Uncertainty - Multiplication and Division

Summary

This lesson focuses on mathematical operations involving uncertainty, specifically multiplication and division. The process starts by calculating the percent uncertainty for each number, then multiplying or dividing the values while adding the percent uncertainties. The result is converted back to a non-percentage form and rounded to match the significant figures of the original measurements. Examples are provided for both operations, illustrating how to handle uncertainties in calculations.

Takeaways

• 📐 When multiplying numbers with uncertainties, first calculate the percent uncertainty by dividing the uncertainty by the measured value and multiplying by 100%.
• 🔢 Multiply the measured values to get the result, but remember to add the percent uncertainties together to find the total uncertainty.
• 🔄 Convert the total percent uncertainty back to a decimal to find the uncertainty in the result by multiplying the result by this decimal.
• ✂️ Round the final result to the appropriate number of significant figures, matching the number of significant figures in the original measurements.
• 🔄 For division with uncertainties, follow a similar process by calculating percent uncertainties for both the numerator and the denominator.
• 📉 Subtract the percent uncertainties when dividing, as the uncertainties propagate inversely with division.
• 📝 Keep additional digits during intermediate calculations to ensure accuracy before rounding at the end to minimize compounding errors.
• 📏 When rounding, consider the significant figures in the original measurements and the number of decimal places in the uncertainties.
• 📉 For division results, add the percent uncertainties to get the total uncertainty, then convert this percentage back to a decimal to find the uncertainty value.
• 🔍 The final answer provides a range within which the true value lies, calculated by adding and subtracting the uncertainty from the measured result.

Q & A

What is the main topic of the lesson discussed in the transcript?

The main topic of the lesson is uncertainty in mathematical operations, specifically focusing on multiplication and division involving uncertain values.

How is the percent uncertainty calculated for the first example in the transcript?

The percent uncertainty is calculated by dividing the estimated uncertainty by the measured value and then multiplying by 100 percent.

What are the measured values and their uncertainties for the first example in the transcript?

The measured values are 15.8 cm with an uncertainty of 0.5 cm and 24.7 cm with an uncertainty of 0.4 cm.

How are the uncertainties added when multiplying two numbers with uncertainties?

When multiplying two numbers with uncertainties, the percent uncertainties are added together, not the uncertainties themselves.

What is the result of multiplying 15.8 cm and 24.7 cm from the transcript?

The result of multiplying 15.8 cm by 24.7 cm is 390.26 cm².

How is the final uncertainty percentage converted back to a non-percentage form in the transcript?

The final uncertainty percentage is converted back to a non-percentage form by multiplying the result of the operation by the decimal form of the uncertainty percentage.

What is the significance of rounding to the appropriate number of significant figures in the context of the transcript?

Rounding to the appropriate number of significant figures ensures that the final answer reflects the precision of the original measurements and maintains consistency in the level of certainty.

What is the process for handling division when dealing with uncertain values as described in the transcript?

For division with uncertain values, the percent uncertainties are first calculated for both the numerator and the denominator, then these percent uncertainties are added together, and finally, the division operation is performed.

What are the rules for converting uncertainty into a percent uncertainty when performing mathematical operations with uncertain values?

The rules for converting uncertainty into a percent uncertainty involve dividing the uncertainty by the measured value and multiplying by 100 for both multiplication and division operations.

How does the transcript suggest rounding the final calculated values to ensure accuracy?

The transcript suggests rounding the final calculated values to three significant figures, focusing on the digit in the fourth decimal place to determine whether to round up or down.

What is the final result of the division example given in the transcript, including the uncertainty?

The final result of the division example is 1.87 cm plus or minus 0.08 cm, indicating the true answer lies between 1.79 cm and 1.95 cm.

Outlines

00:00

📐 Multiplication of Uncertain Values

This paragraph introduces the concept of uncertainty in measurements and how it affects mathematical operations, specifically multiplication. The example given involves multiplying two numbers with uncertainties: 15.8 ± 0.5 cm and 24.7 ± 0.4 cm. The process starts by calculating the percentage uncertainty for each number. The uncertainty is then added as a percentage after the multiplication of the measured values. The final step involves converting the total percentage uncertainty back into a non-percentage form and rounding the result to the appropriate number of significant figures, aligning with the original measurements' precision.

05:04

🔍 Division of Uncertain Values

The second paragraph continues the discussion on uncertainty but focuses on division. It presents an example where 28.4 ± 0.6 cm is divided by 15.2 ± 0.3 cm. Similar to multiplication, the uncertainty for each number is first converted into a percentage. The division operation is then performed, and the uncertainties are added as percentages. The resulting percentage uncertainty is converted back to a non-percentage form. The final result is rounded to three significant figures, which is consistent with the precision of the original measurements. The process highlights the importance of rounding at the end of calculations to minimize propagation of rounding errors.

10:05

📏 Final Calculation and Rounding

The final paragraph concludes the division example by rounding the calculated value to one significant digit in the uncertainty, resulting in a final answer of 1.87 ± 0.08 centimeters. It explains the significance of this result, indicating that the true value lies within the range defined by the calculated value and its uncertainty. The paragraph emphasizes the importance of understanding how uncertainties affect the reliability and interpretation of measurement results.

Keywords

💡Uncertainty

Uncertainty in the context of the video refers to the degree of doubt or inexactness associated with a measurement. It is a fundamental concept in experimental science and engineering, where no measurement can be made with absolute precision. In the script, uncertainty is represented by 'plus or minus' a certain value, indicating the range within which the true value likely lies. For example, '15.8 plus or minus 0.5 centimeters' suggests the measurement could be as low as 15.3 cm or as high as 16.3 cm.

💡Percent Uncertainty

Percent uncertainty is a way to express the relative uncertainty of a measurement as a percentage of the measured value. It is calculated by dividing the absolute uncertainty by the measured value and then multiplying by 100. In the video, this concept is used to standardize the way uncertainties are compared and combined during mathematical operations. For instance, '0.5 divided by 15.8 and then multiplied by 100 percent' gives the percent uncertainty, which is used to determine the combined uncertainty after multiplication or division.

💡Significant Figures

Significant figures are the digits in a number that carry meaning contributing to its precision. This includes all digits except leading zeros. The video emphasizes the importance of rounding results to the same number of significant figures as the original measurements to maintain consistency in the precision of the data. For example, when calculating '390.26 plus 18.67', the final result is rounded to '390 ± 20' to match the significant figures of the original measurements.

💡Multiplication

In the context of the video, multiplication is one of the mathematical operations discussed in relation to handling uncertainty. When multiplying two measurements, each with its own uncertainty, the uncertainties are added together to find the total uncertainty of the result. The video script demonstrates this by multiplying '15.8 cm' by '24.7 cm' and then adding their percent uncertainties to find the overall uncertainty of the product.

💡Division

Division, like multiplication, is a mathematical operation that requires special consideration when dealing with uncertain measurements. The video explains that when dividing two numbers with uncertainties, the percent uncertainties are added together to estimate the total uncertainty of the quotient. This is illustrated when the script divides '28.4 cm' by '15.2 cm' and combines their percent uncertainties to determine the uncertainty of the result.

💡Rounding

Rounding is the process of adjusting a number to a certain number of significant figures or decimal places. The video script discusses rounding in the context of finalizing calculations involving uncertainty. It is important to round at the end of calculations to avoid cumulative errors. For example, the script rounds '18.67' to '20' to match the significant figures of the original measurements.

💡Measured Value

The measured value is the result obtained from a scientific measurement, which is the best estimate of the true value based on the data collected. In the video, measured values are given as '15.8 cm' or '24.7 cm', and they are the starting points for calculations involving uncertainty. These values are then used in mathematical operations like multiplication and division.

💡Estimated Uncertainty

Estimated uncertainty is an approximation of the uncertainty in a measurement, often provided as a fixed value associated with the measurement. In the video, this is represented by 'plus or minus' a certain number, such as '0.5 cm' or '0.4 cm'. These values are used to calculate percent uncertainty, which is then used in further calculations.

💡Decimal Places

Decimal places refer to the digits in a number that come after the decimal point. The video script discusses the importance of considering decimal places when converting percent uncertainties to non-percentage form and when rounding final results. For example, converting '4.784 percent' to a decimal involves moving the decimal point two places to the left, resulting in '0.04784'.

💡True Value

The true value is the actual value of a quantity that is being measured. In the context of the video, the true value is the value that the measured value and its uncertainty are intended to estimate. The script explains that the true value lies within a range defined by the measured value plus or minus its uncertainty, such as '1.87 plus or minus 0.08 centimeters'.

Highlights

Introduction to the concept of uncertainty in mathematical operations, specifically multiplication and division.

Explanation of how to calculate percent uncertainty by dividing the estimated uncertainty by the measured value and multiplying by 100%.

Example calculation of percent uncertainty for the number 15.8 cm with an uncertainty of 0.5 cm.

Example calculation of percent uncertainty for the number 24.7 cm with an uncertainty of 0.4 cm.

Multiplication of two numbers with uncertainties: 15.8 cm and 24.7 cm.

Adding percent uncertainties (3.1646% and 1.6194%) to get the total percent uncertainty for the multiplication.

Conversion of total percent uncertainty back to a non-percentage form for the result of 390.26.

Rounding the final result to three significant figures, as per the original measurements.

Rounding the uncertainty to one significant digit to match the original uncertainties.

Final answer for the multiplication example: 390 cm with an uncertainty of 20 cm.

Introduction to the division example with numbers 28.4 cm and 15.2 cm.

Calculation of percent uncertainty for the division example: 2.1127% for 28.4 cm and 1.9737% for 15.2 cm.

Division of the two numbers with uncertainties: 28.4 cm divided by 15.2 cm.

Adding the percent uncertainties for the division operation to get the total percent uncertainty.

Conversion of the total percent uncertainty for division into a non-percentage form.

Rounding the final result of the division to three significant figures: 1.87 cm.

Rounding the uncertainty for the division to one significant digit: 0.08 cm.

Final answer for the division example: 1.87 cm with an uncertainty of 0.08 cm.

Explanation of the range of the true answer based on the calculated uncertainty.

Transcripts

in today's lesson we're going to talk

about uncertainty

mathematical operations associated with

uncertainty specifically multiplication

and division so let's begin with our

first example

let's say we have

15.8

plus or minus 0.5

centimeters

and we have another number 24.7 plus or

minus

0.4 centimeters

so what we're going to do in this

example is we're going to multiply

these two numbers

so how can we do this

well before you multiply you need to get

the percent uncertainty

so we need to take the

estimated uncertainty

divided by the measured value and

multiply by a hundred percent

so we're going to change the operation

so if we take 0.5

divided by

15.8

and then multiply that by 100 percent

that will give us the percent

uncertainty

which is three point one six

four six percent

now let's do the same thing

with the other number

so it's going to be 0.4 divided by 24.7

times 100

so that's going to be 1.6194

percent

all right so now we're going to multiply

these two numbers

so let's use a calculator to multiply

15.8

and 24.7

so that's going to be 390.26

now for the uncertainties

we're going to add them we're not going

to multiply them

for this to work you need to add the

percentages

so 3.1646

plus 1.6194

that's going to be 4.784

percent

now there's still some other things we

need to do

so now that we have the percentage we

need to convert it back

into

a non-percentage form

so what we need to do is find out what

is

4.784 percent of 390.26

to figure that out take 390.26 multiply

it

by this value as a decimal

if you take four point seven eight four

percent and divided by a hundred

the decimal value is point

zero four

seven eight four

basically move the decimal two units to

the left

so 390.26 times that number

will give us

18.67

so we have

390.26 plus

18.67 as our answer but now

let's round it to the appropriate number

of significant figures

so looking at our original numbers

we see that

they have three significant figures

so to round 390.26 to three significant

figures we could say it's simply 390

with the decimal point which will make

the zero significant

now looking at our uncertainty

the 0.5 or the 0.4 it only has one

significant digit

so let's round 18.67 to a number with

one significant digit and that's going

to be 20

without a decimal point so the zero

is not significant

and then we need to put the unit

centimeters

so this is the answer for this problem

so now you know how to multiply two

numbers

that contain uncertainty values

now let's try an example

with

division

so let's say we have this number

28.4

plus or minus

0.6 centimeters

and the next number is going to be

15.2 plus or minus

0.3 centimeters

so this time we're going to use division

we're going to divide

the top number by the bottom number

but now just like before we need to

convert

the uncertainty

into a percentage value or we need to

get the percent uncertainty

so let's start with the top number it's

going to be

0.6

divided by 28.4 times 100

so 0.6 times 100 is 60 60 divided by

28.4

that's

2.1127 percent

now let's do the same thing with the

other number

so we're going take point three

and we're gonna divide it by

fifteen point two and then multiply that

by a hundred percent

so point three times a hundred is thirty

thirty divided by fifteen point two

that is going to give us 1.9737

now let's rewrite the problem

so this is going to be 28.4

plus or minus

2.1127

percent

and then a bottom number

15.2 plus or minus

1.9737 percent

and we're still dealing with division

all right so now we can get rid of this

28.4

divided by 15.2

so that's one point

eight

six

eight four

i like to get a few additional digits

because i'm gonna round it later and

plus

i'm going to get a percentage of that

number so i want to get a more accurate

answer and then round at the very end

when you round in the middle of the

operations

it'll compound

and create a bigger error at your final

answer so it's always good to round at

the very end

so now let's add

the two percentages

just like we did before so the rules

multiplication and division are the same

you need to convert the uncertainty into

a percent uncertainty

and then when you perform the operation

be it multiplication or division

in both cases you need to add

the percent

uncertainty

2.1127

plus 1.97 three seven

that is going to be four point

zero eight six four percent

all right

so now that we've done the operation

let's convert the percent uncertainty

into a non-percentage uncertainty value

so what is 4.0864

of

that number 1.8684

that's going to be 1.8684

times

4.0864 percent times divided by 100

percent

so this works out to be point

zero

7

6 3 5

so this is going to be 1.8684

plus or minus points

and let's pay attention to the units

this is in centimeters

all right so now let's focus on rounding

our measured values

they're rounded to three significant

digits

in our answer

we have five significant figures but we

need to round it to three

so how should we do this

well we need to focus on rounding this

number the six

looking at the number after it it's an

eight

if it's five or more we need to round up

so we got around

1.8684 up

to 1.87

now

both uncertainty values they only

contain one significant digit so we're

going to round

our uncertainty to one significant digit

so this is the first significant digit

the second one it's a six so we're gonna

round the seven up to an eight

so this is going to be plus or minus

point zero eight centimeters

so that's our final answer 1.87 plus or

minus 0.08 centimeters

so what this means is that

the true answer is somewhere between

these two values if we take 1.87 and

subtract it by 0.08

on the low side this could be

1.79

on the high side if we add 0.08 it could

be 1.95

so the true answer is somewhere

between those two extremes