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Numerical expressions are mathematical sentences involving numbers and operation symbols but without an equality sign. These expressions can be as simple as a single number, or as complex as a combination of addition, subtraction, multiplication, division, and even exponents.

For example, \( \frac{36}{-4} \) is a numerical expression. It consists of two integers and a division operation. Understanding how to evaluate such expressions is important because it forms the foundation of algebra.

To evaluate a numerical expression, you follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). However, when an expression consists solely of a division, like in our example, you simply perform that one operation. The aim is to simplify the expression to find its value, which, in this case, is -9.

Negative numbers represent values that are less than zero. They are an essential part of the number system and allow us to express quantities like debt or temperatures below freezing. In numerical expressions, the rules for operations with negative numbers are crucial for proper evaluation.

When dividing integers, if the numerator and denominator have the same sign, the result is a positive number. Conversely, if they have different signs, the result is a negative number—this is why \( \frac{36}{-4} \) equals -9. The positive numerator 36 divided by the negative denominator -4 gives us a negative result.

It's important to keep track of the signs, as neglecting them can lead to incorrect answers. A helpful tip is that whenever you divide or multiply even numbers of negative signs, the result is positive, and with an odd number, it's negative.

Rational numbers are numbers that can be expressed as a fraction where both the numerator (top number) and the denominator (bottom number) are integers, and the denominator is not zero. They include positive numbers, negative numbers, and zero.

A rational number represents a ratio or a division of two integers. For instance, \( -9 \) is a rational number as it can be written as \( \frac{36}{-4} \) or \( \frac{-36}{4} \). Rational numbers can be in the form of proper fractions, improper fractions, or integers. They can be found on the number line, and they can also be represented in decimal form, which can either terminate or repeat periodically.

An essential aspect of rational numbers is that they allow for parts of a whole to be expressed and manipulated in a mathematical form, broadening the spectrum of solvable problems beyond just integer arithmetic.