decreasing monotonic
Definition
- Adjective: Consistently decreasing; describing a function or sequence where each successive value is less than or equal to the previous one. In mathematics, it is a property of a function whose value never increases as its input increases.
Usage
- As an adjective modifying a noun: The term is used to describe mathematical functions, sequences, or trends that show a steady decline.
- The economist studied a decreasing monotonic trend in unemployment figures over the quarter.
- In calculus, it is crucial to identify whether a function is decreasing monotonic to analyze its behavior.
Advanced Usage
- In formal mathematics: A function ( f ) is said to be decreasing monotonic (or monotonic decreasing) on an interval if for any ( x2 ) in that interval, ( x2 ) implies ( f(x2) ). If the inequality is strict ((>)), it is strictly decreasing monotonic.
- The proof required the lemma that the sequence was decreasing monotonic and bounded below.
Variants and Related Words
- Monotonically decreasing: An adverbial form often used interchangeably with "decreasing monotonic."
- The algorithm ensures the error converges monotonically decreasing to zero.
- Monotonic decreasing: A common alternative phrasing with the same meaning.
- Non-increasing: A synonym in many mathematical contexts, meaning the values do not increase (they can stay the same or decrease).
Synonyms
- Non-increasing: Not increasing; values remain constant or fall.
- Descending: Moving or extending downward (often used more generally).
- Declining: Becoming smaller, fewer, or less; decreasing.
Antonyms
- Increasing monotonic: Consistently increasing.
- Non-decreasing: Not decreasing; values remain constant or rise.
Related Phrases and Concepts
- Monotonic function: A broader term for a function that is either entirely non-increasing or non-decreasing.
- Monotonic sequence: A sequence of numbers that is either non-increasing or non-decreasing.
Adjective
-
consistently decreasing
-
a decreasing monotonic function
-
