Examples of vocabulary that have different meanings in Mathematics compared to "everyday" English

I am asking this question here to obtain a nice reference list. I am asking for examples of words which have different connotations or meanings in Mathematics compared to English as taught in literature class.

Examples:

Even: can be written as 2n, where n is an integer

Even (if): make flat or smooth, or to show extreme degree

Series: the sum of the terms of an infinite sequence

Series: a set of related television or radio programs, especially of a specified kind

Edit: The purpose of this question is to help my teaching to low level ESL students; nothing to do with mathematics is a language debate. I am asking for examples to help me be more aware of my language when teaching AP Calculus and AP Statistics to low functional second language students. My students are at grade or above mathematical ability in their native language but most function between third to sixth grade (American grade levels) English ability.

• Don't almost all terms borrowed from a natural language have a technical difference in their mathematical definitions? (Prime, derivative, uniform, integral, divisible, space, etc; not to mention more modern terms like variety, sheaf,...) Then there's one (pronoun), which might be everyday English borrowing from mathematics, or not. Apr 6, 2016 at 10:52
• @MichaelE2 I teach ESL students and vocabulary issues are a huge stumbling block for my students. I feel a bit too close to the issue and it is really frustrating me, so I am asking for some outside examples, so I can be more mindful in my class, and to help prevent turning my class to an English corner because I used the phrase "even if" and the students are confused upon how that fits into a lesson on summations. I know there are specific vocabulary like "chord" that I must teach before the lesson. I am trying to prepare beforehand, so I can teach the students better.
– Papa
Apr 6, 2016 at 13:17
• Even: In Chinese culture where "even" sometime indicates "lucky", when people say a number is "even" in non-mathematical conversation, they mean an integer $n$ satisfies $n=m\cdot 10^x$ where $2|m$, $5\nmid m$, $m\in \mathbb{N}$, $x\in \mathbb{N}\ \cup \{0\}$. For example, 3600 is "even", 1700 is not.
– user2139
Apr 7, 2016 at 3:48
• This reminds me of an earlier answer I had for a different MESE question. Apr 8, 2016 at 15:25
• You might be interested in this book: cwru.edu/artsci/math/wells/pub/html/abouthbk.html Apr 10, 2016 at 17:23

This is a problem for some English language learners: The triangle on the left is also a right triangle.

"In general" or "generally"

In mathematics, if we say a specific result holds in general, we mean there are no exceptions to the result.

In every-day non mathematical discussions, if someone makes a claim and says it is true in general, they mean it is true most of the time but with possibly a few exceptional cases. Exactly the opposite of the mathematical meaning!

Mathematicians often use "generically" to mean essentially what nonmathematicians mean by "generally".

• To make matters more confusing, algebraic geometers often talk about objects in "general position", which has a similar meaning to "generically". Apr 15, 2016 at 15:53
• Haha, it gets more confusing when I am negating the word: 'generally not' could mean either never or seldom (depending on whether the mathematical or non-mathematical meaning is being understood), while 'not generally' could mean either "often" or "sometimes" (due to the same ambiguity). Feb 5, 2023 at 18:57

This can get quite some list…

Here are a few:

• function: a mapping in mathematics vs. a "feature" a "purpose" and "functionality" in everyday language

• root: e.g. a point where a function is zero vs. the vegetable or a source of something

• character: see here vs. "person"

• matrix: while this is probably not used very much in "everyday" English, this can have quite some different meanings (see here), especially the movie with the same name may not refer to the mathematical meaning but probably more to meaning in biology or geology.

• or: In everyday language this often means the exclusive or while is mathematics this is always non-exclusive.

• real: You know what a real number is, but real numbers are in no sense any more or less real that other numbers.

• complex, rational, natural: Same as above for these numbers.

• I'll add "or" in daily conversation often indicates both of them have some chance to happen. People rarely say $P$ or $Q$ when $P$ is almost always true and $Q$ is almost always false.
– user2139
Apr 7, 2016 at 3:51

The term "statistically significant" is significant in this regard.

In the previous sentence, the first example of "significant" means "unlikely to have occurred by chance", and the second denotes importance.

This could be a huge list but Il add one :

Volume. How loud something is or the number of cubes.

I think an interesting connected question is which maths words are used by the public in a mathematical sense but incorrectly?

• It might also be worth mentioning that "volume" may not actually refer to three dimensional volume. Particularly in upper division and graduate level work, "volume" may just mean the measure of a set (i.e. the "volume" of a $s\times r$ rectangle in the plane is $sr$; the volume of the four dimensional cube with sides of length 2 is $2^4$; etc). Feb 6, 2019 at 19:47

For many calculus theorems, you'll need to refer to open or closed sets. Those terms have nothing to do with the English meanings. I've found that even native English speakers can be confused because the math terms (unlike the English terms) aren't mutually exclusive, i.e. a set can be open, closed, open and closed or neither open nor closed where a door has to be one or the other.

• I'd say "nothing to do with the English meanings" is exaggerated. We not only apply "open/closed" to doors but also to questions, areas, relationships etc. where I interpret "open" as "not conclusive" or "not having a clear end/boundary". This fits well with the topological interpretation, since a point moving in an open subset of $\mathbb{R}^n$ cannot reach the boundary/end. Nov 5, 2020 at 8:53
1. Angle - Formed by 2 rays joined at a vertex, vs. a viewpoint, vs. tilt
2. Yard -A unit of measure equal to 3 feet vs the grassy area around a house
3. Constant - A number that doesn't change (not a variable) vs occurring continuously over a period of time vs. a situation that doesn't change
4. Digit - Any of the numerals from 0 through 9 vs. a finger or toe.
5. Gross - 12 dozen vs. yuck
6. Key or Legend - List of labels on a graph or items in a map vs. what you use to open the door or a great story.
7. Orientation - the angle of an object relative to the axes vs. training for new students vs. feelings and beliefs.
8. Power - the number of times a base is multiplied by itself vs. the capacity to direct others.
9. Similar - having the same shape but not the same size vs. resembling without being exactly the same.
10. Table - Mathematical information organized in columns and rows vs furniture with a flat top and legs.
• The mathematical meaning of all of these words can be traced to their everyday meaning. I doubt someone would be confused by their use in mathematics. Nov 5, 2020 at 9:00
• @MichaelBächtold having taught elementary students who knew the English words and not the mathematical meaning, I would have to disagree. I had to work on getting them to accept the idea that the mathematical word I was teaching was different from the word they were used to. Nov 7, 2020 at 19:24

The adjective simple is heavily overloaded (to use programming terminology) in mathematics.

A simple polygon is non-self-intersecting. It can be arbitrarily complicated, but still called "simple." Analogously, a simple geodesic is non-self-intersecting. So a closed geodesic is a geodesic that forms a simple, closed curve. A Jordan curve is a plane, simple, closed curve.

• A simple polytope in $\mathbb{R}^d$ has each vertex adjacent to $d$ faces. So in $d=3$, every vertex is trivalent.

• A simplicial polytope has every facet a simplex. So in $d=3$, this is a triangulated polyhedron.

• A simple group has no normal subgroups (except the trivial subgroup and the group itself).

• A simple graph has no loops or multiple edges.

• A simple root of a polynomial is one with multiplicity $1$.

• A simple pole is a pole of order $1$.

• A simple algebra is one with no nontrivial ideals.

One could easily extend this list...

• Great example, this is why I asked this question there are just so many that I would have overlooked.
– Papa
May 19, 2016 at 23:11

I'm going with this.

When dealing with teens, someone is bound to make a remark regarding asymptotes since its pronunciation starts with ass. This one, on the other hand, inevitably causes a bit of uproar. Clearly a different meaning than expected.

• "The rectum is the concluding part of the large intestine that terminates in the anus." :)
– JRN
Apr 9, 2016 at 0:33

This is a rather good collection. It is an essay by Reuben Hersh entitled Math Lingo vs. Plain English: Double Entendre.

• that is a very good paper, I was more looking to build a dictionary list of terms, which I could consult when planning lessons for teaching mathematics to ESL students.
– Papa
Apr 6, 2016 at 12:06
• Sorry I don't have any others. I figured I'd share the paper with you anyhow...as it really helped me understand the colossal difficulties my students face in trying to understand mathematical language. Apr 6, 2016 at 17:41
– J W
Apr 8, 2016 at 18:37
• I think J W was asking for a small excerpt or example. Apr 10, 2016 at 16:47
• @JoeTaxpayer: yes, I was requesting something along those lines, although the edit certainly improved the answer,
– J W
Apr 15, 2016 at 11:45

By sphere a mathematician means a two-dimensional surface. In everyday speech, a solid three-dimensional ball is often called a sphere. This difference sometimes causes confusion in introductory calculus classes.

By a manifold a mathematician means an abstract notion generalizing a two-dimensional surface. In everyday speech, a manifold means part of the intake or exhaust system on a car.

I'm quite surprised no one has mentioned groups, rings and fields, along with the other algebraic concepts like category and algebra.

"For an arbitrary x": in mathematics, usually "for all x";in English, usually "for a given x".

• I've never heard of 'arbitrary' to mean 'given' though. 'Arbitrary' and 'any' are synonymous in plain as well as mathematical English; when universally generalised, "arbitrary $x$" becomes "each/every $x$". Feb 5, 2023 at 21:43

Strategy -- in everyday usage it's a general plan of action to deal with uncertainty, whereas in game theory it's an integer sequence or an algorithm.

Neighborhood -- open ball vs. well, neighborhoods.

To say nothing of Group, Ring, Field...

If you're interested in how we connect everyday language to formal mathematical terms, I highly recommend Where Mathematics Comes From by Rafael Nunez and George Lakoff (2000) or even Nunez' "Mathematical Idea Analysis" plenary address from 1999.

Also, “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.”― Johann Wolfgang von Goethe

factor -- "this is a factor of the polynomial" versus "this is a factor that contributes to poverty"

The next few are more about how these are used by mathematicians (rather than having a technical meaning):

obvious -- "you can figure it out ten minutes and pencil" versus "everyone would immediately agree this is the case"

trivial -- see above

intuitive -- "if you've taken the standard math sequence, this should make sense" versus actual intuition

The word "multiply" in English almost always means "increase".

Many people believe that $$\frac{3}{7}\times9$$ must be bigger than $$9$$ because $$9$$ is getting multiplied.

The words rational and irrational mean very different things in math and in real life.

• But the difference in this case is so gargantous that is doesn't cause problems ... Nov 7, 2020 at 19:31

The word imply has significantly different meanings in ordinary speech and mathematics, and has caused me some confusion in the past.

According to this dictionary entry, the word indicates a suggestion (rather than a definite conclusion) that some statement is true. This gives rise to expressions that cover different degrees of strength in a connection between statements - 'weakly imply', 'imply', 'strongly imply' etc.

To my understanding, this usage of 'imply' only relates (in a way) to the likelihood of something. In stark contrast, the usual math definition is of a conclusive nature - A implies B means that whenever A is true, B is definitely true. No clauses.

In fact, the above dictionary entry goes on to state:

Imply and infer do not mean the same thing and should not be used interchangeably: see infer

...and the provided definition of 'infer' is somewhat closer to the mathematical sense in which 'imply' is normally used.

Here's an example of a Quora page (about correlation implying causation) in which the two meanings come into conflict. There's a wide variety of answers saying 'no', 'maybe' and 'yes', and I'm sure some of the misunderstanding stems from the dual meanings of 'imply'.

There are terms I have never liked because their meaning conflicts with what one would expect.

The one that comes most readily to mind is 'whole numbers'. It sounds to me like it should include the negative whole numbers (the ones that don't have fractions), but of course it doesn't. Integers means what I want whole numbers to mean.

With fractions, I often say top and bottom instead of numerator and denominator, though perhaps I shouldn't.

• I did not know "whole numbers" mean the natural numbers and not the integers. In Finnish, the literal translation "kokonaisluvut" certainly includes the negative ones. I do the same as you with fractions, at least to the extent that I never assume students know what a "denominator" is, for example. I try to use it and actively translate it into a more familiar term, so that they can learn. Jan 9, 2019 at 6:08
• So interesting to know that it works differently in Finnish. In English "whole numbers" is defined to be {0,1,2,3,...}. Jan 10, 2019 at 16:51
• German is the same as Finnish. We have natürliche (natural) numbers ([0], 1,2,3,...) and ganze (whole) numbers with negatives. Jan 11, 2019 at 13:20

Two things come in my mind: continuity and circles:

Continuity :
In normal English, a curve is continuous when you can draw it without lifting your pencil.
In mathematics, there's a whole definition with environments, basic environments, or epsilon-delta formulas to describe continuity, but no mathematics teacher has ever shown me the relation between those definitions and the fact that you don't need to lift your pencil while drawing the function.

Circles :
In normal English, a circle is the typical round thing, and yes, in mathematics, a circle, being defined as all the points, located at the same distance of one point (the centre) also reveals the typical round thing.
But you can also use different definitions of "distance", e.g. the so-called 1-distance (or Manhattan distance) :

D(point_1,point_2) = abs(X2-X1) + abs(Y2-Y1)


When drawing a circle, based on that distance, you end up with a square, which means that a square is a circle :-)

Here is an example that is simple, but that causes many problems in math classes: or.

Example: Do you prefer coffee or milk? In everyday English, the "or" is interpreted as to choose only one option: only coffee, only milk but not both. In Mathematics, the "or" allows both choices: only coffee, only milk, coffee and milk.

There are many examples from statistics, one already mentioned in other answer (significant versus statistically significant). Words from English in daily use, hijacked for a highly technical meaning:

sufficient statistic, efficient estimator, unbiased estimator (more troublesome since unbiased in daily use have value connotations), regression another case with a big divergence in meaning!, likelihood is a technical concept very distinct from probability while in daily use more or less synonymous with chance or probability, ...

In ordinary speech, the adjective random describes something that occurs without a definite pattern or predictability.

In mathematics, its meaning is, in a sense, inverted: random/stochastic processes do have underlying structures over time/space and can be used for forecasting.

Two aspects of randomness