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:
“Whoever influences the child’s life ought to try to give him a positive view of himself and of his world. The child’s future happiness and his ability to cope with life and relate to others will depend on it.”
—Bruno Bettelheim (20th century)
“The negative always wins at last, but I like it none the better for that.”
—Mason Cooley (b. 1927)
“I’m an expert in hookers. I’m an expert in doormats. I’m an expert in victims. They were the best parts. And when I woke up—sociologically, politically, and creatively—I could no longer take those parts and look in the mirror.”
—Shirley MacLaine (b. 1934)