Unit 1 Measurements

Objectives 1.1 Convert between scientific notation and standard notation 1.2 Define and identify significant digits including being able to round and perform mathematical operations (add, subtract, multiply and divide) 1.3 Know and use the metric system including their prefixes and symbols 1.4 Use dimensional analysis to convert between units 1.5 Use measuring devices with precision and accuracy 1.6 Create and interpret graphs by calculating a line of best fit and using the line to extrapolate and interpolate data 1.7 Define and calculate density

1.1 Convert between scientific notation and standard notation Often in science, numbers are either very large or very small. We are used to having numbers in standard notation. However, scientific notation can be a easier way to represent these large or small numbers.

Scientific Notation A number in scientific notation uses a mathematical computation to make a number easier to write. Numbers in scientific notation will always have one number in front of the decimal point. The remaining significant digits are listed after the decimal point. To represent the magnitude of the number, the number is multiplied by ten to some power.

Converting a number from standard notation to scientific notation Assume you wanted to convert 5600 to scientific notation. First, determine the sig. digits and put the decimal point after the first sig. digit 5600 has two sig. digits so it should begin with 5.6 Second, determine how many times you had to move the decimal point. To get to 5.6, the decimal point was moved 3 times to the left. Third, multiple the number by ten to the number of times the decimal point was moved. 5600 in scientific notation would be 5.6 x 103

1.2 Define and identify significant digits Significant digits show precision of a number. Each measurement device precise only to a certain extent. Being able to accurately read measurement devices requires an understanding of which digits are significant.

Identify when zeros are considered significant Determining what makes a digit significant can be decided by four guidelines. All numbers 1-9 are significant. Any zero in between two nonzero numbers is significant. Any zero before a nonzero number is not significant. Any zero at the end of a number is significant if a decimal point is written in the number. Examples

Significant figure examples 25 has two significant figures because both numbers are nonzeros. 205 has three significant figures because the zero is in between nonzero numbers. 0.0025 has two significant figures because all of the zeros come before the nonzero numbers 2500 has two significant figures because there is no decimal point. 250.0 has four significant figures because there is a decimal point. Return

Round numerical values to correct significant digits Rounding to a specific significant digit requires the identification of the first significant digits in a number. Assume we wanted to round 25653 to two significant digits. We first must decide what the first two significant digits would be. In this case, it would be the 2 and the 5. We then look one more digit to the right. This will tell whether we round up or down. If the number is 0-4 round down. If the number is 5-9 round up. Since our number is 6, we will round up. Therefore, the number to two significant digits would be 26000 Practice

Rounding Significant figures practice Round the following numbers to 3 significant figures: 34689 0.5059 0.00426 825089 4.11206 0.008253 34700 0.506 0.00426 825000 4.11 0.00800 Return

Calculate using correct significant digits Mathematical calculations are often performed in science The main four computations are adding, subtracting, multiplying, and dividing. It is important to make sure your answer to these calculations represents the correct precision of the computation. i.e.: If I add 10. ml of water from one beaker to another beaker that holds 2500 ml.

Calculate using correct significant digits Adding and subtracting. Determine the number of significant digits in each of the original numbers. Looking at the original numbers, determine which one its last significant digit furthest to the left. Perform the mathematical computation. Round your answer to the place that was determined from step two. Example

Adding and subtracting significant digits Example Add 5.678 to 5.2 Step 1: Determine the significant digits in each number. 5.678 has 4 sig. digits 5.2 has 2 sig. digits Step 2: Determine the place of the last significant digit 5.678: thousandth place 5.2: tenths place (round your answer to this place) Step 3: Perform the mathematical computation 10.878 Step 4: Round to the correct place 10.9 would be the final answer Return

Calculate using correct significant digits Multiplying and dividing Determine the number of significant digits in each number. Decide which has fewer significant digits. Perform the mathematical computation Round your answer to the significant digit determined in step two. Example

Multiplying and dividing significant digits example Divide 5.02 by 3.557 Step 1: Determine sig. digits 5.02 has 3 sig. digits 3.557 has 4 sig. digits Step 2: Determine which one was fewer 5.02 has fewer Step 3: Perform the computation 5.02/3.557 = 1.411301659 Step 4: Round to the correct digit 1.41 would be the correct answer. Return

1.3 Know and use the metric system To give consistency to science measurements, the scientific community has agreed to use the metric system. The metric system is a base 10 system and uses a series a prefixes to distinguish the magnitude of each measurement. Prefixes

Metric Prefixes Kilo 1000 k Hecta 100 h Deka 10 D Base 1 Deci 0.1 d Centi 0.01 c Milli 0.001 m Micro 0.000001 µ Nano 0.000000001 n Return

The base units of the metric system Length meter Volume liter Mass kilogram (1000 grams) Temperature Kelvin Time Seconds Amount mole

Converting metric units What makes the metric system useful is that converting can be as simple as multiplying or dividing by a factor of 10. Using the prefixes, you can determine the conversion factor and then use dimensional analysis to convert. Example

Metric Conversion Example Suppose you wanted to convert 4500 meters to kilometers. By looking at the prefix list, we know that 1000 meters = 1 kilometer Because the conversion factor is set, we do not use these numbers to determine the number of significant digits in our answer. Return

1.4 Use dimensional analysis to do conversions Often when working with measurements, it is necessary to use different units. For instance, if I want to measure a book, I would probably use inches but if I wanted to measure a road, I would use miles. At times, it will be necessary to convert between different units and the technique we use to do that is known as dimensional analysis.

Dimensional Analysis Dimensional analysis uses the idea of multiplying by 1. If you multiply anything by one you get the same thing back. Any conversion factor can be thought of as a fraction that equals one. For instance, 1 ft = 12 inches so I can also flip the fraction to get one, too

Dimensional Analysis Based on dimensional analysis, if you want to convert from one unit to the next, all you have to do is multiple by the conversion factor written as a fraction in which the unit you want is the number on top and the unit you have is the number on the bottom. For instance: Convert 5 feet to inches.

1.5 Use measuring devices with precision and accuracy Science uses several different measuring devices. It is important to know how to read each to the correct precision and accuracy.

Use measuring devices with precision and accuracy When reading a measuring device, there are two aspects that you should look for To what place can you read with 100% certainty To what place can you estimate This reading is one digit to the right of the digit with 100% certainty.

Reading Glassware Liquid in glassware will form a curved surface. You want to read the bottom of the curve (meniscus) Next, determine what digit you can read with 100% accuracy. For this piece of glassware, the increments are increasing by 1 Therefore, I can say with 100% certainty that I can read this piece of glassware to the one’s place.

Reading Glassware The estimated reading is one digit to the right of the place with 100% certainty. For this piece, since the one’s place is 100% certain, we can estimate to the tenth’s place. Therefore, the reading on this piece would be 32.6 units of volume.

Reading Glassware Not all glassware increases in increments of 1, 10, 100, etc. The piece to the right increases by 25. Looking for100% certain reading In this case, it would be the 100s place. It is not possible to read the 10s place with certainty the way this piece is marked. The estimated reading would be the 10s place. The reading on this piece of glassware would be 30 units of volume.

Reading rulers Reading a ruler is same as glassware. Look for the place with 100% certainty Determine the estimated reading With a ruler, it is important to check the end. Some rulers get worn and the first segment is no longer accurate. In this case, you would want to align the object you are measuring at one and not at zero.

Reading rulers For the ruler below, we know: The place with 100% certainty is the one’s place. The estimated reading is the tenth’s place. The object is ends at 2.5 units but it starts at 1 so the length would be 1.5 units of length.

1.6 Create and interpret graphs Often in science, data is collected and graphed. When looking at these graphs, it is the trend that scientist are looking for. To determine the trend, a line of best fit is aligned to the data, and the equation for this line is calculated.

Graphing data points Steps to creating a graph: Title the graph Label the y-axis and x-axis Assign numerical values to each axis Plot each point. Concentration Graph 2.8 Concentration 2.0 1.2 0.4 1 2 3 4 5 6 Sample

Adding a line of best fit Drawing the line: Draw a line to fits the trend. It can but does not have to touch any data point. It must be straight. Concentration Graph 2.8 Concentration 2.0 1.2 0.4 1 2 3 4 5 6 Sample

Calculating the equation A line of best fit will follow the equation y=mx+b. m represents the slope of the line. b represents the y-intercept of the line. To calculate this equation, two data points are required. These points should be directly on the line. It is best to choose two brand new points. It is also a good idea to select your points where two of the gridlines meet.

Calculating the equation Look at the graph and line from two slides ago. Notice, the line crosses the gridlines at twice. Once at point (1, 0.4) and once at point (6, 2.8) Data points are written with the x-coordinate first and the y-coordinate second. (x, y) Using these two data points, it is possible to calculate the line of best fit.

Calculating the slope To calculate the line of best fit, first the slope is required. The slope is calculated using the equation: Using (1, 0.4) as (X1, Y1) and (6, 2.8) as (X2, Y2), we can calculate the slope to be 0.4. Now that the slope is determined, we can put it into the equation to get y=0.4x+b. 𝑌2−𝑌1 𝑋2−𝑋1 =𝑚

Calculating the y-intercept With the equation y=0.4x+b, we can now calculate the y-intercept. To do this, we will take one of the data points we used to calculate the slope and substitute it for y and x. For this example, we will choose (1, 0.4) We put the 0.4 in for y and 1 in for x so our equation looks like: 0.4 = 0.4(1)+b This would be solved to so b=0 which means the y-intercept is zero.

The line of best fit. Now that we know what the slope and y- intercept are equal to, we can write the equation for the line of best fit. The equation should still contain the y and x variables and will look like: y=0.4x + 0 or y=0.4x With the line of best fit equation, you can now use it to solve for unknown variables.

Using the line of best fit Generally in science, the line of best fit is created using a set of known standards. The purpose is to calculate unknowns using the line. Assume, a sample was analyzed and was determined to have a concentration of 1.5 and I wanted to know how close it was to a known sample. We can use our line to determine the answer.

Using the line of best fit The line of best fit gave the equation y=0.4x. We know the concentration is 1.5 and that concentration is on the y-axis. Therefore, we will put the 1.5 in for y: 1.5=0.4x Solve for x and we get 3.75 but with significant digits, it would be 4. Therefore, a concentration of 1.5 is close to sample #4.

1.7 Define and calculate density Density describes how much mass can be placed in a certain volume. Density (D) compares the ratio of mass (m) to volume (V) using a mathematical calculation: 𝑚 𝑉 =𝐷

Define and calculate density To get a better understanding of density, observe the two containers to the right. Notice they have the same volume. Notice how the top container holds 20 particles and the bottom only holds 10. Because there are more particles in the same volume, the top container is more dense.

Define and calculate density Still using the containers to the right, let’s assume: the height and width of the box is 1.0 units (width not drawn), the length is 1.4 units, each red circle is 1.0 grams. We can calculate the density by determining the volume and mass of each container.

Define and calculate density To determine the volume, we will take length x width x height: 1.0 units x 1.0 units x 1.4 units 1.4 units3 The mass for the top would be 20. grams and the bottom would be 10. grams. Since density is mass divided by volume, the top has a density of 14 g/units3 the bottom has a density of 7.1 g/units3.

This concludes the tutorial on measurements. To try some practice problems, click here. To return to the objective page, click here. To exit the tutorial, hit escape.

Definitions-Select the word to return to the tutorial Accuracy The closeness of the measurement to the correct value Precision The number of significant figures a device can be reliably read to The repeatability of a measurement Standard notation The common way to write a number i.e.: 25