I am not a native speaker, but for me "the minimum angle" and "the optimal solution" sound correct, but only because I hear and read them more often.

Why are "the minimal angle" and "the optimum solution" not used as often or wrong? Are there rules?

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    I'm guilty of not checking a dictionary, but I'd suggest that 'optimal' and 'minimal' are expressions of degree, requiring some qualification, Whereas 'minimum' and 'optimum' are definitive (actual) values. An 'optimal' solution may be the closest to the ideal 'optimum' as can be achieved. The financial impact may be 'minimal', but may not quite be the 'minimum' desired.
    – JHCL
    Commented Oct 1, 2015 at 15:09
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    (It's the difference between 'subjective' and 'objective', in other words)
    – JHCL
    Commented Oct 1, 2015 at 15:30
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    If you asked this at math.SE (which may not be the right audience), they would say that -um means the single extreme value and -al means a value that cannot be made better. 'Minimum' means the lowest of all, 'minimal' means can't get lower (from that point) there may be a better solution but you can't get there from here.
    – Mitch
    Commented Oct 1, 2015 at 15:46

8 Answers 8


There's a valid reason for this, which is linked to the fact that both mathematics (as a field) and minimum (as a word) are defined by quantities. Per Oxford Dictionary, minimum: (Note the use of "quantity").

The least or smallest amount or quantity possible, attainable, or required.

As defined by the Oxford Dictionary, minimum is a quantitative representation of the smallest amount needed; thus, making it suitable for math and the term minimum angle.

That brings us to minimal, which can be both a qualitative and quantitative characteristic. Thus, depending on whether the user intends to use it qualitatively or quantitatively he or she may be correct. But regardless of the user' intent, using minimal in this sense is subject to misconception.

Per Merriam Webster...

minimal: barely adequate

Per Merriam Webster: (note the use of "quality")

adequate: good enough : of a quality that is good or acceptable

Therefore, minimal is a qualitative characteristic, which contradicts the quantitative nature of math.

To conclude, I admit that minimal can in some instances be synonymous with minimum; however if you consider the dilemma people will encounter (is minimal qualitative or quantitative?), you'll understand why minimum (only quantitative) better represents mathematics. This explains why many have chosen to use minimal angle rather than minimum angle; however, minimum is more technical.

  • +1 for pointing out that minimal has a connotation of good enough.
    – Ooker
    Commented Jun 26, 2018 at 12:46

Minimum is fairly absolute and solid, and refers to the smallest number or amount possible.

Minimum: the least or smallest amount or quantity possible, attainable, or required.

Minimal is a little more flexible, where it refers to being the smallest amount or degree in non-absolute terms.

Minimal: of a minimum amount, quantity, or degree.

"Of a minimum amount", without specifying the minimum amount.

An example of both might be:

Jane has minimal interest in going to the movies; mostly because it seems the ticket prices rise at a minimum of $1.50 every time.

I keep the amount of furniture in my room at a minimum because I like my room to be minimal. I'm a minimalistic person.


According to Dictionary.com

Minimal is an adjective and Minimum is a noun. They have different meanings and purposes.


[min-uh-muh l]


1. constituting a minimum:

a minimal mode of transportation.

2. barely adequate or the least possible:

minimal care.


[min-uh-muh m]

noun, plural minimums, minima [min-uh-muh] (Show IPA)

1. the least quantity or amount possible, assignable, allowable, or the like.

2. the lowest speed permitted on a highway.

3. the lowest amount, value, or degree attained or recorded.

4. an arbitrary amount set by a restaurant, nightclub, etc., as the least amount to be charged each person for food and drink. Compare cover charge.

5. Mathematics. Also called relative minimum, local minimum. the value of a function at a certain point in its domain, which is less than or equal to the values at all other points in the immediate vicinity of the point. Compare absolute minimum. the point in the domain at which a minimum occurs.

The word Minimal comes from the word minimum consisting of synonyms nominal and minimum.

The word Minimum originates from the Latin word minimus meaning smallest or least. Minimum's synonyms consist of minimal and merest.

Based on your sentences above, The correct way of using the words minimum and minimal would be as follows:

The minimal angle-correct because it's being used as an adjective

The minimum angle-incorrect usage using a noun as an adjective

(The same would follow for optimum/optimal)

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    Thanks! If this is true, why does google suggest "minimum angle" if I search for "minimal angle"? "minimal angle" has only almost half of the hits as "minimum angle". It seems "minimum angle" is used more often. Any explanation for that?
    – Rob
    Commented Oct 2, 2015 at 9:17
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    I can explain it if you tell me where you got that information. Google answers are asked and answered by people like you. The one who doesn't quite know the more proper way to put it. I suggest avoiding sites using google or just answers answered by google. So yes, the more proper way to say that would be my above answer.
    – anonymous
    Commented Oct 2, 2015 at 14:42
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    On google! I mean just the number of search results for both terms. Not a forum or whatever. Google for "minimum angle". You get around 186.000.000 hits. Then google for "minimal angle" --> just 114.000.000 hits and you are asked if you meant "minimum angle" instead.
    – Rob
    Commented Oct 5, 2015 at 9:02
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    @anonymous - According to dictionary.com, the very reference you have used, minimum can be either a noun or an adjective. Your distinction between minimum and minimal is therefore incorrect, and minimum angle is perfectly grammatically acceptable.
    – AndyT
    Commented Oct 7, 2015 at 11:30
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    As AndyT points out, if you scan past the noun definitions of minimum on Dictionary.com, you'll see that minimum can also be an adjective. In fact, Dictionary.com gives four definitions of minimum as an adjective. The split between minimal and minimum is certainly not as basic as one is a noun and the other is an adjective, as you seem to suggest.
    – Sven Yargs
    Commented Oct 10, 2015 at 1:49

In mathematics, in particular in combinatorics, these words have a very specific contrastive usage as adjectives.

  • A minimal solution to a problem can't be made any smaller by "shrinking" it. If we shrink it, it is no longer a solution. There may be other, distinct smaller solutions though.

  • A minimum solution to the problem has the smallest possible size among all solutions. No smaller solutions exist.

This is a very specific, technical usage in certain branches of mathematics. It does not apply to the everyday use of these words.

For example, take these numbers connected with arrows (i.e. a "graph", another technical term):

enter image description here

There are ways to go around in a closed cycle by following the arrows, e.g. 4 -> 7 -> 9 -> 10 -> 5 -> 4:

enter image description here

Which arrows do we need to remove so that there are no such cycles left?

For example, we could remove 6 -> 2, 10 -> 5, 4 -> 7 to break all cycles. This is a minimal solution because not removing any of these three would leave some cycles intact. Thus the solution can't be made smaller. However, it is not a minimum solution because smaller solutions exist. 5 -> 4, 6 -> 2 would be a minimum solution.

In general, all minimum solutions are also minimal, but the converse is not true.

The problem I described above is called the Feedback Arc Set Problem. You will find several such usages of the words minimal and minimum on its Wikipedia page.


In mathematical terms generally minimum means the lowest possible, it is unique. Minimal on the other hand can be sub-optimal.There can be several minimal solutions but only one minimum solution.

  • 1
    I disagree. Minimal can mean minimum. See Merriam Webster which has as a definition for minimal: "relating to or being a minimum: as the least possible". Minimal does not have to mean sub-optimal.
    – AndyT
    Commented Oct 12, 2015 at 15:23
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    Yeah I meant it need not be optimal Commented Oct 12, 2015 at 15:24
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    Why the downvote? This is absolutely correct for mathematics. Commented Oct 12, 2015 at 15:39
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    I edited it @PeterShor previously it was not complete. Commented Oct 13, 2015 at 15:49
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    This analysis in the context of mathematics is wrong. In a set with an order, this is a relation that allows to say between pairs of elements that one is not larger than the other (although not necessarily for every two pairs), a minimum (which turns out to be unique but doesn't necessarily exist) is an element that is comparable to all elements and in that comparison is not larger than any other element. A minimal element is any element for which all other elements are either incomparable or larger.
    – mama
    Commented Feb 2, 2019 at 18:00

To me, minimum indicates a single, smallest value, where minimal indicates a range of values that are approaching the minimum (and may in fact be the minimum value).

In the same vein, optimum is a single best solution/value, where optimal is a set of solutions/values approaching and including the optimum solution.


Minimum tends to be a definite thing, such as 'the minimum number of items you can buy is 7'

Minimal tends to be be an idealogical thing, such as 'the war was over quickly, and there were a minimal number of casualties'. This means the person speaking thinks that the number of casualties was the lowest possible, or approximately the lowest.

If you said 'the war was over quickly, and there were the minimum number of casualties' then it would mean it was not possible in any circumstances for there to have been any less casualties. The person speaking knows for sure it was the minimum number.


I can't find a reference for this but my feeling is that minimum is often used in mathematical contexts while minimal has much wider usage. While it remains true that minimum is a noun, it can legitimately be used in adjectival contexts.

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