Let R be a ring. An element u of R is called a unit if it has a multiplicative inverse. That is, there exists v in R such that uv=1 and vu=1.

If u is a unit then the v in the definition is unique. For if uw=1 then vuw=v and since vu=1 this says that w=v. The inverse of a unit u is usually denoted by u-1.

The collection of all units of R form a group, the multiplicative group of R.

Examples

• 1,-1 are the only units in the ring of integers Z.
• In a field every nonzero element is a unit.
• An nxn matrix with complex number coefficients is a unit in the ring of all such matrices if and only if it has nonzero determinant.
The two most widely used systems of units in the scientific community are the MKS (or SI) system, and the Gaussian system. There are several other systems of units, including for example the natural units and atomic units of quantum theory (or the Heaviside-Lorentz system of subatomic quantum theory), but these are the most prevalent, seen in every field of physical theory.

Every system of units includes a set of base units. In the MKS system, these are meters, kilograms, and seconds. It is a surprising fact (for the uninitiated or non-scientist) that every measurable physical quantity can be reduced to some algebraic combination of these quantities.

The reason for the persistence of these different sets of units is that they give (at least superficially) different expressions for Maxwell's equations. This is not much of an annoyance as long as one is dealing with the simpler equations of electrostatics, but as soon as one becomes interested in time varying electromagnetic fields, the gaussian system becomes preferable because of the symmetric way in which both electric and magnetic contributions to the field come in.

In Type Theory, the type that has exactly one element. This element is called, not surprisingly, the unit element. We usually represent this type as 1, and the unit element as () or <>. We write () : 1.

Since it is the only element of type 1, the unit element carries no information. When we don't want a function to return anything useful (perhaps it has useful side effects, such as I/O or messing with ephemeral structures), we have it return (). This is somewhat analogous to functions in C returning void, but isn't exactly the same thing.

Similarly, if we want a function to take no useful input (say it returns the same thing every time or returns something dependent on ephemeral structures), we have it take () as its only argument.

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No shit, there I was: My sophomore year in college at Georgia Tech, I started getting really active in working on Slackware Linux. Around this time, my friend David was working on porting Slackware to the SPARC. He only had an SS1 so he was working very slowly. I decided that porting the entire distribution was a good challenge. Plus, it'd give me something to do besides go to those pesky classes. I decided it'd be cool to try a port to the Alpha. The only problem was how damned expensive the machines were. I wouldn't just be able to go out and buy one.

A little time passes and I find myself out in San Jose (one of my least favorite towns in the country) working at a trade show. David and I skipped out one afternoon to go eat lunch. As luck would have it, several guys from Alpha Processor Inc. (now API NetWorks, or maybe they're out of business by now) were sitting right behind us. I got up, walked over, and introduced myself. Some shit was shot and I happened to mention wanting to port Slackware but couldn't afford a machine and maybe they could help me out. I ended up with some business cards and promised to get back to them.

A few weeks later, I decided to send them an email. I mailed one of the marketing guys at API about getting a loaner machine. He sent me some paperwork and I faxed it back to him (these guys always like to fax things around). The description of the machine talked about dual processors, lots of RAM, and so forth. "Cool", I told myself, thinking I'd be getting some sort of mid tower case.

About a week later, I opened my Georgia Tech post office box to find a slip of paper inside. When you get a package that won't fit in the tiny little boxes, they stick a piece of paper in your box telling you to come pick it up. I'd gotten such a piece of paper. On the bottom in nice big letters was written: Big, big, big, big, big, big box!

So, I walked up to the counter with a big smile on my face and handed the person the slip. They looked at it for a second and told me to come around back into the place where they store all the packages. As three of my friends and I walked around the back, everyone who worked there was mumbling something about the size of this package and sort of staring at us. Sure enough, my new Alpha had come in a box about five feet by five feet by three feet. Knowing there was no way I'd ever get that thing back to my house, I called up Ted and asked him to bring his car on by. While he was on the way, the four of us struggled to get the box to the loading dock. Later, we'd learn the computer alone weighs about 95 pounds.

By the time Ted got to the loading dock, we were working on getting the computer out of the box. Of course, there was a lot of cursing going on both at the weight of the thing and at just how big it really was. There was also a little comparison of the machine to various body parts. We're not very quiet people, so we said all this loud enough for pretty much anyone to hear. After five or six minutes of this, someone from the post office came back and asked us to keep it down.

Well, eventually we got the thing into Ted's trunk and tossed all the packing peanuts it'd been shipped in. We then took the long way around campus, windows down, yelling and cursing at everyone along the way, letting them know that we had a big damn computer in the trunk and they didn't. Let me tell you that moving a 95 pound machine around when it's four feet tall, four feet deep, and two feet wide is no easy task.

Of course, I named it unit.

U"nit (?), n. [Abbrev. from unity.]

1.

A single thing or person.

2. Arith.

The least whole number; one.

Units are the integral parts of any large number. I. Watts.

3.

A gold coin of the reign of James I., of the value of twenty shillings.

Camden.

4.

Any determinate amount or quantity (as of length, time, heat, value) adopted as a standard of measurement for other amounts or quantities of the same kind.

5. Math.

A single thing, as a magnitude or number, regarded as an undivided whole.

Abstract unit, the unit of numeration; one taken in the abstract; the number represented by 1. The term is used in distinction from concrete, or determinate, unit, that is, a unit in which the kind of thing is expressed; a unit of measure or value; as 1 foot, 1 dollar, 1 pound, and the like. -- Complex unit Theory of Numbers, an imaginary number of the form a + b-1, when a2 + b2 = 1. -- Duodecimal unit, a unit in the scale of numbers increasing or decreasing by twelves. -- Fractional unit, the unit of a fraction; the reciprocal of the denominator; thus, unit of the fraction -- Integral unit, the unit of integral numbers, or 1. -- Physical unit, a value or magnitude conventionally adopted as a unit or standard in physical measurements. The various physical units are usually based on given units of length, mass, and time, and on the density or other properties of some substance, for example, water. See Dyne, Erg, Farad, Ohm, Poundal, etc. -- Unit deme Biol., a unit of the inferior order or orders of individuality. -- Unit jar (Elec.), a small, insulated Leyden jar, placed between the electrical machine and a larger jar or battery, so as to announce, by its repeated discharges, the amount of electricity passed into the larger jar. -- Unit of heat Physics, a determinate quantity of heat adopted as a unit of measure; a thermal unit (see under Thermal). Water is the substance generally employed, the unit being one gram or one pound, and the temperature interval one degree of the Centigrade or Fahrenheit scale. When referred to the gram, it is called the gram degree. The British unit of heat, or thermal unit, used by engineers in England and in the United States, is the quantity of heat necessary to raise one pound of pure water at and near its temperature of greatest density (39.1° Fahr.) through one degree of the Fahrenheit scale. Rankine. -- Unit of illumination, the light of a sperm candle burning 120 grains per hour. Standard gas, burning at the rate of five cubic feet per hour, must have an illuminating power equal to that of fourteen such candles. -- Unit of measure (as of length, surface, volume, dry measure, liquid measure, money, weight, time, and the like), in general, a determinate quantity or magnitude of the kind designated, taken as a standard of comparison for others of the same kind, in assigning to them numerical values, as 1 foot, 1 yard, 1 mile, 1 square foot, 1 square yard, 1 cubic foot, 1 peck, 1 bushel, 1 gallon, 1 cent, 1 ounce, 1 pound, 1 hour, and the like; more specifically, the fundamental unit adopted in any system of weights, measures, or money, by which its several denominations are regulated, and which is itself defined by comparison with some known magnitude, either natural or empirical, as, in the United States, the dollar for money, the pound avoirdupois for weight, the yard for length, the gallon of 8.3389 pounds avoirdupois of water at 39.8° Fahr. (about 231 cubic inches) for liquid measure, etc.; in Great Britain, the pound sterling, the pound troy, the yard, or -- Unit of power. Mach. See Horse power. -- Unit of resistance. Elec. See Resistance, n., 4, and Ohm. -- Unit of work Physics, the amount of work done by a unit force acting through a unit distance, or the amount required to lift a unit weight through a unit distance against gravitation. See Erg, Foot Pound, Kilogrammeter. -- Unit stress Mech. Physics, stress per unit of area; intensity of stress. It is expressed in ounces, pounds, tons, etc., per square inch, square foot, or square yard, etc., or in atmospheres, or inches of mercury or water, or the like.