What is the mathematical definition of "minimum" when describing a real number?

Question

In mathematics, is the word "minimum", when used to describe a real number, conventionally taken with respect to magnitude only or magnitude and sign?

For example, a question asks for the "minimum displacement" of a particle over a certain time period. In this context, would $-100$ be "smaller" than $0.01$?

Answer 1

First of all, this is mathematics, so we don't really know or care what "particles" are. A joke, of course, but still keep in mind that mathematics works with mathematical definitions, so anything that is not strictly mathematically defined cannot really be talked about within the scope of mathematics. That said, let's move to the actual answer.

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$-100$ is not just "smaller" than $0.01$, it is also smaller than $0.01$ in the sense that the inequality

$$-100 < 0.01$$

is true.

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Some more context:

In mathematics, by definition, if you have any partial order $\leq$ (i.e. an antisymmetric, reflexive, transitive relation) defined on any set $A$, then the minimum of $A$ is the element $m\in A$ (if it exists) which satisfies the following condition:

$$\forall a\in A: m\leq a$$

Answer 2

• there is a thing called direction of movement.Let's say rightward movement is positive and leftward movement is negative.Then,in this context ,$-100$ means moving to left $100$ unit and $0.01$ means moving to right 0.01 unit.So,in this sense smaller displacement is $0.01$.
• but if you talk about numbers, then $-100~\lt~0.01$

Answer 3

For real numbers minimum refers to the combination of magnitude and sign. However...

Displacement is a vector. In general there is no ordering defined for vectors. In the special case of 1-d motion where a positive direction has been defined we can use the real number model. However 1m West and 1m East cannot be compared in this way without a positive direction.

In a more general vector setting "minimum" can only refer to magnitude, and the minimum possible magnitude is zero. The expression "minimum displacement" would be incorrect, but would usually be understood to mean "minimum distance".