#mathwords

[kathleenwcurry]

Easily Confused Words: Differential vs. Deferential

Easily Confused Words: Differential vs. Deferential

Differential and deferential are easily confused words.

The spell-check application of most word processing software programs would not catch a slip-up of these two words. Spell-check is looking for words that aren’t in its dictionary, and words that resemble words in its dictionary, but are possibly spelled wrong. Spell-check isn’t perfect. It doesn’t know and can’t guess what word you wanted,…

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[language-of-math-blog]

Discriminants

I have NO idea what a discriminant is. I’ve been skipping these questions on Khan Academy because I just have absolutely no clue what to solve for. So let’s see if we can shed some light on this.

Speaking of Khan, he tells us what discriminants are in this article:

The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.

In other words, the discriminant tells us if and where the function touches or crosses the x-axis.

(Image source)

Pat Ballew gives a similar explanation:

discriminant To most algebra students the discriminant is a number related to a quadratic equation that determines the nature of the solutions. The value of the discriminant in a quadratic is given by the expression B2 - 4AC. A positive discriminant indicates two real distinct roots, a zero indicates a single real root, and a negative value indicates two complex roots. The actual polynomial discriminant has a denominator which is a power of the leading coefficient, but as the most common use is to determine the number of roots, the (always positive) denominator is ignored. (Source)

However, Ballew also expands on the origins of the word discriminant itself:

The origin of the word is from the union of dis, and the very early Indo-European root skeri which referred to separating by sifting, scraping or cutting, through the Greek krinein to separate. Descendants of the skeri root show up in diverse words today including script, crime, decree, secret, and endocrine.

In other words, the “discriminant” of a function helps us to “separate” out the solutions. But this etymology is a bit difficult to understand. Through Google we can trace the true etymology and find some friends in the process:

discriminant 19th century (in the sense ‘showing discernment’): from Latin discriminant- ‘distinguishing between,’ from the verb discriminare (see discriminate).

discriminate early 17th century: from Latin discriminat- ‘distinguished between,’ from the verb discriminare, from discrimen ‘distinction,’ from the verb discernere (see discern).

discern late Middle English: via Old French from Latin discernere, from dis- ‘apart’ + cernere ‘to separate.’

I like this flow better. The discriminant is the part of the quadratic formula that helps us to discern the number of solutions for a given equation.

Now I know!

[language-of-math-blog]

“Hypotenuse”

Hypotenuse comes from the common Greek root hypo(for under, as in hypodermic -under the skin) and the less common tein or ten, for stretch. This last is the source of our modern word tension. The hypotenuse was the line segment "stretched under" the right angle. (Source)

Okay, I’m totally confused by this. How can the hypotenuse be “stretched under” the right angle? Maybe I’m just not visualizing things correctly. Perhaps an image will help?

(Image courtesy of Mathspace)

Alright, this is the first image I found that makes it clear that the hypotenuse can, in fact, be “stretched under” the right angle. I guess I’m just so used to see it on the side, with the right angle touching the “floor,” that I didn’t realize what other configurations could be valid.

That purple triangle in particular makes it very clear where this word originates from!

[language-of-math-blog]

Bowls

I just learned something amazing: “parabola” shares the same root word as “hyperbole.” Here’s how it breaks down (quotes from this source):

The word parabola literally translates "to throw beside". The Greek root bole, which is the ancestor of ball, bowl, bubble, and many such words today was closer to throw or place in the Greek usage.

Parabolas do kind of look like bowls, don’t they? In fact, in machine learning one of the ways I was taught to understand gradient descent was to think of it as a bowl that we’re trying to find the bottom of.

Pat Ballew goes on to say:

Para is a root for beside, as we see it in parallel. Parallel refers to two lines that are "beside" each other. Think of a parabola as the shape obtained when the cutting plane is "thrown" alongside the slope of the cone. A closely related word is parable, a story that is not true, but true-like, or "thrown beside" the truth. The parathyroids are so named because they are beside the thyroids.

This explanation requires an understanding of conic sections, which I got pretty quickly after looking at this image from Wikipedia:

where (1) shows a parabola, (2) shows an ellipse and a circle, and (3) shows hyperbolas. I’m sure I’ve seen hyperbolas before (and after googling images of them on a graph I do, indeed, remember having seen them before) but this is the first time I was able to truly understand it.

Anyway, back to parabolas. They are the shape that we get when a cutting plane is, as Pat Ballew says, “‘thrown’ alongside” the slope of the cone.

Since we’ve got that image up there, let’s go ahead and see what Ballew says about the origins of the other conic sections:

Ellipse

The ellipse is a shape formed when the cutting plane of a cone falls short of the slope of the cone, and the meaning, as you expected, is to fall short. The use of three dots or asterisks to mark where part of a statement has been left out is called an ellipsis for the same reason.

Wikipedia notes that the circle in image 2 is a special form of an ellipse, and that many consider it to be a fourth conic section in its own right.

Hyperbola

The hyperbola shares the common bola root, but this time the throw is "more than enough" or beyond the slope of the cone, creating a hyperbola. Hyper as more than enough or an overabundance appears in the names of objects with more than three dimensions, and the fourth dimension equivalent of a cube is called a hypercube. Hypertension means too tense. The opposite of hyper is hypo, as in hypodermic, which translates to "under the skin", and of course, Hypotenuse.

The English term, hyperbole derives from the same root words as hyperbola, and is used for an obvious exaggeration or extension of the truth.

Didn’t know that about the hypotenuse! Think I’ll go check that one out next ;)

[language-of-math-blog]

“Quadratic”

QUADRATIC is derived from the Latin quadratus, meaning "square." In English, quadratic was used in 1668 by John Wilkins (1614-1672) in An essay towards a real character, and a philosophical language[London: Printed for Sa. Gellibrand, and for John Martyn, 1668]. He wrote: "Those Algebraical notions of Absolute, Lineary, Quadratic, Cubic" (OED2) (Source)

Quadratic is the Latin root for "to make square". The word square was itself derived from this root (Source)

Square is derived from the Latin phrase Exquadrare, like a quadratic. Over time the term was contracted into its present form, and came to mean the regular quadrilateral. The word is also the root of the military term squadron, which originally meant a fighting square from the early practice of fighting in square formations. (Source)

Apparently some people call squares “quadrangles,” to match with triangles, and others call them “tetragons” to match with pentagon, hexagon, octagon, etc. I would have much preferred these uses to “square” which never seemed to fit in with the other shapes (i.e. triangle, rectangle).

[language-of-math-blog]

Logarithm

This is a word that I never truly understood the meaning of. I’ve struggled to connect the concept to exponentiation, but this line from Wikipedia is helping solidify it in my brain:

In mathematics, the logarithm is the inverse operation to exponentiation, just as division is the inverse of multiplication. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. (Source)

In other words, the logarithm is the inverse of the exponent. The exponent is the power to which a root number is raised. The logarithm is the exponent to which the root number must be raised to produce the number in question.

In a way, the operation to the right of “log” actually looks a little bit like an exponent, doesn’t it? We’ve got the number 4 nudged down a bit, and if you take the bottom of the number 4 as the base of a line, then the (16) seems to be raised a bit up and away from the number four. I want to make a suggestion for how to look at this operation, but first lets see if the etymology of “logarithm” can give us a bit more insight.

From etymonline:

1610s, logarithmus, coined in Modern Latin by Scottish mathematician John Napier (1550-1617), literally "ratio-number," from Greek logos "proportion, ratio, word" (see Logos) + arithmos "number" (from PIE *erei-dhmo-, suffixed variant form of root *re- "to reason, count")

Wiktionary asks us to compare:

rational +‎ number, ultimately from Latin rationalis + numerus. Compare analogous logarithm, from Ancient Greek. (Source)

So logarithm means “ratio-number,” but I still don’t understand why. Let’s see if the math words sites can give us a bit more insight.

At first glance it appears that the coined term is as confusing as I believed:

According to the OED2, "Napier does not explain his view of the literal meaning of logarithmus. It is commonly taken to mean 'ratio-number', and as thus interpreted it is not inappropriate, though its fitness is not obvious without explanation. Perhaps, however, Napier may have used logos merely in the sense of 'reckoning', 'calculation.'" (Source)

But it also states, later on in that same page:

According to Briggs in Arithmetica logarithmica (1624), Napier used the term because logarithms exhibit numbers which preserve always the same ratio to one another.

If we look at it from this perspective, I can kind of see why he would coin this term. There is a fixed relationship between the base, the exponent and the power. The ratio of the power to the base and exponent will always be the same.

[language-of-math-blog]

Radicals

I googled the etymology of “radical” and the explanation that Google provided immediately made the concept clearer to me:

late Middle English (in the senses ‘forming the root’ and ‘inherent’): from late Latin radicalis, from Latin radix, radic- ‘root.’

“Forming the root.” The number that, when multiplied by itself, forms the base of the exponent. It is the “root,” the “base,” of that operation.

We can take this word, “root,” and deepen our metaphor:

The Indo-European root werad was used for the branches or roots of plants. Later it was generalized to mean the origins, or beginnings of something whether it was physical or mental. In arithmetic the root of a number is the number that is used to build up another number by repeated multiplication. (Source)  

According to the above source by Pat Ballew, the Indo-European word for root “werad” was used by al-Khowarizmi, and then translated into Latin as radix, which is where we get the word “radical.”

The “root” of a square number (i.e. 16) is called its “square root,” and the root for the squared-number 16 is 4. Similarly, the root for a cubed number is called the cube root, and the root for the cubed-number 8 is 2.

[language-of-math-blog]

Exponent

The term EXPONENT was introduced by Michael Stifel (1487-1567) in 1544 in Arithmetica integra. He wrote, "Est autem 3 exponens ipsius octonarij, & 5 est exponents 32 & 8 est exponens numeri 256" (Smith vol. 2, page 521). (Source)

When I was looking for the etymology of exponent, here is what I found:

From etymonline:

1706, from Latin exponentem (nominative exponens), present participle of exponere “put forth”

From Wiktionary:

From Latin expōnēns, present participle of expōnō (“to expose; to exhibit, display, set out; to explain”), from ex- (“out, away”) + pōnō (“to lay, place, put”).

Reading through these definitions (and having no knowledge of Latin), I tried to reason through what choosing this word for the concept of exponentiation might have meant. It seemed fitting that we would “put out” the number that would raise the base, that we would “place it away” from the base so that it could be seen.

After searching online I ran into this website by math instructor Pat Ballew, who appears to agree with this interpretation:

Exponent is the union of the Latin roots exo(out of) + ponere (place). The literal interpretation is to make something visible or obvious. That seems to be what happens when the index is raised "out of" the line. The English word expound, from the same source, means to make clear. An exponent is also used in English to describe a person who explains or interprets.

I love how he makes the connection to the second meaning of exponent, which is to expound. I was unable to draw the line myself, but I like how it has that additional meaning of “to make something clear.” You make the exponent clear by raising it away from the base. (He notes that this interpretation may not sit well with everyone, as can be seen on this page by Dr. Peterson, one of the Math Doctors at the Math Forum. However, I agree with it wholeheartedly, as it was the connection that made the most intuitive sense to me.)