abstractmath.org

GLOSSARY

# A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

### maximize

To maximize a function is to find values of its argument for which the function has a maximum. Minimize is used similarly.

##### Example

The number x = 0 maximizes the function  (discussed here.)

The metaphor behind this usage seems to be: vary the input until you find the largest value.  This is the way most functions are maximized or minimized (using sophisticated methods of varying the input  Newton’s Method is an unsophisticated example of this).   The method you learned in calculus of setting the derivative to 0 and solving explicitly is rarely of any use in science and engineering.

### mean

#### To form a definition

"Mean" may be used in forming a definition.

##### Example

"To say that an integer is even means that it is divisible by 2."

#### Implies

To say that an assertion P means an assertion Q may signify that P implies Q.

##### Example

"We have proved that 4 divides n. This means in particular that n is even."

Math authors do not always make it explicit whether they are using “mean” to give a definition or to describe an implication.

#### Average

Of course, mean is also a technical term, referring to the arithmetic average.

### minus

The word minus can refer to both the binary operation on numbers, as in the expression a  b, and the unary operation of taking the negative: negating b gives b. In current usage in American high schools, a  b would be pronounced “a minus b”, but b would be pronounced "negative b". The older usage for b was "minus b" and many old fuddy duddy college teachers like me forget and call it “minus b” sometimes.

Don’t assume a minus sign before an expression makes it negative.  The expression “t” denotes a positive number  if t happens to be 42.   Don’t be misled by the fact that we call it “negative t”.  See also subtraction.

“Minus” may also be used to denote set subtraction.

### modulo

The phrase “x is the same as y modulo E” means that x and y are elements of some set, E is an equivalence relation on the set, and x E y. There are variants to this usage:

¨  x is the same as y mod E.

¨  x = y modulo E ( or mod E).

¨  x is the same as y up to E.

The word arose from a special equivalence relation in number theory, which has two conflicting special notations in pure math and computing science that I have seen cause confusion among students.

#### Mod in number theory

For integers a, b and n, the expression  means that a  b is divisible by n.  (This relation is an equivalence relation.  See the number theory chapter).  For example,  and .  In this usage, the symbol “mod” occurs as part of a three-place assertion.

#### Mod in computer languages

In most computer languages and in computing science texts, the expression a mod n means the least nonnegative remainder obtained when a is divided by n.  For example, 23 mod 3 = 2 and

-4 mod 5 = 1.  (See number theory chapter for details.)  In this usage, mod is a binary operation on the integers.

#### Colloquial mod

People in math-related fields use “mod” in phrases such as “The administration kept my salary the same modulo [or mod] inflation”. Presumably the equivalence relation here is something like: “One dollar in 2002 is equivalent to \$1.02 in 2003.”

In particular, a statement such as “The alternating group on three letters is the same as the cyclic group of order 3 up to isomorphism” (less often, modulo isomorphism) refers to the equivalence relation of two structures being isomorphic.

### must

You often see "must be" used in math English when "is" would give the same meaning. It is used with verbs other than "be" in the same way. This may be to emphasize that the fact being asserted can be proved from facts known in the context of the discussion.

Other uses of "must" in mathematical discourse are generally examples of the way the words is used in ordinary discourse.

#### Example

"If m is a positive integer and 2m -1 is prime, then  m must be prime."

#### Example

"Let .  If , then one of A and B must contain two elements of C.