3.1 Integration: Technical Terminology
In this second activity, your task is to choose the better description for each term by selecting one of two provided statements. As with the previous activity, ensure that you are familiar with the pronunciation and the spelling of each term. This information can be found for all terms in this activity in the Glossary, Section 3.7.
The aim of this activity is to extend your familiarity with the field of integration by introducing an additional set of key terms.
Language Focus
In several of the examples above, we have labelled the components of the string as follows: ‘secondary adjective, primary adjective, noun‘. Examples of these are proper rational function and improper rational function. Why have the two adjectives been distinguished in this way?
In phrases like proper rational function, the adjectives are not independent but form a hierarchical relationship:
- Primary adjective: Defines the core category (e.g., rational: a ratio of polynomials).
- Secondary adjective: Refines that category (e.g., proper: numerator degree is less than denominator degree).
You cannot reverse the order because the meaning depends on this hierarchy. For example:
- ✅Proper rational function A rational function that is proper.
- ❌ Rational proper function: Nonsense, since proper function is not a core category.
Mathematical terms often build meaning like a pyramid:
[Function]
↓
[Rational Function] ← Primary Adjective
↓
[Proper Rational Function] ← Secondary Adjective
Reversing adjectives breaks this logical structure.
Compare this with everyday adjectives. Often, in non-mathematical phrases like a thin rough carpet, adjectives are independent properties and order doesn’t matter:
- ✅ a thin rough carpet
- ✅ a rough thin carpet
Both mean the same because white and happy describe separate traits. But this is not always the case with non-mathematical phrases. Consider, for example, the phrase a little red car. In this case, the adjectives describe separate traits, but conventionally they follow a certain order:
- ✅ a little red car
- ❌ a red little car
Learn more about the order of everyday adjectives here.
Note that there are also cases in mathematics where the order of the adjectives doesn’t matter. For example:
- ✅ a positive even integer
- ✅ an even positive integer
Both refer to the same set of integers, because positive and even are independent characteristics of integers.
In the next section, you will shift your focus from individual terms to how these terms are used in complete sentences and paragraphs. You will see how mathematical terminology combines with other words to create meaning in authentic mathematical writing. This will help you understand how mathematicians actually discuss and explain integration in textbooks, lecture notes, and academic papers.
