Introduction to Interval Notation
Interval notation is a mathematical shorthand used to represent subsets of real numbers, particularly those that satisfy an inequality. Instead of writing 'all numbers x such that x is greater than 3 and less than or equal to 7,' it offers a more compact and standardized form using parentheses and brackets. This notation is widely adopted in algebra, calculus, and other areas of mathematics for its clarity and efficiency in expressing ranges of values.
Key Symbols and Their Meanings
The notation primarily uses two types of symbols: parentheses () and square brackets []. A parenthesis '(' or ')' indicates that the endpoint is *not* included in the set (an 'open' interval). For example, (3, 7) means all numbers between 3 and 7, but not including 3 or 7. A square bracket '[' or ']' indicates that the endpoint *is* included in the set (a 'closed' interval). Thus, [3, 7] means all numbers between 3 and 7, including 3 and 7.
Representing Different Types of Intervals
Various types of intervals can be expressed. An open interval like (a, b) includes numbers strictly between 'a' and 'b'. A closed interval [a, b] includes 'a', 'b', and all numbers in between. Half-open or half-closed intervals combine these, such as (a, b] (not including 'a', but including 'b') or [a, b) (including 'a', but not 'b'). Infinity symbols ∞ and -∞ are always paired with parentheses, as infinity is not a number that can be included. For instance, (3, ∞) means all numbers greater than 3.
Applications in Mathematics
Interval notation is crucial for describing the domain and range of functions, specifying solution sets for inequalities, and defining continuity or differentiability over certain regions. It streamlines the communication of complex numerical ranges, making it easier for students and professionals to understand and work with mathematical expressions. Mastery of this notation is fundamental for advanced studies in analysis and other quantitative fields.