Introduction:
Regression analysis is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. In regression analysis, when we have a quantitative variable with m categories, we need to introduce dummy variables to represent these categories in the regression model.

Explanation:
To introduce dummy variables for a quantitative variable with m categories, we need to follow certain rules:

1. Number of Dummy Variables:
We need to introduce m - 1 dummy variables for a quantitative variable with m categories. The reason behind this is the concept of "dummy variable trap" or "multicollinearity".

2. Dummy Variable Trap:
The dummy variable trap occurs when we include a dummy variable for each category of a qualitative variable in the regression model. Including a dummy variable for each category can lead to perfect multicollinearity, where the independent variables are highly correlated. This can result in an unstable regression model with unreliable coefficient estimates.

3. Multicollinearity:
Multicollinearity refers to the situation where two or more independent variables in a regression model are highly correlated with each other. In the case of introducing a dummy variable for each category, one category becomes the reference or baseline category, and the remaining m - 1 categories are represented by dummy variables. As a result, the dummy variables are perfectly correlated with each other, leading to multicollinearity.

4. Baseline Category:
By introducing m - 1 dummy variables, we implicitly define one category as the baseline or reference category. The baseline category is the category for which the coefficients of dummy variables are compared and interpreted. The coefficients of the dummy variables represent the difference in the mean value of the dependent variable between each category and the baseline category.

Conclusion:
In regression analysis, when a quantitative variable has m categories, we introduce m - 1 dummy variables to avoid the dummy variable trap and multicollinearity. By doing so, we can accurately model the relationship between the dependent variable and the quantitative variable with multiple categories.