Question

Which of the following are exact numbers? a. There are 100 cm in 1 m. b. One meter equals 1.094 yards. c. We can use the equation $$^\circ \mathrm{F}=\frac{9}{5}^{\circ} \mathrm{C}+32$$ to convert from Celsius to Fahrenheit temperature. Are the numbers \(\frac{9}{5}\) and 32 exact or inexact? d. \(\pi=3.1415927\)

Short answer

a. Exact b. Inexact c. Exact (both \(\frac{9}{5}\) and 32) d. Inexact

Step-by-step solution

Statement a: 100 cm in 1 m

This statement tells us there are exactly 100 centimeters in one meter. Since this is a precise and definite value, the number 100 is an exact number in this case.

Statement b: One meter equals 1.094 yards

This statement provides a conversion factor between meters and yards. The number 1.094 is a rounded version of the actual conversion factor, and therefore, it is an inexact number.

Statement c: Celsius to Fahrenheit conversion equation

The Celsius-to-Fahrenheit conversion equation is given as: \[^{\circ} \mathrm{F}=\frac{9}{5}^{\circ} \mathrm{C}+32\] In this equation, both the fraction \(\frac{9}{5}\) and the number 32 are used as exact conversion constants. Therefore, both numbers are exact in this case.

Statement d: \(\pi = 3.1415927\)

The value of pi in this statement is given as 3.1415927. We know that the actual value of \(\pi\) cannot be accurately represented by a finite decimal or simple fraction. Therefore, the number 3.1415927 is an inexact representation of \(\pi\).

Key concepts

Measurement Conversion

Measurement conversion is crucial in everyday calculations, especially when dealing with units from different measurement systems. An exact number in a conversion is defined as a value with complete certainty and precision.

• For example, "100 centimeters in 1 meter" is an exact number because it is a universally agreed-upon standard. There is no variation or approximation involved here.
• On the other hand, "1 meter equals 1.094 yards" is an inexact number since it’s a rounded figure. The exact conversion is 1 meter equals approximately 1.09361 yards, showing that the given value is an approximation.

Understanding whether a number is exact or inexact helps in ensuring precision in calculations and avoids potential errors, especially in scientific and engineering contexts.

Celsius to Fahrenheit

Converting temperatures between Celsius and Fahrenheit involves a straightforward formula: \[^{\circ}\mathrm{F} = \frac{9}{5}^{\circ} \mathrm{C} + 32\] In this formula, the constants \(\frac{9}{5}\) and 32 are exact numbers. This means they are used universally without any error or approximation in this context.
The formula is derived based on the freezing and boiling points of water: 0°C equals 32°F and 100°C equals 212°F.

• The \(\frac{9}{5}\) component scales the Celsius degree to match the Fahrenheit scale.
• The "32" shifts the scaled value to align the freezing points on both scales.

Therefore, when converting these values, we rely on these exact constants to maintain consistency and accuracy across calculations. This conversion is essential not only in daily weather reports but also in scientific experiments where temperature readings are pivotal.

Pi Approximation

Pi, represented by the Greek letter \(\pi\), is a mathematical constant that represents the ratio of a circle's circumference to its diameter. Its value is approximately 3.14159, but as a true mathematical constant, \(\pi\) is an irrational number. This means it can't be exactly represented by a finite decimal or a common fraction.

• Whenever you see \(\pi\) written as 3.14 or 3.1415927, these are merely approximations to make calculations more manageable for practical purposes.
• The exact value of \(\pi\) goes on forever without repetition, making it essential to note that any decimal representation is not exact.

In mathematics and physics, using more digits of \(\pi\) allows for more accurate results in calculations. It's pivotal in areas involving geometry, trigonometry, and calculus, demonstrating its fundamental importance in mathematics.