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Understanding factorial notation is crucial when dealing with mathematical expressions involving factorials. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers from 1 to a number 'n'. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

It's important to note that the factorial of 0 is defined to be 1, or \(0! = 1\). This becomes particularly useful when simplifying expressions involving several factorials. As factorials can quickly become very large numbers, understanding how to manipulate these expressions without having to calculate the large values themselves is a valuable mathematical skill.

Simplifying factorial expressions is a method that reduces the complexity of the calculation by identifying and canceling out common factors. When you have a division of two factorials, as in the exercise \(\frac{100!}{98!}\), you can recognize that \(98!\) is a part of the \(100!\) sequence.

Thus, \(100!\) can be rewritten as \(100 \times 99 \times 98!\). This step is beneficial as the \(98!\) in the numerator and denominator cancel each other out, leaving a much simpler expression. The key to simplifying factorial expressions is to break down the larger factorial until it matches the smaller one, allowing for a cancellation of terms.

Evaluating numerical expressions involves performing the arithmetic operations and simplifying the expression to find its value. Once we've simplified the factorial expression, as shown in 'Simplifying Factorials', we can then perform the remaining mathematical operations.

In the given exercise, after cancelling out common factorials, we're left with \(\frac{100 \times 99}{2!}\). With \(2! = 2 \times 1\), the equation simplifies to \(\frac{100 \times 99}{2}\). The last step is to conduct the multiplication and division, yielding the final result. Consistently applying these skills in numerical expression evaluation is essential, as it helps unravel complex-looking problems into simple, solvable calculations.