Blake Stacey’s “Survey For Curmudgeons”.
Efrique Ecstathy on “Fighting Mathiness”.
Timothy Chow on “Out-of-print math books” (at What’s New).
Mark C-C’s take on repeated addition.
Oh, and Carnival #37, at Logic Nest.

If It Ain’t Repeated Addition…. Over 100 comments.
How versus Why at the Unapologetic.
The Intute Mathematics Gateway: free books online.

Feynman annoys Quomodocumque less.
Crackpot e-mail at the Unapologetic.

When considering divisibility properties it is convienient to partition the set into subsets— each with some “nice” property. The reader is of course familiar with the sets and of even numbers and odd numbers. Together these sets form a partition of —this means that each element of occurs in exactly one of the given subsets (in other words, “Every natural number is either even or odd (but not both).”).

The even numbers share the property of being divisible by two (in symbols, “If , then ”). This is exactly what it means to be even. The odd numbers obviously share the opposite property of not being divisible by two. Another way to say the same thing is that when any element of is divided by two, there will always be a remainder (namely 1). For that matter, we could say that the even numbers all have remainder 0. The point here is that the “even-or–oddness” (the parity) of a natural number can be thought of in terms of what happens when you divide by two—specifically, what remainder you get.

If natural numbers and have the same parity (both are even or both are odd), we could say they are “equivalent in ‘two-mode’.” The formal symbolism for this is This is pronounced “ is equivalent mod 2 to ” (or, as some authors prefer, as “ is equivalent to (mod two)”). It means simply that the numbers and have the same remainder when you divide them by two.

Footnote: The notation is essentially that of C. F. Gauss in his masterpiece Disquisitiones Arithmeticae published in 1801. Gauss wrote in Latin and used the word modulus (small measure) for the “number-to-divide-by”.

Thus, for example we have , , and .

Suppose we were to partition instead into three sets. Let’s call the sets , , and . Imagine going through the natural numbers one at a time and “counting off”:

X Y Z
0 1 2
3 4 5
6 7 8 …

So to say that two natural numbers belong to the same set of the partition is the same as saying that they have the same remainder when divided by three. The notation for this is (”a is equivalent mod three to b”). For example, , , and .

The same process we have used for the numbers (or moduli—the plural of modulus) two and three can of course be used for any larger natural number. The general notation is of course meaning that and have the same remainder when you divide each of them by .

All of this probably seems pretty abstract and you might very well be wondering if any of it can be used for anything. But I now claim that equivalence mod twelve is very much an everyday idea! Consider the following problem: “I start work at 9 o’clock and work for 8 hours. What time is it when I finish work?” Of course the answer is not “17 o’clock” but rather “5 o’clock”, the point here being that To make the connection with our earlier discussion more explicit, we can once again partition the natural numbers into sets:

12:00 = {0, 12, 24, 36, … }
01:00 = {1, 13, 25, 37, …}
02:00 = {2, 14, 26, 38, … }
…

Readers familiar with a certain amount of geometry might recognize another application of modular arithmetic in the concept of adding angles:
for example, because degree measures () are defined using mod 360.

Notice that these examples involve the concept of addition. How does this connect with our earlier examples? Well, the rules for addition mod two are actually quite well known: (”Even plus even is even, …”). Any even number added to any even number gives an even answer. The point here is that whenever we have to add two numbers, we can tell which set the answer is in whenever we know which sets the original numbers are in.

The same idea can be applied for the “mod 3” sets we have called , , and : for example, we could say that meaning that whenever we add an element of and an element of , the answer will be in .

Look at enough examples to become convinced or consider the following
Proof:
Let and .
Then there exist and such that and (i.e., is one more than a some multiple of 3 and is two more than some multiple of three).
We then have


.
This number is a multiple of three, so .

The first mathematics we learned as children was how to count: “One, two, three. . .” (or “Un, deux, trois. . .” or what have you). Only later did we learn about other kinds of numbers: fractions and negative numbers, for example. The counting numbers are thus the most “natural”, and the set is sometimes called the set of natural numbers. However, for certain technical reasons it’s convenient to include zero and so we define the set of natural numbers as .

The set brackets “” and “” tell us to consider the collection of things between them as one object—namely a set. The symbol “” is just an abbreviation to allow us to refer to this object concisely. The elements of a set are the “things” in the collection. The symbol “” is pronounced “is an element of”, so for example is an abbreviation for the sentence “Thirty-seven is an element of the set of natural numbers.”.

The ellipses ( … ) in the definition of tell us to continue the pattern displayed. The result is an infinite set—one with no last element. Ellipses are also sometimes used in defining finite sets, for example the familiar

The usual convention is to use capital letters to refer to sets. To read the definition out loud, you could say, “ is the set whose elements are zero, one, two, and so on forever.” The formal way to pronounce the symbol is “the set of natural numbers”. Usually, unless there is some need to be precise, one simply says “en”.

This can be trickier than it might appear because it’s common to use “n” as a variable. The convention here is that lower-case letters refer to numbers. So one can say, for example, “If , then must be either even or odd”. This should be pronounced “If en is a natural number, then …”.

The slash “/” is used to denote negation, as in . This idea is common even outside of mathematics, where we find “” (meaning “no”) on street signs, for example. So it shouldn’t come as any surprise that stands for the (true) sentence “One-half is not an element of the set of natural numbers.” (or just “one-half isn’t in ” or some such variation).

There are several other common ways to say “”: for example, “Two doesn’t go into one” or “One is not a multiple of two”. Here is another way: This is pronounced “Two does not divide one”. In general, whenever we have natural numbers n and m we can say that n divides m, and write , whenever . We can also say in this case that n is a factor of m, or that n is a divisor of m.

Thus, for example, , because and . This can be rephrased slightly as “because and 19 is a natural number”. But because there is no natural number with . In other words, because .

Two special cases involve zero. Suppose we have and . Then and To confirm these facts, look at the definition of “”: since and , we get as advertised. But . In fact isn’t a number at all! Be careful when working with 0 to avoid thinking of the wrong special case.

In the original version [~1995] of this piece, several exercises follow (of course!). As of this moment, they’re too much trouble to reproduce here. Exercise: find an expert user of the various notations and discuss.

The 36th Carnival of Mathematics, at Rigorous Trivialities.

Brian Rude doesn’t know where to put the hyperlinks. So what. Plenty of good math-ed stuff there anyhow. I first spotted this a couple years ago while reading HOLD’s Ten Myths page.
I already cited Jerome Dancis a while back. So I’m putting the link here with the others so I’ll know where to find it later.
Then there’s Lee Lady. I seem to’ve started with this longish memoir; you could do worse. Spotted at D. R. Wilkins’s links page.
I found out about Steele’s Rants over at Isabel’s a while back.

“…ascribing more credibility to practices that most mathematicians would be horrified by”: Lefty on “reform” (& Andrew Hodges).
Google’s equation editor at Teaching College Math Technology Blog.
Gowers has come back to life.

The Arte y Pico award (aka an Interesting Blogroll) at WWIUT?.
Comments on an announcement of the Riemann Hypothesis at GPD.
Denise Let’s Play! surveys Math History on the Net.
Here in honor of the upcoming ICME are newsletters for the HPM Group (History and Pedagogy of Maths). Spotted at convergence.

Typing and math at Ivory Tower Tales (recently moved to The Alternative Scientist).
A Hidden Heptagon at 360.
Jonathan on teaching to the test (at ).
Alexandre Micromath on the shrinking middle class.
No carnival this fortnight.

An announcement for “The All-New Online Math League”, by The Elementary Educator.

I’ve just printed out No Common Denominator: The Preparation of Elementary Teachers in Mathematics by American’s Education Schools, a new report from the National Council on Teacher Quality. I’ll read it over the weekend. I learned about it in this KTM thread. (I dropped ‘em from the blogroll … never said I stopped reading ‘em …)

A deathly ill department: “Mathematics at Flinders” (Terrence Tao in Maths In Oz).
Mathematica turns 20; Wolfram gloats.
Ruben Hersh’s (PDF) “Ethics For Mathematicians” (in Philosophy of Mathematics Education Journal #22) deserves a wide audience.

Michael J. South’s fulcrum.org appears to’ve been mostly about math ed; the little I’ve looked at so far looks pretty useful. It’s in good old-fashioned HTML, so you’ll have to adjust the size of your window to read it easily (this seems to have become something of a lost art). He’s currently blogging at Unconventional Instructional Design (where as far as I’m concerned, he might as well be doing Theology). LeftoverPi claims him as a brother in a sidebar (and I found about the Fulcrum project in this post at that blog…check out “Education’s Bug #1″ while you’re there).

“LaTeX as a word processor?” (Robert Talbert).
Mythic narrative considered harmful (Alice Mercer).
Jacobs’ Human Endeavor and homeschooling (Maria Miller).
Jonathan takes issue with an ill-designed test (JD2718).
I spotted this new entry on Dedekind in the Stanford Encyclopedia of Philosophy at LogBlog.
Chaim Goodman-Strauss plugged his new book in The Math Factor last month.

Barry Garelick’s “It Works For Me” (Parts 1, II, and III) dates from last year. But I just found it. The type’s way too small.
Alternative assessment at Coffee and Graph Paper. Wow. If I wanted to try anything like this, I’d have to lie about it.
In other Dan-Meyers-related news, here’s “Math-Itude”, by Jackie Continuities.
Charlotte Mulcare’s “Maths, madness[,] and movies” in the new +Plus.

Here’s a recent accidental post on “Fuzziness” in Kitchen Table Math. I dropped KTM from the blogroll last week: there’s very little math there these days and you’ve gotta draw the line somewhere (one needs a reason to endure all the self-righteously overprivileged whining …). Evidently this post was intended for a new blog: Math Without Tears.

Which is itself almost entirely devoted to political matters … but it’s actual politics-of-math-ed stuff (rather than, let’s say for instance, boilerplate freemarketeer union-bashing). We’ll see how it goes.

Carnival of Mathematics #35.

A cool thread inspired by “conservapedian math” (at GPD).
A new magazine about maths for the general public (in the UK): iSquared.

This blog turns one year old tomorrow. The fact that my regular audience is in the low single digits is a tremendous disappointment to me. Of course it doesn’t surprise me to be reminded that the world and I have wildly different opinions about what’s interesting … I’m a math major, for heck sake. And sure, I’m not doing it for the greater good of the internet: there ought to be some satisfaction in feeling that I’m the world’s foremost scholar in Early Blathospherics, and indeed there is. Some.

So. I’m not quitting just yet. But once I get the access to the tools, don’t blame me if I go back to just running off a few dozen hardcopies of The Ten Page News and trading ‘em in the mail to get my regular fix instead of messing about with this high-tech stuff that’s just gonna disappear like broadcast TV any day now anyhow. We now return you to my personal bookmarks.