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This is a small, growing, compressed dictionary of elementary mathematic definitions I have found useful in my life.
This page is currently being translated from it's original in Swedish.
What is the meaning of the formula in the top row?
Some useful information in the Internet:
Arctan page :)
Derivatives, table
of
Graph plotter,
also derivative, (pow(x#2) for 
),
Integrator, Wolfram's
Absolute value,
abs, is written between two plumb beams | x |, and denotes
an always positive number , i.e.
x if x > 0,
0 if x = 0 and
-x if x <0. (so if x is -2 it is defined as the positive
value 2 because -(-2) = 2...)
| x |
0 equality only when x = 0. Comp absolute
value for complex numbers.
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Algebra (contr. arithmetic)
is to use letter symbols in counting which
leads to better validity.
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Algebraic (contr.
transcendent) is a number that is the root
of an algebraic equation of whole number koefficients. Ex.: 3, square
root of two.
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Angle is an ordered pair of half
beams emanating from the same point (= apex). The size of an angle is
the arc of the circle between the legs in a circle of radius = 1. The
unite is radians. An arc including the whole
circle then is the circumference of the circle
(because
the radius is set to r = 1).
It is natural to position the angle in the origin (origo) and one leg
in the x.axis. The angle is the arc x in the figures.
(From of old the circumference was parted in degrees, points etc., mostly
counted anticlockwise. If a turn is set to = 360 degrees, we have the
connection degrees/180 = radians/pi i.e.
and
180 degrees then is = pi, 45 degrees is pi/2 etc.)
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Arctan (arcus tangens, arctg) is
the inverse of tan (tangens) i.e. if tan(x) = y then
arctan y = x. This is used in calculating angles. Consider the slope
of 1,25 for a line, then it's inclination is
arctan(1,25) = 51,34 degrees (0,8961 radians).
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Area. Area of a square is length
* width , area of a triangle is (height * width)/2. Area (Y) of a circle
is
,
Area of a sphere is 
.
The area belov a funktion can often be calculated through means of an
integral.
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Arithmetic (contr.
algebra) to use natural numbers in counting.
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Arithmetic mean is given by
the formula
,
considering 3 (n = 3) numbers we get
, e.g. 3,
4 and 7 gives the arithmetic mean of (3+4+7)/3 = 4,67
Asymptote is a straight line
or curve to which another curve (the one being studied) approaches closer
and closer as one moves along it
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cathetus (pl catheti) The sides
of a triangle that is not opposite to the rectangular angle (where the
opposite side is called hypotenuse)
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Circle. Circumference 
,
area
The equation of a circle in implicit form i.e.
,
in explicit form 
The positive resp negative values denote the upper and lower half circles.
Compare angle. You get the equation of any
circle by 
where
it's center is at (h, k) and it's radius is r.
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Complex numbers (comp imaginary
unit)
. Often visualized in a quadratic system where x is the real part (a)
and y is the imaginary part (ib containing the imaginary unit).
real part 
,
imaginary part 
,
absolute value 
angle (or whatever) 
,
in polar form 
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Derivative (contr. integral)
is the slope of a function in a certain point.
A few important derivatives.
,
,
(example 
),
,
,
,
,
A Table is coming up here:
| Function | first derivative | comment |
| y = c | 0 | just a constant |
| y = x | 1 | slope is 1 |
Difference of two squares
(conjugate rule) 
= 
Just learn by heart! comp sum of two
squares.
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Distance between any two points
(x1, y1) and (x2, y2) in a two-dimensional system is readily seen to be
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Dividend is the upper part in
a fraction, e.g. in a/b a is the dividend and
b is divisor.
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Division is the operation of
parting dividend/divisor in a fraction. In Integer
operations there may be a remainder e.g. 5/4 = 1 remainder = 1. This is
also used in the modulo operation.
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Divisor is the lower part in a
fraction, e.g. in p/q p is dividend and q is divisor.
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Domain of a function
(jmfr range of a function) are the possible
values that an independant variable can be in some function.
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e is the base of the natural logaritm.
A transcendent number of the value 2,71828...
This number is the lime of
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Euler's formula 
where
i = imaginary unit,
ny = any real number (here an angle),
e base of natural logaritm.
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the Elementary functions are
polynome, rational functiones, the exponential function, the
potential function, logaritmic functions, trigonometrical functiones (sin,
cos, tg, cot) and arcus funktions (arcsin, arccos, arctg, arccot).
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Explicit (contr. implicit)
is an equation written so that the dependent variable stands alone to
the left like in y = 3x + 5.
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Factor (contr term)
is something separated by multiplication signs.
Ex.: 2x, y/2, abcd, pi*z are the factors 2, x, y, ½, a, b, c, d,
pi, x
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Factorial is the product of 1
* 2 * 3...n and is written n!. 0! = 1 as also is 1!
Ex.: 3! = 6, 4! = 24, 7! = 5040
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Fraction is two numbers in a
division situation, nominator/denominator,
legio in math.
Ex.: 
are five examples of fractions (may also be written a/b, 1/d, 3/6 etc),
are five
nominators and
are
five denominators.
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Function is a rule which assigns
how numbers are dependent of each other (like a number of cog-wheels in
mechanics). Often commonly denoted as f, u, v, g (f(x), u(x), v(x), g(x).
If a number is changed also other numbers change. These are called variables.
The one you change is called independent variable, the
other one that thus changes is called dependent variable.
A function f dependant of x is often written .y = f(x)
where x is the independant variable and y is dependant. To express the
value of a function for a certain value of the independent variable you
can write 
or commonly for all x 
Ex.: Consider function f where y
= 2 x + 3 (or f(x) = 2 x + 3). If you set x = 2, y = 7 we can as well
write f(2) = 7
If for instance the function t = u(s) all values
that s can have make the definition area of the function
and all values that follow for t is the domain of
the function.
When you write the dependant variable alone to the left side of the equation
you write it in explicit form. If the dependent
variable itself is written inside the equation it is said to be written
in implicit form. See circle
for an example of explicit och implicit forms.
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Graph. (same as locus)
A large number of points that visualize the values of a function.
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Greek alphabet. Just learn it!
You always have use for it. Here also in a rythm that works
top
 |
epsilón is pronounced with an accent on the last syllable. |
 |
omikrón is also pronounced with an accent at the last syllable |
 |
Also ypsilón is pronounced with an accent at the last syllable |
 |
l'Hospitals rule. Sometimes when
we study a fraction of two functions (e.g. f(x)/g(x)), we get meaningless
results (like 0/0) in limit operations, we can try this rule that simply
says that if the nominator (here f(x))and denominator (g(x)) have own
derivatives, they can be used (several times) in order to obtain a reasonable
limit value. (more to come here)
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Hypotenuse is the side against
the right angle in a rectangular triangle. The other two smaller sides
are called catheti.
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Imaginary unit, mostly written
i or j (in electrotechnical context), the most important quality of which
is, that it multiplied by itself will be -1, i.e. 
.
Hence it satisfies the equation 
.
Fundamental element in the theory of complex numbers. Comp complex
numbers.
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Implicit form (contr.
explicit) is an equation so written that the dependant
variable not stands alone.
Ex.: The equation of the circle is often written
where
y usually is the dependant variable..
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Inclination of a line having
the slope m is arctan(m) (in radians
or degrees).
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Inverse. Two functions
are inverse when their graphs are symmetrical through
the line y = x. You can create inverse functions by changing x and y and
then write it explicit again.
Ex.: what is the inverse of y = 2x - 1? (answer in next number...)
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Integers are "whole"
numbers like 1, 2, -8, 6, 7, -25, 2033 etc.. In many computer languages
obtained through the word int.
Ex.: INT(4,56) will generate the number 4. Se also sets
of numbers.
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Integral (contr. derivative)
is the area under a function inside a certain field.
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Intercepts are the distances
from the origin where a curve intersects the coordinate axes.
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Isosceles (two legs are equal)
and equilateral (all sides are equal) rectangular triangles
have simple defined values of the sides and thus of some trigonimetric
functions. In the equilateral triangle (to the left below) the height
is (pythagorean theorem) the square root of three, in the isoscele triangle
the hypotenuse is (pythagorean theorem) the square root of 2.
Limit (lat. limes) very important
operations when studying functions.
denoted like 
.
Given a function f(a) where a is changing towards a
certain value b. The function will, if possible, finally reach
the value "Limit value".
Values changing towards zero or infinity are commen and denoted 
respectively 
where
f(x) is some function of x. It is very important to check if the function
in question really has a limit, or value at all, in the investigated area.
Ex.: What will be the limit
for the function 
?
If we simply put in infinity for every x, the denominator will go against
1 - infinity, which is a meaningless expression. However, if we divide
all terms by x we get
,
now when we let x increase to infinity we see that the expression becomes
.
Answer: 
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Line. y = kx + b (standard form
of a straight line) where k, the slope of the line, is sometimes called
the slope coefficient of the line. Two lines are perpendicular when their
slope coefficients are minus unit 
.
Th inclination of the line is arctan(k)
Ex.: The slope of a line through the origin and (4,3)
is 3/4 and the inclination is 36,0 degrees.
The slope at a certain point in more complicated functions is obtained
through derivatives.
The equation of a straight line is easily obtained through the two
point formula.
| A few straight lines | |
| x = 0 | y-axis |
| y = 0 | x-axis |
| y + 5 = 0 | horisontal line through y = -5 |
| x = 3 | plumb line through x = 3 |
| y = x | 45 degrees inclination line through origin |
| y = -x | 135 degrees inclining line through origin |
Logarithm is written log, ln,
etc. If you rise the number 10 up to two you get the value 100 (10*10=100).
This number 2 is said to be the 10-logarithm for the number 100 and 10
is the base of the logarthm. 10-logarithm can be written
but nowadays often simply as log. The number e
is often used as the base and this may be written 
eller ln, this logarithm is said to be the natural logarithm.
In older books in higher mathematics natural logarithms were always written
simply as log, and if you explicitly wanted to use the 10-logarithmen
this was written
The logarithm of its own base is always unit (ln e = 1,
10 = 1 or for any base 
a
= 1). The logarithm of 1 is zero.
0,00001 = -5
Before modern calculators came into fashion tables or slide-rules were
used in calculations of logarithms. Most programming languages (perl,
C, visual basic) use log for the natural logaritm, modern calculators
mostly use ln for natural and log for
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Modulo is denoted mod
in many computer programming languages (e.g. perl, visual basic, C) and
means the remainder from an integer
division.
Ex.: 5 mod 2 = 1, 17 mod 5 = 2, 9 mod 3 = 0, 17 mod 9
= 8.
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Normal is a line perpendicular
to another line at a certatin point.
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ON-system (rectangular
coordinates). A coordinate system where the angle between the coordinate
axes x and y is ninety degrees (pi/2) and the axes are unit vectors of
the size 1.
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Origin (origo) is the point where
axes cross in a coordinate system.
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pi (grek. 
)
is a transcendental number of the value 3,141592...
(near 22/7 or yet better 355/113). Among other things the quotient between
the circumference and diameter of a circle. The
value 3,14 is satisfactory for most numerical calculations.
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Pythagorean
theorem In any rectangular triangle, the area of the square
whose side is the hypotenuse (the side of a rectangular triangle opposite
to the right angle) is equal to the sum of areas of the squares whose
sides are the two legs (i.e. the two sides other than the hypotenuse).
Using H = hyphotenuse, K = smaller sides, i = nearest and m = opposite,
you get
Quadratic ekvation 
is solved as
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Random values are often used in
various computer situations. In computer programs some value must first
be initiated e.g. by means of time. Then you get the numbers between 0
- 0,99 using RND or similar program words.
Typically generate integer values in a certain range by
where r is an integer random number, min, max are minimum and
maximum desired values, rnd is the computer word for generating random
numbers. Actually all random numbers generated this way are only pseudo
random, if you start with same number all following numbers will
run through the same series. Admittedly there's only a very small chance
that a start number is generated in exactly the same (micro)second of
the day.
Ex.: r = 5 + int((23 - 5) + 1)rnd will generate (pseudo)random
numbers between 5 and 23.
Range of a function (compare
domain of a function) is all the values
that a dependent variable can have (like x in some equation).
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Rational numbers (compare sets
of numbers) are numbers expressed exactly through the quotient of
any two integers where division of zero is not defined m/n like 1/4, 7/16,
-8/3, other numbers on the line of numbers are real numbers.
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Rectangular coordinates.
(ON-coordinate system). A plane is divided in four quadrants
around the axes X and Y.
From the picture below you also see the signs of the values e.g. in the
second quadrant the X-values are negativ and the Y-values are positive,
in the third quadrant all X-values are positive while the Y-values are
negative.
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Remainder is what may be left after
an integer division. e.g. 17/6 = 2, remainder is 5. This is used in the
practical modulo operation.
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Root of a function. Numbers satisfying
an equation.
Ex.: Write the equation y = x -5 as y - x + 5 = 0
Now, can x be given a value so that y really becomes zero? Yes, if x gets
the value of 5 y will be y - 5 + 5 = 0. Then the number 5 is said to be
the root of the equation.
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Root of a number. The square root of a real number x is such that the number multiplied by itself is x.
Ex.: Square root of 4 is written 
and is a number that multiplied by itself is 4, i.e. 2. 
means the number that multiplied by itselt tree times will be 8, thus
2*2*2 = 8 and
= 2 .
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Sine graph is formed from the
angle in a rectangular triangle having the radius (hypotenuse) = 1. Sine
then is the value of the opposite side that here is the same as the y-value
in the rectangular coordinate system.
From the figure is apparent that the signs of the upper two quadrants
are positive (as y is positive there), and the lower two quadrants are
negative.
Ex.: sin 0 = 0, sin 90 = 1, sin 180 = 0, sin 270 = -1
och sinus för ett helt varv är noll.
When the Y-values are drawn as a function of x, you obtain the graph (time
for instance):
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Slope of a line in a two-dimensional
system with axes x-y is x/y where y
0. Comp inclination.
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Sphere . The surface of the sphere
is ,
the volume of the sphere is 
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Sum of two squares (quadratic
rule) 
= 
Just
learn by heart! Comp. conjugate rule.
Sets of numbers in mathematics
are for example
natural numbers 1 , 2, 3, ...
or positive integers used in counting.
whole numbers..., -3, -2, -1, 0, 1, 2, 3, ...,
(see Integer)
rational numbers b/a where a
och b are whole numbers and a 0
(i.e. no division by zero!)
real numbers, comprising rational but also numbers that
cannot be expressed by fractions of whole numbers likeoch
.
However, all these numbers can be represented on the line of numbers.
.
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Symmetry of a function. Replace
x by -x, if the equation remains the same, the graph is symmetrical with
respect to the y-axis.
Replace y by -y, if the equation remains the same, the graph is symmetrical
with respect to the x-axis.
Replace x by -x and y by -y, if the equation remains the same,
the graph is symmetrical with respect to the origin.
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Term (contr. factor) is any number
separated by plus or minus.
Ex.: 2, 3 + a, x + 4b, g - p + 1 are the terms 2, 3,
a, x, 4b, g, p, 1
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Top line
formula.
(the first) derivative of the natural logarithm of the absolute
value of x is 1/x. Thus D is an altenative notation for the derivative.
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Transcendent is a number
that cannot be expressed agebraicly.
E x.: numbers e, pi
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Trigonometry is the science
of how angles are related to the sides of a rectangular
triangle. Any angle in the triangle have the following common functions:
sine for the angle alpha (see figure) is the opposite
side Km/hypotenuse, cosine is the nearest side Ki/hypotenuse,
tangens is the opposite side Km/nearest small side Ki,
(cotangens is the nearest side /opposite side). The funcions
are written sin, cos, tg
(tan). From this follows that tangens for 90 degrees has no definition
as the denominator never can be zero.
H =hypotenuse, K = smaller side, m = opposite, i = nearest.
An important and very useful identity for some trigonometric functions
is the trigonometric "unit" 
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Two point formula.(compare line)
Knowing two points 
and 
in a two dimensional ON-system the equation of
a straight line can be expressed as
The fraction lined part in the expression is the slope of the line (often
denoted k or m), and the inclination of the line (in
radians or degrees) is arcsin of the slope.
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Vertex (lat. corner, pl verticis). Corner of a geometrical figure (e.g. polygone, triangle), node (e.g. max or min) in a graph.