# Power

I have rewritten the entry to “power” in the abstractmath.org Glossary:

## POWER

Here are three variant phrases that say that $125=5^3$:

• “$125$ is a power of $5$ with exponent $3$”.
• “$125$ is the third power of $5$”.
• “$125$ is $5$ to the third power”.

Some students are confused by such statements, and conclude that $3$ is the “power”. This usage appears in print in Wikipedia in its entry on Exponentiation (as it was on 22 November 2016):

“…$b^n$ is the product of multiplying $n$ bases:

$b^n = \underbrace{b \times \cdots \times b}_n$

In that case, $b^n$ is called the $n$-th power of $b$, or $b$ raised to the power $n$.”

As a result, students (and many mathematicians) refer to $n$ as the “power” in any expression of the form “$a^n$”. The number $n$ should be called the “exponent”. The word “power” should refer only to the result $a^n$. I know mathematical terminology is pretty chaotic, but it is silly to refer both to $n$ and to $a^n$ as the “power”.

Almost as silly as using $(a,b)$ to refer to an open interval, an ordered pair and the GCD. (See The notation $(a,b)$.)

Suggestion for lexicographical research: How widespread does referring to $n$ as the “power” come up in math textbooks or papers? (See usage.)

Thanks to Tomaz Cedilnik for comments on the first version of this entry.

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# Variations in meaning in math

Words in a natural language may have different meanings in different social groups or different places.  Words and symbols in both mathematical English and the symbolic language vary according to specialty and, occasionally, country (see convention, default).  And words and symbols can change their meanings from place to place within the same mathematical discourse (see scope).

This article mostly provides pointers to other articles in abstractmath.org that give more details about the ideas.

## Conventions

A convention in mathematical discourse is notation or terminology used with a special meaning in certain contexts or in certain fields. Articles and books in a specialty do not always clue you in on these conventions.

Some conventions are nearly universal in math.

#### Example 1

The use of “if” to mean “if and only if” in a definition is a convention. More about this here. This is a hidden definition by cases. “Hidden” means that no one tells the students, except for Susanna Epp and me.

#### Example 2

Constants or parameters are conventionally denoted by a, b, … , functions by f, g, … and variables by x, y,…. More.

#### Example 3

Referring to a group (or other mathematical structure) and its underlying set by the same name is a convention.  This is an example of both synecdoche and context-sensitive.

#### Example 4

The meaning of ${{\sin }^{n}}x$ in many calculus books is:

• The inverse sine (arcsin) if $n=-1$.
• The mult­iplica­tive power for positive $n$; in other words, ${{\sin }^{n}}x={{(\sin x)}^{n}}$ if $n\ne -1$.

This, like Example 1, is a definition by cases. Unlike Example 1, calculus books often make it explicit. Explicit or not, this usage is an abomination.

Some conventions are pervasive among math­ematicians but different conventions hold in other subjects that use mathematics.

• Scientists and engineers may regard a truncated decimal such as 0.252 as an approximation, but a mathematician is likely to read it as an exact rational number, namely $\frac{252}{1000}$.
• In most computer languages a distinction is made between real numbers and integers;
42 would be an integer but 42.0 would be a real number.  Older mathematicians may not know this.
• Mathematicians use i to denote the imaginary unit. In electrical engineering it is commonly denoted j instead, a fact that many mathematicians are un­aware of. I first learned about it when a student asked me if i was the same as j.

Conventions may vary by country.

• In France and possibly other countries schools may use “positive” to mean “nonnegative”, so that zero is positive.
• In the secondary schools in some places, the value of sin x may be computed clockwise starting at (0,1)  instead of counterclockwise starting at (1,0).  I have heard this from students.

Conventions may vary by specialty within math.

Field” and “log” are examples.

## Defaults

An interface to a computer program may have many possible choices for the user to make. In most cases, the interface will use certain choices automatically when the user doesn’t specify them.  One says the program defaults to those choices.

#### Examples

• A word processing program may default to justified paragraphs and insert mode, but allow you to pick ragged right or typeover mode.
•  I have spent a lot of time in both Minne­sota and Georgia and the remarks about skiing are based on my own observation. But these usages are not absolute. Some affluent Geor­gians may refer to snow skiing as “skiing”, for example, and this usage can result in a put-down if the hearer thinks they are talking about water skiing. One wonders where the boundary line is. Perhaps people in Kentucky are confused on the issue.
• There is a sense in which the word “ski” defaults to snow skiing in Minnesota and to water skiing in Georgia.
• “CSU” defaults to Cleveland State University in northern Ohio and to Colorado State University in parts of the west.

Math language behaves in this way, too.

### Default usage in mathematical discourse

#### Symbols

• In high school, $\pi$ refers by default to the ratio of the circumference of a circle to its diameter.  Students are often quite surprised when they get to abstract math courses and discover the many other meanings of $\pi$ (see here).
• Recently authors in the popular literature seem to think that $\phi$ (phi) defaults to the golden ratio.  In fact, a search through the research literature shows very few hits for $\phi$ meaning the golden ratio: in other words, it usually means something else.
• The set $\mathbb{R}$ of real numbers has many different group structures defined on it but “The group $\mathbb{R}$” essentially always means that the group operation is ordinary addition.  In other words, “$\mathbb{R}$” as a group defaults to +.  Analogous remarks apply to “the field $\mathbb{R}$”.
• In informal conversation among many analysts, functions are continuous by default.
• It used to be the case that in informal conversations among topologists, “group” defaulted to Abelian group. I don’t know whether that is still true or not.

#### Remark

This meaning of “default” has made it into dictionaries only since around 1960 (see the Wikipedia entry). This usage does not carry a derogatory connotation.   In abstractmath.org I am using the word to mean a special type of convention that imposes a choice of parameter, so that it is a special case of both “convention” and “suppression of parameters”.

## Scope

Both mathematical English and the symbolic language have a feature that is uncommon in ordinary spoken or written English:  The meaning of a phrase or a symbolic expression can be different in different parts of the discourse.   The portion of the text in which a particular meaning is in effect is called the scope of the meaning.  This is accomplished in several ways.

### Explicit statement

#### Examples

• “In this paper, all groups are abelian”.  This means that every instance of the word “group” or any symbol denoting a group the group is constrained to be abelian.   The scope in this case is the whole paper.   See assumption.
• “Suppose (or “let” or “assume”) $n$ is divisible by $4$”. Before this statement, you could not assume $n$ is divisible by $4$. Now you can, until the end of the current paragraph or section.

### Definition

The definition of a word, phrase or symbol sets its meaning.  If the word definition is used and the scope is not given explicitly, it is probably the whole discourse.

#### Example

“Definition.  An integer is even if it is divisible by 2.”  This is marked as a definition, so it establishes the meaning of the word “even” (when applied to an integer) for the rest of the text.

### If

Used in modus ponens (see here) and (along with let, usually “now let…”) in proof by cases.

#### Example(modus ponens)

Suppose you want to prove that if an integer $n$ is divisible by $4$ then it is even. To show that it is even you must show that it is divisible by $2$. So you write:

• “Let $n$ be divisible by $4$. That means $n=4k$ for some integer $k$. But then $n=2(2k)$, so $n$ is even by definition.”

Now if you start a new paragraph with something like “For any integer $n\ldots$” you can no longer assume $n$ is divisible by $4$.

#### Example (proof by cases)

Theorem: For all integers $n$, $n^2+n+1$ is odd.

Definitions:

• “$n$ is even” means that $n=2s$ for some integer $s$.
• “$n$ is odd” means that $n=2t+1$ for some integer $t$.

Proof:

• Suppose $n$ is even. Then

\begin{align*} n^2+n+1&=4s^2+2s+1\\ &=2(2s^2+s)+1\\ &=2(\text{something})+1 \end{align*}

so $n^2+n+1$ is odd. (See Zooming and Chunking.)

• Now suppose $n$ is odd. Then

\begin{align*} n^2+n+1&=(2t+1)^2+2t+1+1\\ &=4t^2+4t+1+2t+1+1\\ &=2(2t^2+3t)+3\\ &=2(2t^2+3t+1)+1\\ &=2(\text{something})+1 \end{align*}

So $n^2+n+1$ is odd.

##### Remark

The proof I just gave uses only the definition of even and odd and some high school algebra. Some simple grade-school facts about even and odd numbers are:

• Even plus even is even.
• Odd plus odd is even.
• Even times even is even.
• Odd times odd is odd.

Put these facts together and you get a nicer proof (I think anyway): $n^2+n$ is even, so when you add $1$ to it you must get an odd number.

### Bound variables

A variable is bound if it is in the scope of an integral, quantifier, summation, or other binding operators.  More here.

#### Example

Consider this text:

Exercise: Show that for all real numbers $x$, it is true that $x^2\geq0$. Proof: Let $x=-2$. Then $x^2=(-2)^2=4$ which is greater than $0$. End of proof.”

The problem with that text is that in the statement, “For all real numbers $x$, it is true that $x^2\geq0$”, $x$ is a bound variable. It is bound by the universal quantifier “for all” which means that $x$ can be any real number whatever. But in the next sentence, the meaning of $x$ is changed by the assumption that $x=-2$. So the statement that $x\geq0$ only applies to $-2$. As a result the proof does not cover all cases.

Many students just beginning to learn to do proofs make this mistake. Fellow students who are a little further along may be astonished that someone would write something like that paragraph and might sneer at them. But this common mistake does not deserve a sneer, it deserves an explanation. This is an example of the ratchet effect.

## Variable meaning in natural language

Meanings commonly vary in natural language because of conventions and defaults. But varying in scope during a conversation seems to me uncommon.

It does occur in games. In Skat and Bridge, the meaning of “trump” changes from hand to hand. The meaning of “strike” in a baseball game changes according to context: If the current batter has already had fewer than two strikes, a foul is a strike, but not otherwise.

I have not come up with non-game examples, and anyway games are played by rules that are suspiciously like mathematical axioms. Perhaps you can think of some non-game occasions in which meaning is determined by scoping that I have overlooked.

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# Tangents

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Tangent Line.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This is an experiment in exposition of the mathematical concepts of tangent.  It follows the same pattern as my previous post on secant, although that post has explanations of my motivation for this kind of presentation that are not repeated here.

### Tangent line

A line is tangent to a curve (in the plane) at a given point if all the following conditions hold (Wikipedia has more detail.):

1. The line is a straight line through the point.
2. The curve goes through that point.
3. The curve is differentiable in a neighborhood of the point.
4. The slope of the straight line is the same as the derivative of the curve at that point.

In this picture the curve is $y=x^3-x$ and the tangent is shown in red. You can click on the + signs for additional controls and information.

Etymology and metaphor

The word “tangent” comes from the Latin word for “touching”. (See Note below.) The early scholars who talked about “tangent” all read Latin and knew that the word meant touching, so the metaphor was alive to them.

The mathematical meaning of “tangent” requires that the tangent line have slope equal to the derivative of the curve at the point of contact. All of the red lines in the picture below touch the curve at the point (0, 1.5). None of them are tangent to the curve there because the curve has no derivative at the point:

The curve in this picture is defined by

The mathematical meaning restricts the metaphor. The red lines you can generate in the graph all touch the curve at one point, in fact at exactly at one point (because I made the limits on the slider -1 and 1), but there are not tangent to the curve.

Tangents can hug!

On the other hand, “touching” in English usage includes maintaining contact on an interval (hugging!) as well as just one point, like this:

The blue curve in this graph is given by

The green curve is the derivative dy/dx. Notice that it has corners at the endpoints of the unit interval, so the blue curve has no second derivative there. (See my post Curvature).

Tangent lines in calculus usually touch at the point of tangency and not nearby (although it can cross the curve somewhere else). But the red line above is nevertheless tangent to the curve at every point on the curve defined on the unit interval, according to the definition of tangent. It hugs the curve at the straight part.

The calculus-book behavior of tangent line touching at only one point comes about because functions in calculus books are always analytic, and two analytic curves cannot agree on an open set without being the same curve.

The blue curve above is not analytic; it is not even smooth, because its second derivative is broken at $x=0$ and $x=1$. With bump functions you can get pictures like that with a smooth function, but I am too lazy to do it.

### Tangent on the unit circle

In trigonometry, the value of the tangent function at an angle $\theta$ erected on the x-axis is the length of the segment of the tangent at (1,0) to the unit circle (in the sense defined above) measured from the x-axis to the tangent’s intersection with the secant line given by the angle. The tangent line segment is the red line in this picture:

This defines the tangent function for $-\frac{\pi}{2} < x < \frac{\pi}{2}$.

The tangent function in calculus

That is not the way the tangent function is usually defined in calculus. It is given by $\tan\theta=\frac{\sin\theta}{\cos\theta}$, which is easily seen by similar triangles to be the same on $-\frac{\pi}{2} < x < \frac{\pi}{2}$.

We can now see the relationship between the geometric and the $\frac{\sin\theta}{\cos\theta}$ definition of the tangent function using this graph:

The red segment and the green segment are always the same length.
It might make sense to extend the geometric definition to $\frac{\pi}{2} < x < \frac{3\pi}{2}$ by constructing the tangent line to the unit circle at (-1,0), but then the definition would not agree with the $\frac{\sin\theta}{\cos\theta}$ definition.

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# Case Study in Exposition: Secant

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code comes from several Mathematica notebooks lists in the References. The notebooks are available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

### Pictures, metaphors and etymology

Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept.  I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of “secant” that use pictures, metaphors and etymology to describe the concept.

The exposition is interlarded with comments about what I am doing and why.  An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!

### Secant Line

The word “secant” is used in various related ways in math.  To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:

The word “secant” comes from the Latin word for “cut”, which came from the Indo-European root “sek”, meaning “cut”.  The IE root also came directly into English via various Germanic sound changes to give us “saw” and “sedge”.

The picture

Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept.  The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points.  You also get a very strong understanding of how the secant line is a function of the two given points.  I don’t think that is obvious to someone without some experience with such things.

This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects.  (Math books are full of such pictures.)  So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds.  This is the sort of claim that is amenable to field testing.

The metaphor

Most metaphors are based on a physical phenomenon.  The mathematical meanings of “secant” use the metaphor of cutting.  When the word “secant” was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor.   In those days essentially every European scholar read Latin. To them “secant” would transparently mean “cutting”.  This is not transparent to many of us these days, so the metaphor may be hidden.

If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.

• The straight line does not really cut the curve.  Indeed, the curve itself is both an abstract object that is not physical, so can’t be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it?  Cut the screen?  The line can’t do that.
• You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
• The metaphor is restricted further by saying that it is determined by two points on the curve.   This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines.  You could define such a family by using one point on the curve and a slope, for example.  This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.

### Secant on circle

Another use of the word “secant” is the red line in this picture:

This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve.  This means it corresponds to a number, and that number is what we mean by “secant” in trigonometry.

To the ancient Greeks, a (positive) number was the length of a line segment.

The Definition

The secant of an angle $\theta$ is usually defined as $\frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.

In many parts of the world, trig students don’t learn the word “secant”. They simply use $\frac{1}{\cos\theta}$.

This illustrates important facts about definitions:

• Different equivalent definitions all make the same theorems true.
• Different equivalent definitions can give you a very different understanding of the concept.

The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of “secant”.  You can tell a lot from its behavior right off (it goes to infinity near $\pi/2$, for example).

The definition $\sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $\sec\theta$.  It also reduces the definition of $\sec\theta$ to a previously known concept.

It used to be common to give only the $\frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it.  That is a crime.  Yes, I know many students don’t want to “understand” stuff, they only want to know how to do the problems.  Teachers need to talk them out of that attitude.  One way to do that in this case is to test them on the geometric definition.

Etymology

This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in “Geometria Rotundi” (1583), where the first known use of the word “secant” occurs.  I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.

It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on.  A cursory search of the internet gave me the current name in Arabic for secant but nothing else.

Graph of the secant function

The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.

### References

Mathematica notebooks used in this post:

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