# Tag Info

38

"The whole numbers" is not a term that professional mathematicians use to describe a certain set of numbers. The term is used in elementary education when fractions are introduced, so that one can distinguish between numbers that have a fractional part and numbers that don't. In the US, this happens in grades 3 or 4. As far as I can tell from the ...

27

I'm no native English speaker, but you can tackle that question from the mathematical point of view as well. The best verb depends on how you view the nature of definite and indefinite integrals. Operators/Functionals Indefinite integrals are operators mapping functions to a set of functions or function (as representative of the equivalence class). ...

25

I’m more curious about incorrect things in them. Yet, this is the first thing I found. There's absolutely nothing "incorrect" about this. As Dave L Renfro noted in a comment: and whole number is rarely used as a precise designation outside of school mathematics there is no agreed-upon rigorous definition of the term, and in fact it's largely ...

24

Short Answer You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term should be introduced as such—emphasize that "real" in mathematics does not mean the same thing that "real" means in everyday vernacular English. ...

23

This is also borderline not-an-answer, but it might be a nice broadening of your students' worldview to know that the "$m$" and the "$b$" are not universally accepted. Showing them this map (even though I do not know its original source, so it may not be accurate in its details) could help their thinking out a bit: Substantial edit: I now no longer believe ...

20

Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes are basic to our understanding of everything. Where mathematics comes from, by George Lakoff and Rafael E. Núñez. Basic Books, 2000. Exploring the role of ...

20

I think, while teaching, the principal way to judge mathematical language is not whether it's standard, but whether it's effective communication. This difference applies principally to communication that's more substantive than "read an equation out loud" where there's only really one right way and not much opportunity for change - but, even on the small ...

19

This is "left involution". ("left" because it doesn't work when you try it on the right.) \begin{align*} x \circ y &= z & \\ x \circ (x\circ y) &= x \circ z & [\text{apply $x \circ -$}] \\ y &= x \circ z & [\text{simplify the involution}] \text{.} \end{align*} I would be shocked to see anyone use that term ...

19

I don't think that "textbooks" decided this, usage did. The term "integer" covers positive and negative, so it would be redundant for whole numbers to refer to that category. And there is an argument to be made for the term linguistically: a negative number is sort of the opposite of having a whole thing. But ultimately, there's not much ...

17

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental theorem of calculus. So the course includes limits, the definition of the derivative, techniques and applications of the derivative including trigonometric and ...

16

Doctor Peterson in The Math Forum refers to the following sources. From D. E. Smith, History Of Mathematics Vol II (1925), footnote page 393: The Arabs also used jidr (dyizr, root), whence the Latin radix. A longer quote from W. W. R. Ball, A Short Account of the History of Mathematics (4th edition, 1908): The algebra of Alkarismi holds a most ...

16

Elementary algebra is the study of a few specific rings: integers, rational numbers, polynomials with rational coefficients, and rational functions. All the rules for "rearranging" equations are applications of the ring or field axioms. Solving equations can be seen in the framework of algebraic or analytic geometry as studying the solution sets in some ...

16

It depends a little on context and how careful I am trying to be: If I were being very careful, I might call such a number "a negative number with a large absolute value" or "a negative number with large magnitude." If I am being a little less careful (which I often am when discussing asymptotic behaviour of, for example, rational functions), I'll generally ...

16

This is a case where you might be looking for a distinction that's pretty subtle. By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered. I'd be ok with a student's answer to "What is the y-intercept?" to be simply the y value, or the $(0,y_0)$ point. If a teacher prefers one, you can ask ...

15

I can think of a few instances where it might be useful: To situate the current piece of the concept among others coming up To call-back to something earlier in the course that really should be easy to them at this point To intentionally make fun of something you know they thought was stupid (e.g. high school? if you're teaching a proofs course) And, ...

15

The main point here is that these words/expressions should not be used as substitute for an argument. They obviously have some negative effects: You evaluate your students by them and not in the positive sense. If you say "obviously", then it has a message that "it better be obvious". And if it is for someone not clear, then it is an evaluation that this ...

14

I would avoid the verb solve as I reserve this for things like equations, inequalities and problems. An integral is equal to a number or a function, so verbs like find, evaluate etc are more appropriate. I'd use compute only for numerical integration methods. evaluate and find are the two verbs that are used in textbooks and exams that I've come across. I ...

14

Express $\cos(x + \pi)$ in terms of $\sin x$ or $\cos x$ (possibly both). In my opinion, the "or" is logically more correct and the parenthetical recalls/stresses/clarifies this. In general, I am in favor of redundancy and details for such things.

14

You ask "Why isn't the term inequation widely used in English?" The answer, however tautological, is that the term isn't used in English because it isn't used in English. Perhaps more usefully, remember that languages change and evolve as speakers of the language (for example) start using new words, stop using old words, alter pronunciation, and play with ...

14

The problem with the word "solution" is that it could mean the final answer or it could mean how the final answer was obtained, so I suggest that you don't use it. How about the following? Clearly show how you got your final answer if you want to get full credit for it. If the phrase "full credit" is not familiar to them, you may use "all the points" or ...

13

Origami has things like lemmas, theorems and corollaries. Definition/axiom: The basic folds such as book, mountain, valley etc Lemma: a folding procedure that is used as part of another, such as the bird base or the water bomb base. Theorem: a folding procedure that produces something you wanted, such as the traditional crane. Corollary: a folding ...

13

One of the most colorful names I have heard is the Chicken Mc Nugget theorem: for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$. link1, link2. From the links: The story goes that the Chicken McNugget Theorem got its name because in McDonalds, ...

13

(From a comment) On American math exams, I see the phrase "justify your answers".

13

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions ...

12

In calculus one calls $x \to ax + b$ a linear function. In linear algebra one calls $x \to ax + b$ an affine transformation, and says that it is linear only if $b = 0$. The first order Taylor approximation to a differentiable map is reasonably called the linear approximation of the map. The terminology is reasonable because one is doing function theory, ...

11

The main reasons both of these are called the same thing are historical. Group theory was more or less invented by Galois to study when one could solve polynomial equations by radicals, and ring theory was more or less invented by Hilbert to solve the main problems of invariant theory, which aimed to find ways to discover when two systems of equations were "...

11

For things in a list, in my experience it is very uncommon to have things indexed by letters other than m, n, i, j, k and sometimes r. Letters whose names end in a vowel sound (like i, j, k) and letters whose names end in a glide (like m, n, r) offer most people no particular problems with pronouncing the "-th" construction. This even goes for unusual ones ...

11

From cooking: Theorem: recipe, something that produces a desired result Lemma: a technique (e.g. creating a roux, proofing dough, preparing a mirepoix, etc.), a procedure that one uses in many different recipes, but might not produce something interesting/tasty by itself Corollary: (really stretching here) extra garnish/styling to slightly modify the dish ...

11

In Central Mexico, the expression \begin{equation} x_{\pm} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{equation} that solves quadratic equations of the form $ax^2 + bx + c = 0$ is called "fórmula del chicharronero" (formula of the chicharronero). The chicharronero is the guy who sells salty snacks made of wheat (called chicharrones). Outside most schools ...

11

I taught low level algebra for a bit and those students really struggled with knowing what variables were and that they actually stood for numbers. They would see $y=mx+b$ and just not have any idea what it meant or stood for. I've heard the "monter" explanation before, but the students didn't find it helpful (they didn't speak french after all...). What I ...

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