in taking a measurement of some quantity no matter how good you are or how advanced are your measuring equipment you cannot assume that your measurement is perfect and that it is the truest value so meaning that your obtained measurement cannot be perfectly accurate so when we do a direct measurement there will always be a doubt that our obtained value is not the True Value so therefore we need to report so in recording our measured value we need to report it together with the uncertainty of the measured value so to Define uncertainty uncertainty is the range of possible values within which the true value of the measurement can be found so for example length of an objects for example so in this case you need to estimate so means for example yeah but however using other other um tools for example other measuring devices so in those cases we need to estimate so is it closer to the next value or is it closer to the previous value so this means that when measuring there is always a doubt in our measurement so that's why uncertainty is also defined as the doubt that exists about the result of a measurement so in reporting a measurement with its uncertainty we use the form Q plus or minus Delta Q so we're in Q is the obtained measurement and Delta Q is the uncertainty of measurement so this expression indicates that the best estimate of the detail the best estimate of the true value is found between the obtained value plus or minus the uncertainty of your obtained value for example so suppose you measure the length of an object to be 10 centimeters long and the terminates uncertainty to be 0.05 centimeter so to report this um measure measured value so we follow this form we're in Q plus or minus Delta q and then the unit in 10 centimeters not 10 plus or minus 0.05 which is our uncertainty so this means so this measured value means that the true value of the length of the object lies between 9.95 centimeters to 10.05 centimeter so the true value is unlikely to be less than 9.95 and more than 10.05 so you improve value nothing diva yeah especially True Value can be found within this range only so given this information 10.05 so this is the lower boundary or the minimum and this is your upper boundary or the maximum so uncertainty is a way of expressing our confidence in our measurements object so you want certainty is is our way of expressing the confidence in our measurement so any measurement that disregards the measure of uncertainty is meaningless so that is we always we we always need to include the uncertainty because the relationship between error and uncertainty is demonstrated by the equation measured value is equal to the quantity of measurement plus or minus uncertainty and then the units measured value is equal to Q plus or minus Delta q and then the unit so we know already that error is the difference between the actual value and the measured value so your uncertainty normally it is the estimate of the range between your your um estimate between the range of your measured value and then the True Value wherein it represents you you uncertainty nothing represents the reliability of the measurement so your measured value becomes more reliable and accurate if it is within the range of your uncertainties so meaning that uncertainty defines how much error for example again so for example uh if we have a measured value that is equal to 12 plus or minus one for example centimeters so this means that you can make an error of up to the quantity minus 1 or the quantity plus one so your measurement move you can only acceptable language within the range of 12-1 which is 11 up to 13. so acceptable error now bus is 11. 11 is acceptable because that because that is within the range of your uncertainty 12.6 for example that is acceptable because that is within the range of your uncertainty however for example nagging 14. so that is outside the boundaries so outside these boundaries the measurement becomes more and more inaccurate and unreliable uncertainty so there are two ways of determining the uncertainty of measurements so first um we can look for the discount and check that we can use the range method to look for the uncertainty which is to look for the standard deviation the actual value is unknown that was given a set of measurements then we do to to look for the uncertainty we sold for the standard deviation so you can also um you can also look for standard deviation of a data set so list count is um the smallest value that can be read from any measuring device so list count is the same as the resolution of your measuring device wherein the resolution is the the smallest the vision or the smallest value that can be or value that can be measured so for example we have here a centimeter ruler okay zero and then one so and that is one centimeter for example but if you look at your ruler so this is equal to one millimeter so meaning that the resolution of this ruler is one millimeter this is the smallest possible value that can be uh measured by your ruler so one millimeter and this is also the list count of your uh measuring device of this ruler probably like so as a general rule uncertainties must be reported with the same number of decimal places as the instrument you assume later on um we'll look at the different examples uncertainty nothing the uncertainty also defines the significant digits in your measurement significant digit so in real life so in application rather so in application included a significant figures don't say measurement by looking at the answer by looking at the the uncertainty or the least count of your measuring device so your uncertainty nothing is given by plus or minus the least terms just indicate their plus or minus and then the value of the list count so in recording your measurement it must contain so the numerical value or the quantity of measurement Qs so and then plus or minus the degree of uncertainty in the measurement so don't forget to include the unit of the measurement also so uncertainty so young uncertainty nothing is given by the least count so quantity of measurement plus or minus the least count is equal also is the same as quantity of measurement plus or minus the uncertainty foreign Ty of a digital measuring device and an analog measuring device so for the digital measuring device the uncertainty is simply the measure of the list analog device analog measuring device nothing the uncertainty is one half of the list count you need to multiply your list count by one half to give so that is the uncertainty of an analog measuring device but others also accept the least count as the I know as the the the uncertainty of an analogue Paramus accurate measurement s you need to multiply your list count by one half and that is equal to your uncertainty digital and measuring device scales so these are the these are the the measuring device that gives you so these are the measuring device that already shows you the the value of the measurements so no need to estimate no need to no need to look for the certain value using a scale so just for example it's a digital balance that is an example of a digital device so they thought the list count of digital device the the the uncertainty of it of a digital measuring device is equal to the list count so if we look at this example so in this digital balance the the smallest possible value so young decimals is only one decimal so only to the tenth place is 0.1 value the lowest possible value of the decimal that is not zero so it is one in this in this example more um digital balance we have the analytical balance so your analytical balance it can give um up to thousands hundreds thousands please so yeah four decimal places analytical device so in this case parasite analytical device Angeles so therefore in this example [Music] weighs 15.4 kilograms a 15.4 grams a will have here the measured Value Union quantity plus or minus 0.1 and then the unit so meaning the true mass of the banana is found um between so 15.4 minus 0.1 is 15.5 15.3 15.4 plus or minus I plus 0.1 is 15.5 so the true Mass of the banana is found between 15.3 to 15.5 grams so on the other hand analog kitchen scale in your in your measuring device that is an example of an analog so when a device is a scale device that is an analog equipment so in this example if we examine the scale scale so the the lowest resolution or the least count is 10 grams 10 grams zero to 100 and then your smallest division is division between 0 and 100 so we have here one two three four five six seven eight nine and then so 100 divided by 10 is equal to 10. so therefore the least count of this uh of this kitchen scale is foreign is equal to 10 grams so since this is an analog we need to multiply it by one half in order to determine the uncertainty of this kitchen scale so the uncertainty of this kitchen scale is plus or minus 5 grams so therefore if we if we report the values [Music] is about 610 610 plus or minus five and then the unit which is grams okay so let us have a concept Builder so this is um concept Builder number so for the following so determine the uncertainty of the volume of the liquid and the measure measured value measured value so here we have here Beaker and this is the scale so always read the volume of your liquid at the lower or the upper meniscus okay so for number one so you may this you may pause this video and just resume watching after you submitted your answer so for number one so first then 10 mL s we have here 20 and then 40 and then another line this must be 30 Dubai is 10 mL the measure of this interval or the smallest division is 10 mL that is our discount so you you uncertainty but this is an analog since scale this measuring device is an analog measuring device so you uncertainty nothing is equal to the list count times one half so therefore the uncertainty here is positive negative or plus and minus 5 ml estimate subscribe so just um refer to measuring with precision so you need to determine the halfway point so between this line is yeah so between 40 and then this one and smallest division at any steps therefore this must be 50. this is 50 ml so we need to determine the halfway point between 40 and 50 ml and then that means so we can estimate it to be from 46 to go to 46 47 or 48 Yen so estimate so so estimate now 48 plus or minus five ml um 46 or 47 plus or minus 5 ml the new best estimate nothing because so the best thing that we can do is to just estimate and then add our uncertainty so for the graduation cylinder young list count nothing so sorry list count nothing is 1 ml so if this line here is 20 so 21 22 23 24 and then this is 25 so therefore the smallest graduation or this yeah the smallest division here is 1 ml so this is our um discount one of Ellen since this is a an analog measuring device so to get the uncertainty so you know uncertainty is equal to the list count times one halves therefore the least count is equal to zero point five zero point zero five zero point zero five ml it's only 0.5 ml 1 divided by sorry sorry yeah so 0.5 ML and then determine the volume so young volume nothing um we know that this is 21 and this is 22. so determine the halfway point between 21 and 22 and estimate consensus so this could be 21.5 plus or minus our uncertainty answer nothing the second method is the range method so range method is a way of determining uncertainties in measurements that only applies to other edges of repeated measures multiple trials and so for example multiple Trials of the same physical quantity so you can solve for the uncertainty by looking for the range of your measurement so when we have a set of values using the range method will give us the uncertainty of measurements so for example different obtained values to determine the uncertainty of repeated measures subtract the lowest measurement from the highest measurement so dito you need to identifying value so here negative 22.3 value and sorry the lowest value here is 20.5 so young range simplest way on identifying Sabina Precision is using the range so we're in we we subtract the highest value and the lowest value so this um method can be used to identify the uncertainty so all we need to do so after after subtracting the the highest and the lowest value we need to divide it by two yeah so the highest value is 23 ML and then the lowest value is 20.5 ml so subtract the lowest from the highest value and then divide it by two so so divide the difference by two so we'll get 0.9 ml so the uncertainty of this data set is zero point positive negative or plus or minus 0.9 ml so to report liano the measurement to report this measurement you need to to get the average and then you mean and then plus or minus the uncertainty we have here 21.5 plus or minus 0.9 ml wherein young lowest boundary nothing is 21.5 minus 0.9 is 20.6 ML and the upper boundary nothing is 20 21.5 plus 0.99 ml which is 22.4 m so there are different types of uncertainties we have absolute uncertainty relative uncertainty so we're in relative uncertainty can also be expressed as a fraction or as a percentage in percent form nothing absolute uncertainty this is the actual size of the uncertainty yeah this is the actual size of uncertainty relative uncertainty it is expressed as a fraction or as a percentage only so young size of the uncertainty in order to know the size of uncertainty you need to relate the value of your fractional uncertainty or the value of your percentage uncertainty to the quantity of your measurement so it is important to know this to know how to get the absolute uncertainty and to know how to get the relative uncertainties of measurement because um fundamental measurements so some of our quantities are derived quantity so we're in we need to combine um other quantities you measurable quantities together with other measurable quantities so in those cases quantities so absolute uncertainty is expressed as the number of in is expressed as a number independent of the original number so meaning that for example the quantities quantity plus or minus the uncertainty Society Hindi given you you quantity measurement nothing uncertainty because your uncertainty nothing is simply the the recorded uncertainty uncertainty using the the list count method and the range method so that is the recorded uncertainty a number that is independent from our quantity while um fractional uncertainty in percentage uncertainty are values of uncertainties that needed to be relate it tells us or it it it um tells us how big of a part of the quantity is our uncertainty while in absolute uncertainty tells us the actual size of uncertainty so to get the fractional uncertainty it is expressed as uh the ratio between the original number and wait love Express us the fraction of the of the uncertainty rather and the quantity so your uncertainty for example and then the quantity or the measured value while the percent and percent uncertainty is simply uh multiplying the fractional uncertainty by 100 so let's say for example barbecue stick that measures 20.1 plus or minus 0.05 centimeters long so just an example so a find the absolute uncertainty B find the fractional uncertainty and C find the percent uncertainty so suppose that l represents our quantity and Delta L represents the uncertainty of our quantity so given by this information here so 20.1 cm is equal to l and then Delta L is equal to plus or minus 0.05 centimeters so to look for the I know so to look for the absolute uncertainty excuse me just simply look at the just simply look if given a young measure measured value nothing which is in the form of positive IQ plus or minus Delta Q so in this case we have your L plus or minus Delta L just look for the value of your uncertainty in your measured value and that will give you the absolute foreign how to determine the uncertainty because your uncertainty is simply the least count or simply the recorded uncertainty that is your absolute uncertainty and to express this to express absolute uncertainty we have here 20.1 hour quantity which is our L plus or minus our absolute uncertainty and then the unit so on the other hand we have here fractional and percent uncertainties your fractional nothing is the ratio between the uncertainty so uncertainty on and the quantity so you uncertainty numerator and the quantities and the denominator so we want to know how big of a part is our uncertainty of our quantities so 0.05 CM divided by 20 cm 20.1 cm is equal to 0.002 so to report this we can write it as 20.1 CM plus or minus 0.002 and unit s this is the same as 20.1 CM plus or minus 0.05 CM young uncertainty expression of uncertainty during the fractional the division of the numbers so to know the actual size of your of your of your fractional uncertainty you need to multiply this by your quantity so in this case we need to multiply 0.002 by the value of L which is 20.1 which will give us the value of 0.05 again so 0.002 times 20.1 cm is equal to 0.05 cm so on the on the other hand the month same percent are certainty nothing it's just the fractional uncertainty multiplied by 100 in order to express it as a percent so meaning is 20 of our quantity so to express it we write it as 20.1 CM which is our quantity L plus or minus the percent so voila again you um you need less young percent uncertainty just percent so to get the actual size so you need to multiply 0.2 percent times the quantity so which will give us all the 0.05 centimeter so as you can see expressions or minus 0.002 is the same as will give us the same value of uncertainty which is 0.05 CM so in 20.1 cm plus or minus zero point two percent will also tell us that the uncertainty the actual size of uncertainty is 0.050 so the only difference is the way it was expressed so dito tells us the actual size of the value while in in the relative uncertainties for fractional and percent tells us how big of a part in comparison to our quantity so that's why it is expressed as a fraction or expressed as a percentage of your um original value or of your quantity so for example original value or you want it is equal to 20.1 so young 0.002 not 10. becomes meaningless to know the actual size of your uncertainty because original value or the quantity because to know the actual size you need to multiply this or you need to relate by multiplying these two the quantity so the same with the person page foreign so the absolute value is the actual measure of uncertainty and shows how large the uncertainty is while fractional and percent uncertainty measures how big of a part of their quantity is the uncertainty and shows how large the uncertainty is in relation to the quantity or in relation to the True Value so the central problem in error propagation or uncertainty propagation is best conveyed in the question how do we report the uncertainties of derived quantities so we know that a derived quantity is dependent on other quantities that can be measured because a derived quantity is a combination of other quantities so if we combine other quantities how do we now combine their uncertainties so for example if I am to measure if I am to solve for the the force of an object so you must now object and the acceleration of an object has its own measure of uncertainty so if I if I multiplied mass times the acceleration which is a derived quantity so calculating with uncertainties describes how to combine measurements with uncertainties so so the the combination is expressed as the sum the sum the difference product the whole Shadow even the power of other quantities so there are rules that are needed to be applied needed to be applied involving um calculations with uncertainties so first is the um first is the addition rule so for addition and subtraction so if data are to be added so if data are to be added or subtracted we need to add their absolute uncertainties to express the uncertainty and measurement we um use the form Q plus or minus excuse me plus or minus Delta Q or the uncertainty recorded uncertainty is our absolute uncertainty which is the actual size of the uncertainty if we are to add or to subtract two values or data we need to add other absolute uncertainty so for example if you want to if we if we add a plus b results so we know that if a if a is a measured value so it also it will automatically have a measure of uncertainty which is given by Delta a so B will also have its own value of uncertainty which is given by Delta B so of course will also have its own um measure of uncertainty uncertainty some of these data so let us express this in this form so Delta a so a plus the plus or minus Delta a and then plus B plus or minus Delta P so this will give us C plus or minus Delta C so to get the value of C we just simply add the quantity of a plus the quantity B to get the value of quantity C is uncertainty which is Delta C so we will base now the value of this to the rule for addition and subtraction so for addition and subtraction if we if we are to add data the absolute uncertainties is the sum of the the uncertainty of your data so if if this is the uncertainty of of a and this is the uncertainty of quantity B the uncertainty of quantity C is given by Delta C is equal to Delta a plus Delta B so the same with subtraction so subtraction among for example A Plus Delta A minus B plus or minus Delta B will give us so C plus or minus Delta C so D2 the same then whether it is to be added or to be subtracted the absolute values are to be added so quantities with their uncertainties derived quantity we need to add the uncertainties so this is for addition and subtraction it's for second rule if data are to be multiplied or divided we need to add their relative uncertainty so therefore we need to look for the relative uncertainties of our data to be multiplied or to be divided so for example if C is equal to a times B so we know that a has its own value of uncertainty and Bs also has a measure of uncertainty so we know also that c as a derived quantity also has its value or measure of uncertainty given by Delta c s that is by um getting by getting the sum of their relative uncertainties and we know that relative uncertainty is the fraction between the uncertainty divided by the quantity so here if we want to know the value if you want to know the the the value of this uncertainty what we need to do is to equate Delta C relative uncertainty now measurement C Delta C over C is equal to Delta a over a plus Delta B over B so this is a relative uncertainty relative uncertainty of quantity B and this is the relative uncertainty of quantity C so we need to get the sum of the relative uncertainty in order to get the relative uncertainty of c and um just derive a an equation a solution to solve for Delta C so later we'll have an example so next is for a number raised to a power fractional or not multiply the relative uncertainty by the power so if we have if a number raised to a certain power for example that is to n or a number raised to 1 over n which is a fraction so the the relative uncertainty so so the uncertainty of the of the derived quantity is um solved by multiplying the relative uncertainty by the power so for example let the young measurement not an a raised to n plus or minus Delta a so cool uncertainty derived quantity nothing is we get the relative uncertainty of our measurement over um um we get the the uncertainty tail Bar measurement over relative uncertainty and then multiply it by the power yeah so for the last rule so for a number to be multiplied by a constant multiply the number and uncertainty if it is absolute constant number we need to multiply the constant to the quantity and to the absolute in and to the uncertainty if and only if the uncertainty is absolute uncertainty and if the uncertainty is relative we only need to multiply the constant by the quantity so using those rules kindly answer the following concept Builders so determine the uncertainty of the following so determine the uncertainty of the following problem so concept Builder number so you may pause this video and then resume watching after you submitted your answers in the given publicly so for number one your number one items ability to the length and width of a rectangle or L is equal to 6.5 plus or minus 0.1 meters and width is 3.4 plus or minus 0.2 meters respectively so we are asked to find the perimeter and the areas for letter A we are asked to find the perimeter so on Perimeter non rectangle is simply by adding all the sides of a rectangle so that is length plus length plus with plus width or simply two times length plus width excuse me so here uncertainty Nito this is an absolute uncertainty because if we if we distribute this unit to this one so we'll get 3.4 meters plus or minus 0.2 meters so since meron you need to uncertainty and this is an absolute uncertainty the same with the length so therefore applying the rule so dito we added we need to add the length and the width so you may apply nothing rule is the addition and subtraction rule for adding uncertain pieces addition and subtraction rule if data are to be added or subtracted we need to add their absolute uncertainties is 3.4 plus or minus 0.2 excuse me to find the the the quantity new quantity we need to add the quantity of the length and the quantity of the width so dito 6.5 plus 3.4 will give us nine point nine nine point nine and then since this is an addition between measured values we need to add their absolute answer to this so 0.1 plus 0.2 will give us 0.3 so we need to multiply this value to 2 so this is a constant number so we will now apply another rule rule number four a number to be multiplied by a constant multiply the number or the quantity and the uncertainty if the uncertainty is absolute if we look at the uncertainty of our given so you you want certain thing yeah so 0.3 is an absolute uncertainty because we added two absolute uncertainties so 0.3 is an absolute uncertainty so we need to multiply the the quantity and the answer and the absolute uncertainty to the constant so it's 9.9 times 2 is 19.8 and then 0.3 times 2 is 0.6 so the perimeter now is 19.8 plus or minus 0.6 meter so this is the uncertainty for this um quantity for the perimeter of the rectangle for letter B so finding the area is 6.5 plus or minus 0.1 meters young with nothing is 3.4 plus or minus 0.2 meters and what we want to know is the the area and the uncertainty of the area so you have here the area plus or minus the uncertainty of the area and we know that the unit of area is square so here is to solve for the area nothing um so using the given quantity of Ln W so so in quantity L is 6.5 is W is 3.4 so 6.5 times 3.4 this give this will give us 22.1 but since we have your two significant figures so 22 squared meter and area so since we multiply since we multiply here the length and the width so completing the nothing rule for multiplication if data are to be multiplied we need to add their relative uncertainties relative uncertainties so therefore don't say given nothing I don't problem nothing we need to look for the relative uncertainties of this value so we need to look for the relative uncertainties that if we have a quantity the relative uncertainty of the quantity is given by the uncertainty the fraction um is given by the uncertainty divided by D given quantity um Delta a over a so Delta a over a is equal to the sum of the values the uncertainties the relative uncertainties of the of the given values we have here length uncertainty the relative uncertainty for length is Delta L over L and then Plus Delta W over w so no need to add here parenthesis so if we plug in the information so you give information so Delta a over a is equal to 0.1 meter that is the uncertainty you want certainty divided by the quantity of the length which is 6.5 and then 0.2 young Delta W nothing or the uncertainty of w of BD is 0.2 and the quantity is 3.4 so if you perform this to 0.1 divided by 6.5 will give us this value and 0.2 divided by 3.4 will give us this value so this is the sum so this one is the sum of the relative uncertainties oh L and w es the uncertainty of our area so here so Delta a over a buffalo tonight by our value of the area which is 22 squared in third and then we need we just need to derive the solution for Delta a so by cross multiplying so 22 squared meter times the sum of the relative uncertainties of length and width will give us 1.64 squared meter and based from our definition of uncertainties but an uncertainties um Express into one significant figure into one significant figure so we will have an uncertainty of 2 squared meter so now the area 22. is 2 squared meter so as expressed in this form if it Expresses in this form or in in this form so a plus or minus Delta a is so 22 squared meter foreign so next the mass of a cubical object is 253.1 plus or minus 0.1 grams so if one side of the cube measures 5.25 plus or minus 0.05 centimeter what is its density so again young given attempts is which is equal to 235 0.1 plus or minus 0.1 grams um one side of a cube and one side of a cube measures new one side over cube is measured uh measured Five Point twenty five plus or minus 0.05 centimeters and what we want to know is the density so in density nothing if we express it in this form together with the uncertainty so we'll have here the density the quantity plus or minus the uncertainty of our density and and the unit for density is grams per cubic centimeter so first density so of value in the density is equal to Mass divided by the volume to get the volume we just need to get the cube of this one side of a cube so it's this is equal to the quantity of mass is 235.1 grams over 5.25 foreign so we'll have here 235.1 grams divided by so 5.25 cubed 2 raised to 3 is equal to 144.5 144.7 oh three one two five cubic centimeter so if we divide this one so 235 .1 divided by the volume [Music] so young density nothing is equal to 1.62 so following the significant digits of our given three significant digits so 1.62 grams per cubic centimeter okay so is equal to 1.62 grams per cubic centimeter so you apply so first 10 million you first rule they apply nothing is um rule number three so for a number raised to a power fractional or not multiply the relative uncertainty by the power because only one side of the cube so we need to solve for the volume of the cube so your volume is um s side raised to three so here so your volume down Cube nothing is equal to side raised to 3. so we have here young side nothing is 5.25 Plus or minus 0.05 over 0.05 CM so we need to get the the cube of this value for a number is top power fractional or not multiply the relative uncertainty by the power so we need to solve for the relative uncertainty of this uh of the cube of this measurement and then multiply it by the power so your power nothing is three so we need to solve for the relative uncertainty of this value cm and then you and CM so get the cube uncertainty so first quantity so 5.25 raised to three is equal to 144.703125 and then cubic centimeter so uncertainty uncertainty so since we have here a power so relative uncertainty which is Delta s over the side so this will give us new relative uncertainty is 0.05 divided by 5.25 0.05 divided by 5.25 is equal to 9.52 times 10 raised to negative three unit foreign and we need to multiply it by the power so according to the rule we need to multiply the relative uncertainty to the power to get the uncertainty of our volume nothing it is volume plus or minus the uncertainty of the volume yeah so 9.5 1952 times 10 raised to negative three times three is equal to so plus or minus zero point zero three so since human certainty nothing is two decimal places is only expressed into two decimal places so your volume nothing will be the volume nothing is equal to 144 point 70 plus or minus 0.03 and then cubic centimeter so that is our volume so now we can solve for the uncertainty we can solve for the for the uncertainty of the density so value so young value nothing for the volume is 144.70 plus or minus is volume of our Cube is equal to 144.70 plus or minus Delta D so plus or minus 0.03 cubic centimeter so since we already know the new uncertainty for the volume density is a density of an object and density of an object is equal to the mass divided by the volume volume and um to get the uncertainty of our density is division rule to get the density we divide mass by the volume so for for division if the data are to be divided we need to add their relative uncertainty so again we need to solve for the um relative uncertainty of each of the values so dito so Delta d the relative uncertainty of our density is equal to the sum of the relative uncertainty of mass so relative uncertainty of mass plus the relative uncertainty of our volume substitute not in detail so Delta d over d s value which is 1.62. uncertainty which is 0.1 grams and then quantity is 200 35 .1 grams so plus you relative uncertainty in the volume which is 0.03 cubic centimeter over 144.70 cubic centimeter NATO so we just need to perform the operation 0.1 divided by 235.1 plus 0.03 divided by 144.70 so just I will just look for the answer excuse me okay so by getting the sum of the relative uncertainties of the mass and the volumes [Music] will be so will be equal to 6.32 or 33 times 10 raised to negative 4. so to get the to solve for the the the uncertainty of our density to solve for the absolute uncertainty of our density we need to cross multiply 1.62 times the the sum of the relative uncertainty so you multiply an attention absolute uncertainty now density is equal to one point zero two times ten raised to negative 3. uncertainty nothing we need to add other we need to include other values [Music] so therefore the density the measure of density together with its uncertainty is equal to 1.62 plus or minus 1.02 grams per cubic centimeter so this is now your density measure of density together with its uncertainty foreign so just remember class that in measurement and in application there is no such thing as a little mistake because um if if the error if the error you made is outside the acceptance the range of accepted values deviation from the acceptable values collectively for example young deviation yeah from the range of acceptable values but collectively of your work so that's all for this video so see you in our next um discussion thank you