conversion factors (There are 12 inches in 1 foot)
• Numbers used in definitions (A ream of paper is 500 sheets)
• All exact values or conversion factors have an infinite (never ending) number of significant
figures.
---- SIGNIFICANT FIGURES AND CALCULATIONS -------------------Adding and Subtracting Significant Figures –
When adding or subtracting with significant figures, the rules for rounding numbers are
somewhat different than those for multiplying and dividing. To understand this difference, you
must remember that there is uncertainty in measurements. This applies in terms of which places
are specified. So....
The least-significant place of a measurement is the smallest (right-most) place specified in
the measurement.
For example, the the least-significant place in the measurement 1230 is the tens place. Other
less significant places (such as the ones place in this case) are unknown and could have any value.
This is extremely important when understanding that there is uncertainty in the results when
adding and or subtracting with significant figures.
Science@Lopezville
Check your Understanding.
Identify the least significant place in the following measurements with an arrow and label
1230
15300
0.160
14250
2.345 x 103
tens place
When adding or subtracting two measurements, the result should be rounded to the same
place as the least-significant place of the measurement.
To grasp this statement, let’s say we're adding 1130 (certain to the tens place) and 452 (certain
to the ones place). Adding 1130 + 452 gives us 1582. So, how confident should we be with this
particular answer? Well, the answer cannot be any more certain than the tens place. If it were,
then we'd be ignoring the uncertainty of the measurement 1130, which is only certain to the tens
place. So, the answer can only be certain to the tens place. Given this, we round to this place
(tens) to reflect the uncertainty of the measurement giving us 1580 with the proper number of
significant figures.
This works similarly with the tenths, hundredths and thousandths place when there is a decimal.
Add or subtract the number as you normally would. Then, round the answer to the LEAST
number of places in the decimal portion of any number in the problem. In adding or subtracting
measurement data, there can be no significant number in any place (except the first) unless all
the numbers have significant digits in that same place.
Example: 227.31 + 14.516 + 3155 = 3396.826
You have no idea what digit belongs in the tenths spot because
there is no tenths digit in 3155. The answer must be rounded to 3397
227.31?
14.516
3155 ???
3396.826
Example: 23.1 + 4.77 + 125.39 + 3.581
Since the number with the least number of numbers in the decimal
position is 23.1, the answer must also be rounded to the tenths place.
The answer is 156.8
23.1??
4.77?
125.39?
3.581
156.841
Example: 123 + 10300 + 1453
We see that two of these numbers are certain to the ones place with the third
only certain to the hundreds place. When added, the answer is 11876. However,
this number can only be certain to the hundreds place - so we round to 11900.
Multiplying or Dividing Significant Figures –
Calculate the values then round to the least number of significant figures in ALL the values.
Example: 3.5293 mol x 34.2 g/mol = 120.70206 g
or
121 g
There are five (5) sig-figs in 3.5293 and three (3) sig-figs in 34.2
Your final answer can ONLY be written with three (3) Sig-Figs.
Example: 125.06 g ÷ 0.023 g = 5437.391
or
5400
There are five (5) sig-figs in 125.06 and two (2) sig figs in 0.023
Your final answer can ONLY be written with two (2) Sig-Figs.
Science@Lopezville
---- COMBINED CALCULATIONS ---------------------------------------------When you have more than one operation in a math problem, you must solve it following the
correct order of operations.
A memory aid to help you with this order is the following Please Excuse My Dear Aunt Sally. The P stands for Parentheses, the E for Exponents, the M and D for Multiply or Divide, the A
and S for Add or Subtract. Do the operations within each level from left-to-right. Do not
're-use' any numbers.
Here's the order we use:
First, do all operations that lie inside parentheses.
Next, do any work with exponents or roots.
3. Working from left to right, do all multiplication and division.
4. Finally, working from left to right, do all addition and subtraction.
1.
2.
Consider the following example. 33.11 - 2.8552 ÷ ( 14.87 - 14.36) First we carry out the operation in the parentheses. Remembering to keep intermediate digits.
33.11 - 2.8552 ÷ ( 14.87 - 14.36)
= 33.11 - 2.8552 ÷ 0.51 Next we do the exponent = 33.11 - 8.1510 ÷ 0.51
Next we do the division = 33.11 - 15.9
Finally we do the subtraction = 17.21 = 17 (2 sig figs)
---- Avoid Round-Off Error in Multi-Step Calculations ----------------------------NOTE: KEEP the Extra Digits in Intermediate Answers
When doing multi-step calculations, keep the digits in intermediate results. Often, far more than needed in
your final answer. Just keep track of what is needed in the end.
For instance, a final answer may require only two significant figures. As you work the calculation, carry
several digits along for the ride. If you round-off all your intermediate answers to only two digits, you are
discarding the information contained in the other digits and, as a result, the second digit in your final
answer might be incorrect. This is known as "round-off error."
For example:
3.50 x 105 x [2.8 ÷ (5.4 - 4.09)]
NOT using extra digits: Rounding intermediate answers. (incorrect answer)
Parentheses 1st (5.4 - 4.09) = 1.31 = 1.3
Then brackets 2.8 ÷ 1.3 = 2.1538462 = 2.2
Then multiply 3.50 x 105 x 2.2 = 770000
USING extra digits: Not rounding intermediate answers. (correct answer)
Parentheses 1st (5.4 - 4.09) = 1.31
Then brackets 2.8 ÷ 1.31 = 2.1374046
Then multiply 3.50 x 105 x 2.1374046 = 748091.6
The problem indicates that the answer should be shown with 2 sig figs..so….. 750000
Big Difference...Huh?
Science@Lopezville