Prerequisite Science Skills Lecture Presentation Prerequisite Science Skills John Singer Jackson College

Measurements All measurements are acquired by use of an instrument. Every measurement has a degree of inexactness, termed uncertainty.

Measurements We will generally use metric system units. These include: the centimeter, cm, for length measurements the gram, g, for mass measurements the milliliter, mL, for volume measurements

Length Measurements Ruler A has 1-cm divisions, so we can estimate the length to ± 0.1 cm. The length is 4.2 ± 0.1 cm. Ruler B has 0.1-cm divisions, so we can estimate the length to ± 0.05 cm. The length is 4.25 ± 0.05 cm.

Uncertainty in Length Ruler A: 4.2 ± 0.1 cm; Ruler B: 4.25 ± 0.05 cm. Ruler A has more uncertainty than Ruler B. Ruler B gives a more precise measurement.

Mass Measurements The mass of an object is a measure of the amount of matter it contains. Mass is measured with a balance and is not affected by gravity.

Volume Measurements Volume is the amount of space occupied by a matter.

Significant Digits Each number in a properly recorded measurement is a significant digit (or significant figure). The significant digits of a measurement include all values known with certainty plus one uncertain digit (the last one).

Significant Digits

Exact Numbers Significant digit rules do not apply to exact numbers. For example, there are seven quarters in the image to the right.

Rounding Off Nonsignificant Digits All numbers from a measurement are significant. However, we often generate nonsignificant digits when performing calculations. There are three rules for rounding off numbers.

Rules for Rounding Numbers If the first nonsignificant digit is less than 5, drop all nonsignificant digits. If the first nonsignificant digit is greater than or equal to 5, increase the last significant digit by 1 and drop all nonsignificant digits. If a calculation has several multiplication or division operations, retain nonsignificant digits in your calculator until the last operation.

Rounding Examples A calculator displays: 15.73849 Three significant digits are justified. The first nonsignificant digit is a 3, so we drop all nonsignificant digits and get 15.7 as the answer.

Rounding Examples A calculator displays: 18.750019 Three significant digits are justified. The first nonsignificant digit is a 5, so the last significant digit is increased by one to 8. All the nonsignificant digits are dropped, and we get 18.8 as the answer.

Rounding Off and Placeholder Zeros Round the measurement 183 mL to two significant digits. If we keep two digits, we have 18 mL, which is only about 10% of the original measurement. Therefore, we must use a placeholder zero: 180 mL. Recall that placeholder zeros are not significant.

Rounding Off and Placeholder Zeros Round the measurement 48,457 g to two significant digits. We get 48,000 g. Remember, the placeholder zeros are not significant, and 48 grams is significantly less than 48,000 grams.

Adding and Subtracting Measurements When adding or subtracting measurements, the answer is limited by the value with the most uncertainty.

Adding and Subtracting Measurements Let’s add three mass measurements. The measurement 114.3 g has the greatest uncertainty (± 0.1 g). The recorded answer should be 115.6 g.

Multiplying and Dividing Measurements When multiplying or dividing measurements, the answer is limited by the value with the fewest significant figures.

Multiplying and Dividing Measurements Let’s multiply two length measurements: (7.28 cm)(4.6 cm) = 33.488 cm2 The measurement 4.6 cm has the fewest significant digits—two. The recorded answer should be 33 cm2.

Exponential Numbers Exponents are used to indicate that a number has been multiplied by itself. Exponents are written using a superscript; thus, (4)(4)(4) = 43. The number 3 is an exponent and indicates that the number 4 is multiplied by itself three times. It is read “4 to the third power” or “4 cubed.” (4)(4)(4) = 43 = 64

Powers of 10 A power of 10 is a number that results when 10 is raised to an exponential power. The power can be positive (for numbers greater than one) or negative (for numbers less than one).

Powers of 10

Scientific Notation Numbers in science are often very large or very small. Scientific notation utilizes the significant digits in a measurement followed by a power of 10. The significant digits are expressed as a number between 1 and 10.

Applying Scientific Notation Step 1: Place a decimal after the first nonzero digit in the number, followed by the remaining significant digits. Step 2: Indicate how many places the decimal is moved by the power of 10. A positive power of 10 indicates that the decimal moves to the left. A negative power of 10 indicates that the decimal moves to the right.

Scientific Notation 2.68 × 1022 atoms There are 26,800,000,000,000,000,000,000 helium atoms in 1.00 L of helium gas. Express the number in scientific notation. Place the decimal after the 2, followed by the other significant digits. Count the number of places the decimal has moved to the left (22). Add the power of 10 to complete the scientific notation. 2.68 × 1022 atoms

Scientific Notation 1.16 × 10–7 m The typical length between a carbon and oxygen atom in a molecule of carbon dioxide is 0.000000116 m. What is this length expressed in scientific notation? Place the decimal after the first 1, followed by the other significant digits. Count the number of places the decimal has moved to the right (seven). Add the power of 10 to complete the scientific notation. 1.16 × 10–7 m

Scientific Calculators A scientific calculator has an exponent key (often EXP) for expressing powers of 10. If your calculator reads 7.45 E-17, the proper way to write the answer in scientific notation is 7.45 x 10–17.

Scientific Calculators To enter the number in your calculator, type 7.45, press the exponent button (EXP) and type in the exponent followed by the +/– key.

Chapter Summary A measurement is a number with an attached unit. All measurements have uncertainty. The uncertainty in a measurement is dictated by the calibration of the instrument used to make the measurement. Every number in a recorded measurement is a significant digit.

Chapter Summary, Continued Placeholding zeros are not significant digits. If a number does not have a decimal point, all nonzero numbers and all zeros between nonzero numbers are significant. If a number has a decimal place, significant digits start with the first nonzero number, and all digits to the right are also significant.

Chapter Summary, Continued When adding and subtracting numbers, the answer is limited by the value with the most uncertainty. When multiplying and dividing numbers, the answer is limited by the number with the fewest significant figures. When rounding numbers, if the first nonsignificant digit is less than 5, drop the nonsignificant figures. If the number is 5 or more, raise the first significant number by 1, and drop all of the nonsignificant digits.

Chapter Summary, Continued Exponents are used to indicate that a number is multiplied by itself n times. Scientific notation is used to express very large or very small numbers in a more convenient fashion. Scientific notation has the form D.DD x 10n, where D.DD are the significant figures (and between 1 and 10) and n is the power of 10.