How is the opposite of -2.5 different from the absolute value of -2.5?
Understand the Problem
The question asks to compare and contrast the concept of the 'opposite' of a number (specifically -2.5) with the 'absolute value' of the same number. The task involves understanding the definitions of these two concepts and explaining how they differ in the context of the given number.
Answer
The opposite of $-2.5$ is $2.5$, and the absolute value of $-2.5$ is $|-2.5| = 2.5$.
Answer for screen readers
The opposite of $-2.5$ is $2.5$, and the absolute value of $-2.5$ is $2.5$. Although they yield the same result for this particular negative number, they represent different concepts: the opposite is the additive inverse (changes the sign), while the absolute value is the distance from zero (always non-negative).
Steps to Solve
- Define the Opposite of a Number
The opposite of a number is the number that, when added to the original number, results in zero. It's also known as the additive inverse. For any number $x$, its opposite is $-x$.
- Find the Opposite of -2.5
To find the opposite of $-2.5$, we simply change its sign. Therefore, the opposite of $-2.5$ is $-(-2.5) = 2.5$.
- Define the Absolute Value of a Number
The absolute value of a number is its distance from zero on the number line. The absolute value is always non-negative. For any number $x$, the absolute value is denoted as $|x|$. If $x$ is positive or zero, $|x| = x$. If $x$ is negative, $|x| = -x$.
- Find the Absolute Value of -2.5
Since $-2.5$ is a negative number, its absolute value is the opposite of $-2.5$, which we already found to be $2.5$. Therefore, $|-2.5| = -(-2.5) = 2.5$.
- Compare and Contrast
In this particular case, the opposite of $-2.5$ is $2.5$, and the absolute value of $-2.5$ is also $2.5$. So, in this specific instance, the opposite and absolute value are the same. However, they are distinct concepts. The opposite changes the sign of a number while the absolute value provides the magnitude (distance from zero) and is always non-negative.
The opposite of $-2.5$ is $2.5$, and the absolute value of $-2.5$ is $2.5$. Although they yield the same result for this particular negative number, they represent different concepts: the opposite is the additive inverse (changes the sign), while the absolute value is the distance from zero (always non-negative).
More Information
The absolute value of a number is always non-negative. The opposite of a number can be positive, negative, or zero depending on the original number. If the original number is positive, its opposite will be negative, and its absolute value will be positive. If the original number is negative, its opposite will be positive, and its absolute value will also be positive. The only number whose opposite and absolute value are the same is zero.
Tips
A common mistake is to assume that the opposite and absolute value are always the same. This is only true for zero and negative numbers when finding the absolute value. For positive numbers, the opposite is negative, while the absolute value remains positive.
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