mathematics

Positive and negative parts

Positive and negative parts is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Positive and negative parts rather than just read about it. In short: In mathematics, the positive part of a real or extended real-valued function is defined by the formula f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise. {\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\text{ if }}f(x)>0\\0&{\text{ otherwise.}}\end{cases}}} Intuitively, the graph of f + {\displaystyle f^{+}} is obtained by taking the graph of f {\displaystyle f} , 'chopping off' the par…

Key takeaways

Positive and negative parts belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Positive and negative parts to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Positive and negative parts from memory before moving on to harder problems.

Reference excerpt

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

f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise. {\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\text{ if }}f(x)>0\\0&{\text{ otherwise.}}\end{cases}}}

Intuitively, the graph of f + {\displaystyle f^{+}} is obtained by taking the graph of f {\displaystyle f} , 'chopping off' the part under the x-axis, and letting f + {\displaystyle f^{+}} take the value zero there. Similarly, the negative part of f is defined as

f − ( x ) = max ( − f ( x ) , 0 ) = − min ( f ( x ) , 0 ) = { − f ( x ) if f ( x ) < 0 0 otherwise {\displaystyle f^{-}(x)=\max(-f(x),0)=-\min(f(x),0)={\begin{cases}-f(x)&{\text{ if }}f(x)<0\\0&{\text{ otherwise}}\end{cases}}}

Note that both f+ and f− are non-negative functions. A peculiarity of terminology is that the 'negative part' is not negative (like the imaginary part of a complex number is not imaginary). The function f can be expressed in terms of f+ and f− as

f = f + − f − . {\displaystyle f=f^{+}-f^{-}.}

Also note that

| f | = f + + f − . {\displaystyle |f|=f^{+}+f^{-}.}

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

f + = | f | + f 2 f − = | f | − f 2 . {\displaystyle {\begin{aligned}f^{+}&={\frac {|f|+f}{2}}\\f^{-}&={\frac {|f|-f}{2}}.\end{aligned}}}

Another representation, using the Iverson bracket is

f + = [ f > 0 ] f f − = − [ f < 0 ] f . {\displaystyle {\begin{aligned}f^{+}&=[f>0]f\\f^{-}&=-[f<0]f.\end{aligned}}}

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.

Measure-theoretic properties Given a measurable space (X, Σ), an extended real-valued function f is measurable if and only if its positive and negative parts are. Therefore, if such a function f is measurable, so is its absolute value |f|, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking f as

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Illustrations

Worked examples

Example 1 — a first encounter with Positive and negative parts

Start with the simplest possible case. Write down what Positive and negative parts claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Positive and negative parts before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Positive and negative parts ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Positive and negative parts

In research: Positive and negative parts appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Positive and negative parts in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Positive and negative parts is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary mathematics, so understanding it makes those chapters shorter.
In everyday life: Look for Positive and negative parts outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Positive and negative parts in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Positive and negative parts means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Positive and negative parts out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Positive and negative parts in simple terms?

In mathematics, the positive part of a real or extended real-valued function is defined by the formula f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise. {\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\text{ if }}f(x)>0\\0&{\text{ otherwise.}}\end{cases}}} Intuitively…

Why does Positive and negative parts matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Positive and negative parts?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Positive and negative parts.