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:
“I am positive I have a soul; nor can all the books with which materialists have pester’d the world ever convince me of the contrary.”
—Laurence Sterne (1713–1768)
“The idealist’s programme of political or economic reform may be impracticable, absurd, demonstrably ridiculous; but it can never be successfully opposed merely by pointing out that this is the case. A negative opposition cannot be wholly effectual: there must be a competing idealism; something must be offered that is not only less objectionable but more desirable.”
—Charles Horton Cooley (1864–1929)
“Forgetfulness is necessary to remembrance. Ideas are retained by renovation of that impression which time is always wearing away, and which new images are striving to obliterate. If useless thoughts could be expelled from the mind, all the valuable parts of our knowledge would more frequently recur, and every recurrence would reinstate them in their former place.”
—Samuel Johnson (1709–1784)