Mr. Hansen’s Abbreviations, Part II

(rev. 9/12/2003, 10/17/2004, 1/9/2005, 5/20/2005, 7/6/2008, 9/7/2013)

Part II: Standard abbreviations and symbols that most mathematicians would understand

for any ...

(Hebrew letter aleph): cardinality of the integers, cardinality of the reals

a.e.

almost everywhere

aka, a.k.a.

also known as

arcsin x, arccos x, arctan x, arccsc x, arcsec x, arccot x

same as sin–1 x, cos–1 x, etc.; these are the inverse trigonometric functions

ASTC

memory aid (“All Students Take Calculus”) for the signs of the Big 3 trig functions:

All 3 are positive in Quadrant I.
Sine is positive in Quadrant II. (Of course, so is the cosecant.)
Tangent is positive in Quadrant III. (Of course, so is the cotangent.)
Cosine is positive in Quadrant IV. (Of course, so is the secant.)

bwoc

by way of contradiction

C

{complex numbers}

C

*constant of integration

nCr or

combinations: the number of ways of choosing n things taken r at a time, where order does not matter

is (is not) a subset of

cf.

compare with

CLT

**Central Limit Theorem

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

lowercase delta (usu. a small positive constant)

uppercase delta (denotes change in whatever follows; e.g.  means “change in x”)

d/dx

*derivative operator wrt x

deg(P)

degree of polynomial function P

det(X) or |X|

determinant of X (defined only if X is a square matrix)

df

**degrees of freedom

Df

domain of function f

diffeq.

*differential equation

DNE

does not exist

Dx

*derivative operator wrt x (same as d/dx)

dy/dx

*derivative function (instantaneous rate of change of y wrt x)

*partial derivative of y wrt x

d2y/dx2

*second derivative function of y wrt x

there exists ... (with a slash through it, means “there does not exist ...”)

!

there exists a unique ...

such that

:

such that

|

such that

epsilon (usu. a small positive constant)

is (is not) an element of
e

base of the natural logarithm; approx. 2.718281828459

e.g.

for example (exact translation from Latin: exempli gratia)

ES

**effect size

et al.

and others (exact translation from Latin: et alii)

et seq.

and onward (loose translation from Latin: et sequens)

EVT

*Extreme Value Theorem

exp(x)

ex

E(X)

**expected value of random variable X (same as mX)

f (x)

function f evaluated at x; never means multiplication

f n (x)

f (x) raised to the nth power; e.g., sec3 x means (sec x)3
EXCEPTION: f –1 always means the inverse of f, not the reciprocal of f

f (n) (x)

*the nth derivative of function f, evaluated at x

FTC, FTC1, FTC2

*Fundamental Theorems of Calculus (textbooks differ; see note [1] below)

GCD, gcd

greatest common divisor

GCF, gcf

greatest common factor (obsolete term; equivalent to gcd)

H0:

**null hypothesis

Ha:

**alternative hypothesis

i

the square root of –1, i.e., the principal complex solution of the equation x2 = –1

i2

should always be simplified as –1

i3

should always be simplified as –i

i4

should always be simplified as 1

i, j, k, m, n

typical integer variable names (e.g., using i as a term index, we speak of the ith term)

i.e.

that is (exact translation from Latin: id est)

iff

if and only if (same as the symbol Û)

indep.

independent

indet.

indeterminate form (e.g., 0/0, /, , 0 · , ,   – , or 00)

IVT

*Intermediate Value Theorem

LCM, lcm

least common multiple

(L’Hôp.)

*use of L’Hôpital’s Rule is justified at the indicated place; you should indicate 0/0 or / as well

LHS

left-hand side of an equation or implication

lim

limit (pay special attention to the subscripts; e.g., since the expression means “take the limit of x2 as x approaches 3 from above”)

ln x

natural logarithm of x (pronounced “el en eks”); i.e., the exponent that you place upon e in order to get x

log x

common (base-10) logarithm of x; i.e., the exponent that you place upon 10 in order to get x

log29 x

(Here, 29 is merely used as an example.) This means the base 29 logarithm of x, i.e., the power to which you could raise 29 in order to get x. In other words, if y denotes this power, we are saying y = log29 x iff 29y = x.

Although the TI-83 calculator does not have a key for computing this, you can find the answer by dividing any convenient log of x by that same log of 29. For example, to find log29 748, you would punch in log(748)/log(29) ENTER to get an answer of 1.965, correct to 3 decimal places. Note that you get the same result if you use natural logs, since ln(748)/ln(29) is also 1.965. Both of these results are true as a result of the Change-of-Base Formula.

LOLN

**Law of Large Numbers

**population mean

**hypothesized mean (i.e., the mean used in H0)

**mean (a.k.a. expected value) of random variable X

min, Q1, med, Q3, max

**minimum, first quartile, median, third quartile, maximum (collectively called the “five-number summary”)

m.o.e. or MoE

**margin of error

MSE

**mean square error (an estimate of the population variance)

mutatis mutandis

“assuming that the necessary changes have been made” (useful when a proof is correct except for some notational change that needs to be applied throughout)
MVT

*Mean Value Theorem
n

term number or number of terms, depending on context

n

**sample size

N

{natural numbers}

N

**population size

N.B.

note (from Latin nota bene: “note well”)

NPP, NQP

**normal probability plot, normal quantile plot (synonyms)

null set; can also be written as { } but not as {}

ODE, o.d.e.

*ordinary differential equation

or

[means that first expression is true, or second expression is true, or both are true; different from xor]

p

**depending on context, can mean either (1) the single-trial probability of success, or (2) the conditional probability (given H0 true and chance alone being the only force at work) of obtaining a test statistic as extreme as, or more extreme than, the observed value
P

**same as second definition of p (also called the P-value)

**sample proportion

ratio of any circle’s circumference to its diameter; approx. 3.14159265358979323846

continued product (similar to , except using multiplication instead of addition)

nPr

permutations: the number of ways of arranging n things taken r at a time, where order matters

P(A)

unconditional probability of event A

P(A  B)

probability that events A and B both occur; there are two formulas for this:
1. P(A  B) = P(A) P(B | A) is always true
2. P(A  B) = P(A) P(B) is true provided A and B are indep. (reason: if A and B are indep., P(B | A) = P(B), so formula #1 can be written in the special form of #2)

P(A  B)

probability of A or B (i.e., the probability that either A or B, or both, will occur); again, there are two formulas:
3. P(A  B) = P(A) + P(B) – P(A  B) is always true
4. P(A  B) = P(A) + P(B) is true provided A and B are disjoint (reason: if A and B are disjoint, P(A  B) = 0, and hence formula #3 can be written in the special form of #4)

P(B | A)

conditional probability of event B given A; from #1 above, there is a formula for this:
5. P(B | A) = P(A  B) / P(A)

PDE, p.d.e.

*partial differential equation

PPV

**positive predictive value

q

**probability of failure on one trial (i.e., 1 – p)

Q

{rationals}

Q1, Q2, Q3

**first quartile, second quartile (usu. called the median), third quartile

QED or Q.E.D.

done! (from Latin quod erat demonstrandum, literally meaning “which was to be shown”)



[Halmos sign] same as Q.E.D.

q.v.

indicates that you should look something up using the indicated reference (from Latin, literally meaning “which see”)

{real numbers}

2-dimensional space, i.e., the Cartesian plane

3-dimensional space, i.e., space described by x-, y-, and z-coordinates

n-dimensional space

re (or in re)

in the matter of (re is a Latin word, and although it is not an abbreviation for “regarding,” you can think of it that way)

resp.

respectively

Rf

range of function f

RHS

right-hand side of an equation or implication

Rule of 72

(a.k.a. Rule of 69)
If an investment (or population, or whatever) is growing at w% per year, then the time required for each doubling is approximately 72/w years. More generally, a quantity that grows exponentially, with a periodic growth rate of w%, will double in approximately 72/w periods. In practice, 72 is used because it has many divisors, but 69 gives better accuracy if the growth is continuously compounded. In the world of finance, for investments having a nominal (usu. annually compounded) yield of w%, the Rule of 72 works quite well and is preferred because it is somewhat conservative.

Rule of 78

(mostly obsolete)
This was a method once used by banks for calculating the interest rebated to the borrower if the borrower paid off the loan early. Modern software and the prevalence of simple-interest loans have made this rule obsolete, though in some states it is still used by unscrupulous lenders trying to boost their interest earnings. By law the Rule of 78 is now prohibited nationwide in mortgages and long-term consumer loans, and many states prohibit the Rule of 78 in all loans.

r.v.

**random variable

s, s2

**sample standard deviation, sample variance

**s.e. of the regression slope
Note: Although your stat textbook contains some fiendishly difficult formulas for calculating , there is an easy way. Since the t statistic for regression is defined to be b/, where b1 denotes the slope, just solve the equation t = b/ for . If you do this, you will get a formula for  that is much easier.

§

section

§§

sections

**population standard deviation, population variance

**standard deviation and variance of random variable X

summation (shorthand for continued addition)

s.d. or std. dev.

**standard deviation

s.e. or SE

**standard error

**standard error of the regression slope (same as ; see entry above)

sgn(x)

signum (sign) function: returns 0 if x = 0, returns 1 if x > 0, and returns –1 if x < 0

Examples:
sgn() = 1
sgn(–15) = –1
sgn(48 – 96/2) = 0
sgn(3 – 3/2 + 3/4 – 3/8 + 3/16 – 3/32 + . . .) = 1

sin

sine function of , where  is usu. in radians

SOHCAHTOA

Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent
More details . . .

SRS

**simple random sample

SSE

**sum of squared errors (i.e., sum of squared residuals)

SSM

**sum of squares about the mean

statistics theorems
--CLT
--LOLN
--Empirical Rule
--Normal Rule
--CT

**
Central Limit Theorem
Law of Large Numbers
same as the 68-95-99.7 rule; applies only to normal distributions
another name for the Empirical Rule
Chebyshev’s Theorem

T

indicates matrix transposition; e.g., AT is the transpose of matrix A
t

**value of the standardized t statistic
t*

**critical t value (pronounced “t star”)

TBA, TBD

to be announced, to be determined

trig

trigonometry, trigonometric

Type I

**inference error caused by rejecting a true H0

Type II

**inference error caused by failing to reject a false H0

union

usu.

usually

var(X) or V(X)

**variance of random variable X (same as )

vbl.

variable

vbls. or vars.

variables

viz.

namely

wlog

without loss of generality

wrt

with respect to

x or xi

**data value or observation

**sample mean

X, Y, Z, etc.

**names of random variables

**value of the chi-square statistic

xor ,

exclusive or [means that first expression is true, or second expression is true, but not both]

Examples:
    “You will clean up your room xor you will get no dinner!”
    (Note that parents usually say or when they should really say xor.)
    0  0 = 0
    0  1 = 1
    1  0 = 1
    1  1 = 0
    10110101  00110110 = 10000011

estimated value for y

*usu. dy/dx (or dy/dt), depending on context

*usu. d2y/dx2 (or d2y/dt2), depending on context

*third derivative

yiv or y(4)

*fourth derivative

y(n)

*nth derivative

z

a complex number (often written in a + bi format)

z

**standard normal score (note: when writing, always cross the z)

z*

**critical z value (pronounced “z star”)

| z |

modulus (generalized absolute value) of z; if z = a + bi, then | z | = (a2 + b2)1/2

complex conjugate of z; if z = a + bi, then = a – bi

Z

{integers}

is proportional to

–1

inverse function (superscript –1 after a function); does not mean reciprocal of the function

[12, )

interval notation for {x } or, equivalently, {x }
(note that  can never be next to a square bracket)

[12, 17]  [–2, 4.3]

a subset of , or to use plain English, a 2D graphing window (hit WINDOW key on TI-83): xmin = 12, xmax = 17, ymin = –2, ymax = 4.3

[ ]

optional argument, as in fnInt(function, X, lowerbound, upperbound [,])

[ ]

matrix (when used to surround any rectangular grid of numbers)

[x]

greatest integer function (greatest integer less than or equal to x)

| x |

absolute value function (see also | z | above)

| X |

determinant of X (defined only if X is a square matrix)

floor of x (same as greatest integer function)

ceiling of x (i.e., least integer greater than or equal to x)

intersection

<< , >>

much less than, much greater than

and

or

~

not

~

similar (geometry)

approximately equal (note: in Statistics, we often use = instead, since  is understood)

is defined as, or is identically equal to (i.e., equal regardless of the values of the variables)

corresponds to (in higher math courses, this symbol indicates a one-to-one correspondence, i.e., a mapping or function that is one-to-one and “onto”)

approaches

grows without bound

contradiction

implication (“only if”)
We read A  B in any of the following ways:
(1) A implies B.
(2) A only if B.
(3) If A, then B.
(4) A is a sufficient condition for B.
(5) B is a necessary condition for A.

reverse implication (“if”)
We read A  B in any of the following ways:
(1) B implies A.
(2) A if B.
(3) If B, then A.
(4) A is a necessary condition for B.
(5) B is a sufficient condition for A.

iff (necessary and sufficient condition)
This means the statement is biconditional: LHS  RHS and RHS  LHS.

contains

\

without
This symbol indicates “subtraction” of sets. For example, to say
Df =  \ {7} means that the domain of f is any real number except for the number 7.

. . .

pattern continues (e.g., 1 + 2 + 3 + . . . + 100 denotes the sum of the first 100 positive integers)

since

therefore

!

factorial

Notes:
* Calculus only.
** Statistics only.
[1] Textbooks differ on what to call FTC1 and FTC2. We shall define them as shown below in order to be consistent with the sequence in the Foerster textbook and the recommended Web resource, Eric Weisstein’s World of Mathematics. However, be aware that many textbooks (including the one by Finney, Demana, Waits, and Kennedy) label FTC1 and FTC2 in reverse order.

FTC1: The definite integral from a to b of a continuous function f is G(b) – G(a), where G is any antiderivative of f.

FTC2: Let f (t) be a continuous function and define a new “accumulator function” H(x) to be the definite integral of f (t) dt, where the lower limit is a constant and the upper limit is x; i.e., H(x) = f (t) dt. Then .

FTC2 in words: “The rate of change of an accumulator function is given by the curve for which the area is being accumulated.” This is a very deep result, but when expressed in this way it sounds almost like a tautology.

The proof of FTC1 is found in virtually all calculus textbooks. A clever proof of FTC2 (thanks to Braxton Collier) and a proof that FTC1 and FTC2 imply each other (i.e., are equivalent) are both worth reading.