Positive and negative parts

 ( Elementary Mathematics ) 

In mathematics, the positive part of a real number or extended real-valued function is defined by the formula

Intuitively, the graph of $f^+$ is obtained by taking the graph of $f$, 'chopping off' the part under the -axis, and letting $f^+$ take the value zero there.

Similarly, the negative part of is defined as

Note that both and 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 can be expressed in terms of and as $$f\; =\; f^+\; -\; f^-.$$

Also note that $$|f|\; =\; f^+\; +\; f^-.$$

Using these two equations one may express the positive and negative parts as

Another representation, using the Iverson bracket is $$\backslash begin\{align\}$$

f^+ &=  [f>0]f \\
f^- &= -[f<0]f.
     

\end{align}

One may define the positive and negative part of any function with values in a linearly ordered group.

The unit ramp function is the positive part of the identity function.

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Measure-theoretic properties Given a sigma-algebra , an extended real-valued function is measurable if and only if its positive and negative parts are. Therefore, if such a function is measurable, so is its absolute value , being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking as $$f\; =\; 1\_V\; -\; \backslash frac\{1\}\{2\},$$ where is a Vitali set, it is clear that is not measurable, but its absolute value is, being a constant function.

The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.

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See also

• Rectifier (neural networks)
• Even and odd functions
• Real and imaginary parts

• Frank Jones (2025). 9780763717087, Jones and Bartlett. ISBN 9780763717087
• (2025). 9789810241919, Singapore; River Edge, NJ: World Scientific. ISBN 9789810241919
• Inder K Rana (2025). 9780821829745, American Mathematical Society. ISBN 9780821829745

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External links

• Positive part on MathWorld

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