Question

True or False? \(5.5 \mathrm{~L}=5.500 \mathrm{~L}\). ______

Short answer

True, 5.5 L equals 5.500 L.

Step-by-step solution

Understanding the Problem

We need to determine if the volume expressed as \(5.5 \mathrm{~L}\) is equal to the volume expressed as \(5.500 \mathrm{~L}\). This is essentially an evaluation of the equality between the two decimal numbers.

Analyzing Decimal Places

The first value given is \(5.5\), which can be rewritten as \(5.50\) or \(5.500\). Adding zeroes after the last significant digit does not change the value of a decimal. Thus, \(5.5\) is equivalent to \(5.500\).

Conclusion on Equality

Since adding zeroes after the decimal in \(5.5\) does not alter the value, we conclude that \(5.5 \mathrm{~L}\) and \(5.500 \mathrm{~L}\) are indeed equal.

Key concepts

Decimal Numbers

Decimal numbers are fundamental in representing values that are not whole, and they play a crucial role in various fields. This type of number uses a decimal point to separate the whole number from the fractional part.
For instance, in the number 5.5, the digit '5' before the decimal point is the whole part, and '.5' is the fractional part.
One important thing to note about decimal numbers is that you can add as many zeroes as you wish after the decimal point without changing the value of the number. For example:

• 5.5 = 5.50 = 5.500, and even = 5.5000.

These representations are equivalent because the trailing zeroes do not affect the numerical value. They merely serve as placeholders to indicate a level of precision often dictated by significant figures. Understanding how decimal numbers work helps us better comprehend numerical values in scientific and everyday contexts.

Significant Figures

Significant figures are the digits in a number that carry meaning and contribute to its precision. This concept is essential in scientific measurements and calculations, where precision and accuracy are critical.
When we say a number has significant figures, we're referring to all non-zero digits, any zeros between them, and any trailing zeros in the decimal portion. For example:

• In 5.500, all three digits (5, 5, and 0) are significant.
• Contrastingly, in 5.5, only the '5.5' is significant unless this is being expressed for precision as 5.500.

By understanding significant figures, we know how precisely a measurement is indicated. If a volume is expressed as 5.500 L, it implies a degree of precision to the thousandth of a liter, despite it being numerically equal to 5.5 L. Therefore, while both values are equal in mathematical terms, the number of significant figures provides an insight into measurement accuracy.

Numerical Analysis

Numerical analysis is a branch of mathematics that deals with algorithms for solving numerical problems. It includes the study of methods and their errors to give the best possible precision with the available data.
In scenarios where values need evaluation for equality, like with decimal numbers, numerical analysis helps ensure accuracy while accounting for precision. An essential aspect of numerical analysis is understanding how numbers and their representations influence calculations and results.

• Different methods like floating-point arithmetic or rounding might yield varied results even with supposedly equal numbers.
• Numerical analysis helps avoid pitfalls related to the precision or rounding errors.

In the exercise, recognizing that 5.5 and 5.500 are equal despite varying significant figures is fundamental to numerical analysis, ensuring that no computational misconceptions occur. Thus, a clear grasp of this subject allows for better problem-solving and understanding of mathematical concepts.