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Chapter 3: Problem 56

True or False? \(2.4 \mathrm{~g}=2.04 \mathrm{~g}\). ______

Short Answer

Expert verified
False. \(2.4\, \text{g} \neq 2.04\, \text{g}\).

Step by step solution

01

Identify the Numbers

Look at the given expressions: \(2.4\, \text{g}\) and \(2.04\, \text{g}\). The first number is \(2.4\) grams, and the second number is \(2.04\) grams. Notice the different digit arrangements after the decimal point.
02

Compare Place Values

Examine the digits after the decimal point. In \(2.4\), there is one digit '4' in the tenths place, making it \(2.4\, \text{g}\) equivalent to \(2.40\, \text{g}\). In contrast, \(2.04\) has '0' in the tenths place and '4' in the hundredths place.
03

Conclusion

Since \(2.4\) corresponds to \(2.40\), showing only one digit '4' at the tenths place, and \(2.04\) has a '0' with a '4' in the hundredths place, the values are not equal. Thus, \(2.4\, \text{g} eq 2.04\, \text{g}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Place Value
Understanding decimal place value is crucial when comparing numbers like 2.4 and 2.04. Each digit in a decimal number has its own place and significance.

For instance:
  • The first position to the right of the decimal point is the tenths place.
  • The second position is the hundredths place.
  • The third position, if needed, is the thousandths place.
To interpret 2.4, we recognize the "4" is in the tenths place, which equals 0.4. However, for 2.04, the "0" immediately after the decimal point means zero tenths, and the "4" is in the hundredths place, which is 0.04.

Thus, each number uses its decimal position to indicate its size and value.
Numerical Equality
Numerical equality refers to different expressions representing the same value. When comparing two decimal numbers such as 2.4 and 2.04, it's essential to analyze their representation to ascertain if they equate to the same numerical value.

Let's break down each number:
  • 2.4 can also be written as 2.40 because adding a zero does not change the value.
  • 2.04 consists of two significant digits post the decimal.
Even though trailing zeros like in 2.40 do not change the value, zeros between significant digits have value implications. Here 2.4 and 2.04 express different magnitude, thus they are not numerically equal.
Mathematical Reasoning
Mathematical reasoning is the thought process required to derive conclusions by using logical steps. Let's apply reasoning to the comparison of 2.4 and 2.04.

Start by:
  • Identifying each decimal's components, translating the structure and value each digit holds.
  • Recognizing that although both numbers share similar digits, they lie in different decimal places, influencing their overall value.
Finally, deduce: since the tens and tenths structure for 2.4 equals 2.40, and differs from 2.04's hundredths, proper reasoning reveals that these numbers are dissimilar in value. Logical breakdown provides this clarity on the difference in their numerical value.

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