Question

Identify the numbers in each of the following statements as measured or exact: a. There are 31 students in the laboratory. b. The oldest known flower lived \(1.20 \times 10^{8}\) years ago. c. The largest gem ever found, an aquamarine, has a mass of \(104 \mathrm{~kg}\). d. A laboratory test shows a blood cholesterol level of \(184 \mathrm{mg} / \mathrm{dL}\).

Short answer

(a) exact, (b) measured, (c) measured, (d) measured

Step-by-step solution

- Identify if the number in statement (a) is measured or exact

Consider the statement 'There are 31 students in the laboratory.' The number of students is counted and represents an exact count. Hence, the number 31 is exact.

- Determine if the number in statement (b) is measured or exact

Consider the statement 'The oldest known flower lived \(1.20 \times 10^{8}\) years ago.' This number is derived from scientific research and measurements. As it is based on estimation and measurement, the number \(1.20 \times 10^{8}\) is a measured number.

- Evaluate the number in statement (c)

Consider the statement 'The largest gem ever found, an aquamarine, has a mass of \(104 \mathrm{~kg}\).' The mass of the gem is determined through a measurement process. Therefore, the number \(104 \mathrm{~kg}\) is a measured number.

- Assess the number in statement (d)

Consider the statement 'A laboratory test shows a blood cholesterol level of \(184 \mathrm{mg} / \mathrm{dL}\).' This value is obtained through a laboratory test and involves measurement. Thus, the number \(184 \mathrm{mg} / \mathrm{dL}\) is a measured number.

Key concepts

Measured Numbers

Measured numbers are values obtained through the process of measurement and observation. These numbers are not always exact due to the limitations of measurement tools and techniques. They often come with a degree of uncertainty. For example, in the statement 'The largest gem ever found, an aquamarine, has a mass of \(104 \mathrm{kg}\),' the mass of the gem is measured using a scale, which can have a margin of error depending on the precision of the instrument.
Another instance is 'A laboratory test shows a blood cholesterol level of \(184 \mathrm{mg} / \mathrm{dL}\).' This value is obtained from a medical test, which involves scientific measurement methods. Hence, these results might fluctuate slightly upon subsequent measurements due to various factors such as equipment calibration or human error.

Exact Numbers

Exact numbers are values that are known with complete certainty. These numbers are typically obtained through counting or are defined quantities. For example, in the statement 'There are 31 students in the laboratory,' the number 31 is an exact number because it is obtained by simply counting the students, leaving no room for uncertainty.
Exact numbers also include defined values, such as 1 dozen equaling 12 items or 1 kilometer equaling 1000 meters. These values are universally accepted and do not change, providing a concrete base for calculations and comparisons.

Scientific Measurement

Scientific measurement is a method used to obtain values with a specified degree of accuracy and precision. This process involves using instruments and techniques to observe and record values. These values can be physical quantities like length, mass, or temperature.
For instance, 'The oldest known flower lived \(1.20 \times 10^{8}\) years ago.' This is determined through methods such as radiometric dating, which estimates the age of objects based on the rate of decay of radioactive elements. While these measurements are highly sophisticated, they still come with certain degrees of uncertainty and are categorized as measured numbers.
In scientific research and experiments, ensuring the precision and accuracy of measurements is crucial. Scientists are always working towards minimizing errors and improving the reliability of their measurements.

Numerical Precision

Numerical precision refers to how finely a measurement is expressed. It indicates the smallest unit to which a measured value can be reliably known.
For example, the mass of an aquamarine gem is given as \(104 \mathrm{kg}\). If the scale used measures up to the nearest kilogram, its precision is limited to whole numbers. On the other hand, the value \(1.20 \times 10^{8}\) years indicates the precision up to two decimal places, reflecting the level of reliability in the measurement of the flower's age.
Precision ensures we understand how detailed our measurement is, while accuracy reflects how close our measured value is to the true value. Both elements are essential in reporting scientific data accurately.