Hello, I'm Kimi, your go-to expert for mathematical concepts and much more. I'm here to provide you with clear and concise explanations, especially when it comes to the fascinating world of numbers. Let's dive right into the topic at hand.

### Rational Numbers: A Deep Dive

Rational numbers are a subset of the real numbers that include all numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where \( p \) is the numerator and \( q \) is the denominator. It's important to note that \( q \) cannot be zero, as division by zero is undefined. Rational numbers are characterized by their ability to be written as a ratio of two integers.

The set of integers includes all whole numbers and their negative counterparts. This set is represented as \( \{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\} \). Integers are a special type of rational number because they can be expressed as a ratio where the denominator is one. For example, the integer \( -3 \) can be written as \( \frac{-3}{1} \), which clearly demonstrates that it is a rational number.

### Negative Numbers and Rationality

When we talk about negative numbers, we are referring to numbers that are less than zero. Negative numbers can also be rational. In fact, any integer, whether positive or negative, is inherently a rational number because it can be expressed in the form of \( \frac{n}{1} \), where \( n \) is the integer itself.

Let's consider the number -3, which is the focus of your question. As mentioned earlier, -3 can be represented as \( \frac{-3}{1} \). This representation satisfies the definition of a rational number because it is the quotient of two integers. The numerator is -3, and the denominator is 1, both of which are integers.

### Misconceptions and Clarifications

It seems there might be a misunderstanding in the statement you provided: "Integers include all whole numbers and their negative counterpart e.g. ---4, -3, -2, -1, 0,1, 2, 3, 4,-- is a rational number but not an integer." This statement is partially incorrect. All integers, including negative ones, are indeed rational numbers. The error lies in the assertion that a rational number is not an integer. In fact, every integer is a rational number, but not every rational number is an integer.

### Conclusion

To answer your question directly: Yes, negative 3 is a rational number. It fits the criteria for being a rational number as it can be expressed as a fraction where both the numerator and the denominator are integers. The misconception that a rational number cannot be an integer is not accurate. Integers are a subset of rational numbers, and they include both positive and negative whole numbers.

Now, let's proceed with the translation as requested.

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Integers include all whole numbers and their negative counterpart e.g. ---4, -3, -2, -1, 0,1, 2, 3, 4,-- is a rational number but not an integer. All rational numbers belong to the real numbers.read more >>