As a domain expert in mathematics, I would like to clarify the concept of absolute value and its application to the number zero.

The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction. It is a non-negative number that represents the magnitude of a number without considering its sign. The concept of absolute value is crucial in various mathematical operations and functions, especially when dealing with inequalities and functions that require non-negative inputs.

When we talk about the absolute value of zero, it is important to note that zero is a unique number in the context of absolute value. It is neither positive nor negative; it is neutral. The absolute value of zero is defined as:

\[ \left| 0 \right| = 0 \]

This is because zero is already at a distance of zero from itself on the number line. There is no need to move it in any direction to reach zero, as it is already there.

The statement "Absolute value makes the expression non-negative, but not everywhere positive" is correct. This is because the absolute value function takes any real number and maps it to a non-negative result. For positive numbers, the absolute value is the number itself. For negative numbers, the absolute value is the positive counterpart of the number. However, for zero, the absolute value is simply zero.

The law of trichotomy, as mentioned, states that every real number is either positive, negative, or zero. This law is fundamental in understanding the classification of real numbers and their behavior under mathematical operations. In the context of absolute value, it reinforces the idea that the absolute value of a number will always be non-negative, aligning with the trichotomy of real numbers.

In conclusion, the absolute value of zero is a straightforward concept that is fundamental to understanding the properties of absolute value as a mathematical function. It is a basic yet essential piece of knowledge that is applied in various areas of mathematics, from algebra to calculus and beyond.

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Absolute value makes the expression non-negative, but not everywhere positive. No, the absolute value of zero is zero. For real numbers, the law of trichotomy states that every real number is either positive, negative, or zero. 0.Dec 28, 2015read more >>