welcome to significant figures often referred to as sig figs and also known as significant digits
a standardized way to determine and communicate the reliability or certainty of a measurement
and a standardized way to determine and communicate the reliability or certainty
of a number calculated from measured values the science and engineering communities are dependent
on communication of the certainty reliability and the precision of measured values which is ensured
by following the rules established for significant figures in this video we look at how to determine
significant figures for measured values the rules for determining significant figures in a number
and how to determine significant figures in calculated numbers
first let's look at significant figures and measured values significant figures are the digits
in a measurement that correspond to the markings on an instrument plus one estimated digit for
example the markings on this thermometer can only tell us the temperature is either 28 or 29 degrees
but we know it is between 28 and 29 degrees and so we add an estimated digit the digits that the
instrument allows us to read we know are certain and including the estimated digit is required we
know the temperature is between 28 and 29 degrees and so that is shown by the estimated digit
our reading of this thermometer results in a measure of three significant figures in other
words the measure is reliable to three digits because the rules for this are standardized anyone
reading this measured value would assume the last digit is estimated but you can also show that it
is estimated by adding a plus or minus sign with the amount of uncertainty in the last digits shown
here plus or minus 0.1 showing the measures is reliable within 28.5 to 28.7 degrees
let's look at a thermometer with tenths of a degree markings on it here we can
see the temperature is between 28.6 and 28.7 degrees and so 28.6 is the certain measure
with an added estimated digit at this time in the hundredths place
this gives us four significant figures with a reliability of plus or minus 0.01 degree
the greater the number of significant figures the greater the precision implied for the measurement
an important point is knowing what to do with a measure that is exactly on a
marking of a measuring device here we see that we have a volume of 44 milliliters
but if that is what we wrote down as the measure the person reading that may believe that the
second four is uncertain and the measure could be reliable within 43 to 45 milliliters we need
to abide by the rules and give an estimated digit 44 milliliters is certain and then adding a point
zero communicates that it is reliable to the tenths of a milliliter our measure has three sig
figs three digits of reliability with an estimated digit in the tenths of a milliliter before going
further we need to understand how to look at a number and know how many significant figures
are being communicated to us in that number in other words how much certainty or reliability is
in that number let's put the sig fig rules in a flow chart starting with whether the number
contains a decimal point if it does have a decimal point then you find the leftmost non-zero digit
that digit and all those to its right including zeros are significant let's see how that works
first we find the leftmost non-zero digit which here is three that digit and all those to its
right including zeros are significant note that any zero between non-zeros is always significant
38.05 has four significant figures what if there are zeros after the last non-zero digit
these are called trailing zeros and there is still a decimal in the number and so these two trailing
zeros are significant so with the decimal present we find the leftmost non-zero digit
that digit and all those to its right are significant 38.0500 has six sig figs now let's
look at a number that has what are called leading zeros zeros before the first non-zero digit
in a number with a decimal leading zeros are not important because they only tell us where
the decimal is they are not significant they are not part of the measured value
here the leftmost non-zero digit is 7 so that is significant as are all those digits to its
right this number has three sig figs if the measured value had a trailing zero
then that would be significant as well according to sig fig rules giving the number four sig figs
let's look at how we can get rid of the leading zeros by changing this to scientific notation
which is especially convenient for expressing numbers that have a large amount of leading zeros
starting with an exponent of zero each digit we move the decimal to the right makes the
coefficient ten times larger and so we have to make the exponent of 10 that many times smaller
by decreasing the exponent by negative 1 for each digit that the decimal is moved and this keeps the
value of the number constant the decimal was moved three places to the right and so the exponent is
negative three we can now drop the leading zeros and the coefficient still shows four significant
figures when expressing numbers in scientific notation it is the coefficient that determines
the amount of significant figures now let's take a look at numbers that do not have a decimal point
find the rightmost non-zero digit that digit and all those to its left are significant for this
number 6 is the rightmost non-zero digit that digit and all those to its left are significant
note that as before any zero between non-zeros is significant this number has four sig figs
often however a number like this is ambiguous it does not show a decimal however the decimal is
implied to avoid ambiguity change the number to scientific notation by putting the decimal after
the first number we know from the original number that these trailing zeros are not significant
and so they are removed we now use the rules for numbers with decimals which still gives
four sig figs and no ambiguity with trailing zeros another rule is that exact numbers have
unlimited significant figures some textbooks say infinite sig figs and that is the same thing
unit relationships are often exact one kilogram is the same as a thousand grams it is an exact
relationship and so has unlimited sig figs one inch is defined as 25.4 millimeters
it is an exact relationship and has unlimited sig figs however when a unit relationship is a
measured amount such as the density of iron 7.874 grams per cubic centimeter
the unit relationship is not exact this is more appropriately written as 7.874 grams per
1 cubic centimeter but the 1 cubic centimeter is ambiguous it came out of a division of some
measured amount of grams and cubic centimeters for example 19.685 grams and 2.500 cubic centimeters
but that information is not available to us and we can assume it is not exactly one cubic centimeter
but we cannot know how many sig figs the 1 cm3 represents and so we ignore the 1 and use the 7.874 which has four sig figs.
Counted amounts are always exact for example if you count 24 students or 328 gerbils
that is an exact amount of students or gerbils you cannot have fractions of students or fractions of
gerbils they have unlimited sig figs how can we apply significant figure rules to a calculation
well it is simply that a calculated number cannot be more certain more precise or more reliable than
the measured numbers used to calculate it think of a chain a chain cannot be stronger than its
weakest link and so a calculated answer cannot be more certain than the least certain number
used in the calculation for multiplication and division the result is rounded to the
same number of significant figures as the factor with the least number of significant figures
for addition and subtraction the result has the same number of decimal places as the
quantity with the least number of decimal places let's use the volume of a box to demonstrate
you measure the three sides and multiply to get the volume but your calculator does not know about
sig figs and so we look to see which measured number has the least sig figs which in this
case is two the answer is rounded to two sig figs but what if the rounding includes a trailing zero
let's look at a box with slightly different dimensions giving this volume from the calculator
just as before the least amount of sig figs is two we have to round the answer to 2 000 to get to two
sig figs but the number 2000 only has one sig fig we can show two sig figs with scientific notation
but the coefficient two by itself is only one sig fig and so we add a zero remember a zero
after a decimal and after a non-zero number is significant the volume of the box is 2.0
times 10 to the 3 cubic centimeters rounded to two sig figs let's do a calculation in
which the determination of sig figs is not as straightforward but still following sig fig rules
let's say we have a cube of iron its density being 7.87 grams per cubic centimeter and occupying a
space of 226 cubic centimeters what is its mass in kilograms the calculation has this setup
with a calculator answer of 1.779524 kilograms as discussed previously the
one in unit relationships that are calculated from measured values has an unknown amount of sig figs
and is therefore ignored the unit relationship between kilograms and grams is exact it has
unlimited sig figs and so is not relevant of the two numbers left three sig figs is the least
amount in the numbers used in the calculation and so we need to round the answer to three sig figs
the answer rounds to 1.78 kilograms and so the iron cube has a mass of 1.78 kilograms
for addition and subtraction the result has the same number of decimal places as
the quantity with the least number of decimal places in adding 224.683 and 87.8 we see that
the least amount of decimal places is 1 and so the answer must be rounded to a single decimal
the answer is 312.5 and has four sig figs let's do a subtraction 187.812 subtracted from 30.09
the answer has to be rounded to two decimal places it is negative 157.72 which is five
sig figs the answer is reliable to five digits this has been a mad scientist production see ya