Summary
IConversion rate and measurementsIIPrecision, greatest possible error and percent errorADefinitionBApplication to area and volumeIIIScientific notationConversion rate and measurements
Conversion rate
A conversion rate is a function which relates units of measurements.
The conversion rate between celsius and fahrenheit is \left(x\times \dfrac{9}{5}\right)+32.
If the temperature outside is 30^\circ celsius, then the temperature outside in fahrenheit is:
\left(30\times \dfrac{9}{5}\right)+32=86^\circ fahrenheit.
A cheetah can run after its prey at 90 kilometers per hour.
There are 1\ 000 meters in every kilometer and there are 60 minutes in an hour. Thus the conversion rate between kilometers per hour and meters per minute is:
\dfrac{1\ 000x}{60}=\dfrac{50x}{3}
Therefore, a cheetah can run after its prey at a rate of:
\dfrac{50\left(90\right)}{3}=\dfrac{4\ 500}{3}=1\ 500 meters per minute.
Precision, greatest possible error and percent error
Definition
Greatest possible error
The greatest possible error of a measurement is the largest possible difference between the measured value and the actual measurement.
We are measuring speeds of planes to the nearest kilometer per hour.
If we measure a plane traveling at 880 kilometers per hour, then the actual speed of the plane is somewhere between 879.5 kilometers per hour and 880.5 kilometers per hour.
Therefore, the greatest possible error is .5 kilometers per hour, or 500 meters per hour.
When a measurement is rounded, the greatest possible error is always half the unit of measurement that was rounded.
The mass of a cat is measured to be 4.5 kilograms. The measurement indicates that the weight was measured to the nearest tenth of a kilogram.
Therefore, the greatest possible error of the measurement is 0.05 kilograms, equivalently 50 grams.
Precision
The precision of a measurement is twice the greatest possible error. Equivalently, the precision of a measurement is the difference between the largest and smallest possible values that the measurement of an object could be.
If we are measuring speeds of planes to the nearest kilometer per hour, then the precision of the measurement is 1 kilometer per hour.
For example, if we measure the speed of a plane to be 850 kilometers per hour, then:
- The lowest possible speed of the plane could be is 849.5 kilometers per hour.
- The highest possible speed of the plane is less than 850.5 kilometers per hour.
The difference between these two values is 1 kilometer. Therefore, the precision is 1 kilometer per hour.
Percent error
The percent error provides us a way to measure the relative accuracy of a measurement. The percent error of a measurement is given by the formula:
\dfrac{\mbox{Greatest possible error}}{\mbox{Measurement}}\times 100\%
We are measuring the speeds of planes to the nearest kilometer per hour. The greatest possible error of any measurement is 0.5 kilometers per hour. If we measure a plane traveling 800 kilometers per hour, then the percent error of our measurement is:
\dfrac{0.5}{800}\times 100\%=.625\%
- A large percent error, such as a percent error of 10\% or greater, is an indication that the measurements being made are not accurate.
- A percent error between 1\% and 10\% indicates that the measurement being made is accurate.
- A percent error below 1\% indicates the measurement being made is very accurate.
In the above example, we computed the percent error of the speed of a plane to be 0.625\%. This number is below 1\%. Therefore, the measurement being made was very accurate.
Application to area and volume
The notions of precision, greatest possible error, and percent error of a measurement can be used to determine the accuracy of measures of area and volume. Specifically, any percent error below 1\% indicates that the measurement being made is very accurate and is a reliable measurement of data.
The surface area of the earth has been measured to the nearest million square kilometer to be 510{,}000{,}000 square kilometers. Therefore:
- The greatest possible error of the measurement is 500{,}000 square kilometers.
- The precision of the measurement is 1{,}000{,}000 square kilometers.
- The percent error of the measurement is \dfrac{500{,}000}{510{,}000{,}000}\times 100 \%\approx 0.1 \%.
Therefore, the measurement of 510{,}000{,}000 square kilometers is a very accurate measurement of the Earth's surface.
The volume of the pacific ocean has been measured to the nearest million cubic kilometer to be 660{,}000{,}000 cubic kilometers. Therefore:
- The greatest possible error of the measurement is 500{,}000 cubic kilometers.
- The precision of the measurement is 1{,}000{,}000 cubic kilometers.
- The percent error of the measurement is \dfrac{500{,}000}{660{,}000{,}000}\times 100\%\approx 0.08 \%.
Therefore, the measurement of 660{,}000{,}000 cubic kilometers is a very accurate measurement of the volume of the pacific ocean.
Scientific notation
Scientific notation
A number is said to be written in scientific notation if it is written as:
y\times 10^n
Where:
- 1\leq |y|\lt 10
- n is an integer
The scientific notation of a measurement of 7\ 110 miles is:
7.110 \times 10^3 miles
Observe that the integer n corresponds to how many places the decimal has to be moved so that the leading term of our number is followed by a decimal.
- Moving the decimal to the left corresponds to positive values of n.
- Moving the decimal to the right corresponds to negative values of n.
If x is a nonzero real number, then there is a unique number y and integer n such that
- 1\leq |y| \lt 10
- x=y \times 10^n
- The number 785 can be written as 7.85\times 10^2.
- The number -0.785 can be written as -7.85\times 10^{-2}
Scientific notation is a useful way of expressing measurements which are either too large or too small to write using conventional notation.
The radius of the Milky Way Galaxy is approximately 5\times 10^{17} kilometers. If we were to write the radius of the galaxy in standard notation, then our number would have 18 digits.
The average size of a proton is 1\times 10^{-16} meters. If we were to write the size of a proton in meters using standard notation, then our number would require writing 16 digits to the right of the decimal.
Adding and subtracting numbers in scientific notation
To add or subtract numbers in scientific notation, one must first convert the number with the smaller exponent on the 10 so that the exponents on the 10 s are the same.
Suppose we wanted to add 7\times 10^3 and 3\times 10^2. We convert 3\times 10^2 to have a 3 as the exponent on the 10 :
3\times 10^2=0.3\times 10^3
Therefore:
7\times 10^3+3\times 10^2=7\times 10^3+0.3\times 10^3=7.3\times 10^3
6\times 10^4-3.2\times 10^5= 0.6\times 10^5-3.2\times 10^5=-2.6\times 10^5
Multiplying and dividing numbers in scientific notation
Suppose x\times 10^n and y\times 10^m are two numbers written in scientific notation. Then:
\left(x\times 10^n\right)\left(y\times 10^m\right)=xy\times 10^{n+m}
\dfrac{x\times 10^n}{y\times 10^m}=\dfrac{x}{y}\times 10^{n-m}
\left(7\times 10^3\right)\left(3\times 10^2\right)=21\times 10^{3+2}=21\times 10^5
However, 21\times 10^5 is not in scientific notation because 21 is not between 1 and 10. Thus we adjust by moving the decimal spot of 21 to the left by 1 and add 1 to the exponent of 10 :
21\times 10^5=2.1\times 10^6
\dfrac{3\times 10^4}{6\times 10^5}=\dfrac{3}{6}\times 10^{4-5}=0.5\times 10^{-1}
However, 0.5\times 10^{-1} is not in scientific notation. Thus, we adjust by moving the decimal to the right by 1 and subtracting a 1 from the exponent of 10 :
0.5\times 10^{-1}=5\times 10^{-1-1}=5\times 10^{-2}