Ideal vs. Optimum — What's the Difference?

This is an ideal opportunity to save money

The swimming pool is ideal for a quick dip

In an ideal world, we might have made a different decision

You're my ideal of how a man should be

The ideal of national unity.

The restaurant is considered the ideal in fine dining.

Optimum conditions for climbing

"those who regarded even a rhetorical commitment to the ideal of disarmament as irresponsibly utopian" (James Carroll).

An ideal work environment.

An ideal world where everything works out fine.

An ideal notion of how businesses are run.

An optimum return on capital

Optimal concentration of a drug

The ideal forms.

The ideal theory or philosophy

Ideal point

An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny - Carl Schurz

Let $\backslash mathbb\{Z\}$ be the ring of integers and let $2\backslash mathbb\{Z\}$ be its ideal of even integers. Then the quotient ring $\backslash mathbb\{Z\}\; /\; 2\backslash mathbb\{Z\}$ is a Boolean ring.

The product of two ideals $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is an ideal $\backslash mathfrak\{a\; b\}$ which is a subset of the intersection of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is a subset of $\backslash mathfrak\{a\; +\; b\}$.

Formally, an ideal $I$ of a given set $X$ is a nonempty subset of the powerset $\backslash mathcal\{P\}(X)$ such that: $(1)\backslash \; \backslash emptyset\; \backslash in\; I$, $(2)\backslash \; A\; \backslash in\; I\; \backslash and\; B\; \backslash subseteq\; A\backslash implies\; B\backslash in\; I$ and $(3)\backslash \; A,B\; \backslash in\; I\backslash implies\; A\backslash cup\; B\; \backslash in\; I$.

The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.

There will always be a wide interval between practical and ideal excellence.

The ideal is to be attained by selecting and assembling in one whole the beauties and perfections which are usually seen in different individuals, excluding everything defective or unseemly, so as to form a type or model of the species. Thus, the Apollo Belvedere is the ideal of the beauty and proportion of the human frame.

A poem or essay may be typical of its period in idea or ideal content