Positive And Negative Parts
In mathematics, the positive part of a real or extended real-valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of, chopping off the part under the x-axis, and letting take the value zero there.
Similarly, the negative part of f is defined as
Note that both f+ and f− are non-negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part).
The function f can be expressed in terms of f+ and f− as
Also note that
- .
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group.
Read more about Positive And Negative Parts: Measure-theoretic Properties
Famous quotes containing the words positive, negative and/or parts:
“The learner always begins by finding fault, but the scholar sees the positive merit in everything.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“Most literature on the culture of adolescence focuses on peer pressure as a negative force. Warnings about the “wrong crowd” read like tornado alerts in parent manuals. . . . It is a relative term that means different things in different places. In Fort Wayne, for example, the wrong crowd meant hanging out with liberal Democrats. In Connecticut, it meant kids who weren’t planning to get a Ph.D. from Yale.”
—Mary Kay Blakely (20th century)
“Beauty that shocks you, parts that none will trust,
Wit that can creep, and pride that licks the dust.”
—Alexander Pope (1688–1744)