Question

Which of the following are exact numbers? a. There are \(100 \mathrm{~cm}\) in \(1 \mathrm{~m}\). b. One meter equals \(1.094\) yards. c. We can use the equation \({ }^{\circ} \mathrm{F}=\frac{9 \circ}{2} \mathrm{C}+32\) to convert from Celsius to Fahrenheit temperature. Are the numbers \(\frac{9}{3}\) and 32 exact or inexact? d. \(\pi=3.1415927\).

Short answer

a. Exact b. Inexact c. \(\frac{9}{5}\) is exact, 32 is exact d. Inexact

Step-by-step solution

a. Determine if 100 cm in 1 m is an exact number

100 cm in 1 m is a well-defined conversion factor, as it is a universally accepted relationship. It is known with certainty that 1 meter is exactly equal to 100 centimeters; therefore, this number is exact.

b. Determine if 1.094 yards in 1 m is an exact number

The relationship between meters and yards given here is subject to some uncertainty or rounding as it is given with limited decimal places. Therefore, 1.094 yards in 1 m can be considered as an inexact number.

c. Determine if the numbers in the conversion equation from Celsius to Fahrenheit are exact or inexact

The conversion equation provided is: \({}^{\circ} \mathrm{F}=\frac{9}{5} \mathrm{C}+32\) The coefficients \(\frac{9}{5}\) and 32 are both well-known and exact conversion factors, so they are considered exact numbers.

d. Determine if the given value of \(\pi\) is exact or inexact

The given value of \(\pi\) is an approximation with limited decimal places: \(3.1415927\). The value of \(\pi\) is an irrational number that cannot be expressed as an exact finite decimal nor a simple fraction, which makes this number inexact.

Key concepts

Measurement Accuracy

Measurement accuracy is crucial in chemistry as it reflects the closeness of a measurement to the true value. High measurement accuracy ensures reliable and valid results that can be trusted for scientific analysis. To achieve this, chemists use precise tools and instruments calibrated to standards.

Take for instance a digital balance that measures mass; its accuracy is determined by how close the displayed mass is to the actual mass of the sample. Accuracy is impacted by various factors such as calibration of the instrument, environmental conditions, and the skill level of the person conducting the measurement. It’s always important to note that no measurement is perfectly accurate due to natural variations and limitations of measuring instruments; there's always a degree of uncertainty.

Conversion Factors

Conversion factors are used to change one unit of measurement to another. They are critical because the scientific community requires common units for clear communication and comparison of data. For instance, converting centimeters to meters or Fahrenheit to Celsius involves the use of conversion factors.

These factors are often based on defined physical constants and internationally accepted units and are therefore considered exact numbers. For example, the exact conversion factor of 1 meter being equal to 100 centimeters allows for precise calculations without additional uncertainty. Conversion factors are the backbone of dimensional analysis and ensure consistency across various scientific disciplines.

Significant Figures

The concept of significant figures is about indicating the precision of a number. Significant figures (also known as significant digits) in a numeric value convey how exact that measurement is. Basically, significant figures include all non-zero numbers, zeroes between non-zero numbers, and any final zeroes in the decimal portion.

The number of significant figures in a measurement is telling of the measurement's precision. For example, if a length is measured as 3.20 meters, it has three significant figures, implying that it was measured to the nearest hundredth of a meter. When calculations are performed, it’s important to keep track of significant figures to ensure that the results are not overrepresented in terms of their actual precision.

Exact and Inexact Numbers

In chemistry, numbers can be classified as exact or inexact. Exact numbers have a definite value that is known with complete certainty. These often come from defined quantities, such as conversion factors or counted objects, where there is no variability or uncertainty (e.g., 100 cm in 1 m is an exact number).

In contrast, inexact numbers are those with a level of uncertainty due to the limitations of the measurement devices or the method of acquiring the number. Measurements are typically inexact because they are subject to some degree of inaccuracy, as seen with the value of 1.094 yards being an approximation. An inexact number’s precision is generally indicated by the number of significant figures it has. Understanding the difference between these types of numbers is essential for correctly interpreting and reporting measurements in science.