A symbol used to distinguish one quantity 
("
prime") from another related 
. Prime marks are most commonly used to denote
1. Transformed coordinates,
2. Conjugate points,
3. derivatives,
4. The complement 
of a set 
,
5. As an alternate notation for transpose.
Of course, the term "prime" also refers to a prime number. The Wolfram Language command giving the 
th prime number is Prime[n].
Bar
A bar (also called an overbar) is a horizontal line written above a mathematical symbol to give it some special meaning. If the bar is placed over a single symbol, as in 
(voiced "
-bar"), it is sometimes called a macron. If placed over multiple symbols (especially in the context of a radical), it is known as a vinculum. Common uses of the bar symbol include the following.
1. The mean
 |
of a set 
.
2. The complex conjugate
 |
for 
.
3. The complement 
of a set 
.
4. A set stripped of any structure besides order, hence the order type of the set.
In conventional typography, "bar" refers to a vertical (instead a horizontal) single bar such as those used to denote absolute value (
) (Bringhurst 1997, p. 271).
Hat
The hat is a caret-shaped symbol commonly placed on top of variables to give them special meaning. The symbol 
is voiced "
-hat" (or sometimes as "
-roof") in mathematics, but is more commonly known as the circumflex in linguistics (Bringhurst 1997, p. 274).
Uses of the hat in mathematics include:
1. To denote a unit vector (e.g., 
) or an estimator (e.g., 
).
2. To denote an estimator, e.g., 
for the sample mean.
3. As the growth rate of 
in hat calculus, e.g., 
(Jones 1965, who however used the symbol 
instead of a hat).
Tilde
The tilde is the mark "~" placed on top of a symbol to indicate some special property. 
is voiced "
-tilde." The tilde symbol is commonly used to denote an operator. In informal usage, "tilde" is often instead voiced as "twiddle" (Derbyshire 2004, p. 45).
1. An operator such as the differential operator 
.
2. The statistical median 
(Kenney and Keeping 1962, p. 211).
The tilde is sometimes used as its own symbol.
1. In asymptotic notation, 
is used to mean that 
.
2. Physicists and astronomers use same notation to mean "
is of the same order of magnitude as 
."
3. In set theory, 
means that there is an equivalence relation between 
and 
.
4. In statistics, the tilde is frequently used to mean "has the distribution (of)," for instance, 
means "the stochastic (random) variable 
has the distribution 
(the standard normal distribution). If 
and 
are stochastic variables then 
means "
has the same distribution as 
.
