HOW TO READ MATHEMATICAL NOTATION IN ENGLISH.

(“in order of appearance”)

Arithmetical Operations (formula manipulation)

Rules, Laws

Algebra deals with numbers, objects and mathematical operations on them

Arithmetic operations on ….matrices

PRELIMINARIES

A Digit

Natural numbers 

Prime numbers 

Integers, whole numbers , 

Rational numbers Q

Irrational Numbers Q^'

Real numbers 

Complex numbers 

positional number system of (with) base n

convert from base n to base m

decimal representation,

every real number can be representaed by a "nonterminating decimal"

-deviation from exact value

2 binary operations

3 ternary

4 quaternary

5 quinary

6 senary

7 septimal

8 octal

9 monary

10 decimal

12 duodecimal

16 hexadecilmal

+ a sum, to add

- a difference, to subtract

* a product, to multiply, times

/ quotient, ratio, to divide, (2/3 : two over three)

written addition, 2+9 is 1 "carry" 1 (w pamieci)

written multiplication,

long division,

A+B = B+A commutative law

A+(B+C) = (A+B)+C associative law

A*(B+C) = A*B+A*C 1-st distributive law

A*B+A*C = A*(B+C) - FACTOR OUT (the) 'A', to pull out a factor A (a term A), extract

-A is negative of A

< less than

less equal than, smaller than

> greater than, larger than

greater equal than

order of operations

reciprocal,( i.e. number: 2, reciprocal:
) (two reciprocal:
)

! factorial, 3! - three factorial

-"n choose k", binomial coefficient

x to the power n , x is the base of the exponent function, n is the exponent

2⁴ - two to the fourth

x² x square

x³ x cubed

n-th root of x

square root

cubic root

extract a root

point at infinity

a sub 1, subscript, eh one

a sup 1, superscript

a bar,

a hat,

f prime ,

identity

expression

( ) parentheses (am.), brackets (brit.)

( { [ round, curly, square brackets

omit brackets, drop, leave, dissolution of brackets

factor an expression, cancel factors

expand an expression

collect

replace

cancel terms, by cancelling from both sides

simplify

fraction,

, least common denominator

three over two

cross multiplication

SETS

union

intersection

element

subset, in

proper subset

empty set, null set, void set

U universal set, universe of discourse

disjoint, not disjoint

and, alternation, disjunction

or, conjunction

if then, implication

not, negation

iff, equivalence

quantifiers

∀ for all, any

∃ there exists, for some

relation

relation for set S is a set of ordered pairs

a pair (a,b) is in relation :aRb

reflexive relation (x,x) Є R

symmetric relation (a,b) Є R →(b,a) Є R

transitive relation

theorem, lemma, corollary, claim, remark

conditions => conclusion

mathematical induction, transfinite induction

induction problem

proof by induction

Intuitively, the inductive (second) step allows one to say, look P(1) is true and implies P(2). Therefore P(2) is true. But P(2) implies P(3) Therefore P(3) is true which implies P(4) and so on. Math induction is just a shortcut that collapses an infinite number of such steps into the two above.

Direct proof: where the conclusion is established by logically combining the axioms, definitions and earlier theorems

Proof by induction: where a base case is proved, and an induction rule used to prove an (often infinite) series of other cases

Proof by contradiction (also known as reductio ad absurdum): where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false.

Proof by construction: constructing a concrete example with a property to show that something having that property exists.

Proof by exhaustion: where the conclusion is established by dividing it into a finite number of cases and proving each one separately

modular arithmetic, 'clock arithmetic'

"mod" operator

congruent

congruent modulo

remainder

a is congruent to b modulo m.

15:4 r.3 fifteen divided by four gives you a remainder of 3.

multiplication, addition (subtraction,division) tables

Greatest Common Divisor (d&-'vI-z&r)

relatively prime, coprime

the decomposition of a number into prime factors (unique apart from order)

Euclidean algorithm

Chinese Remainder Theorem

GEOMETRY

point lies on a straight line

plane passes through points…

points on a coordinate diagram

right-angled triangle, legs

straight angle, full angle

hypotenuse, opposite side , adjacent side (&-'jA-s&nt-)

isosceles, equilateral triangle

inclined at an angle of 45 deg.

obtuse, acute

bisector, bisectrix

vertices of triangle, square

altitude, altitude to the corresponding side

sine, cosine, tangent: One familiar mnemonic to remember these definitions is SOHCAHTOA. It reminds one that "SOH", sin = opposite/hypotenuse, "CAH", cos = adjacent/hypotenuse, and "TOA", tan = opposite/adjacent.

Pythagorean Theorem

circle with center P and radius r

coordinate system, origin of …

inscribed in a circle, interior of a circle,

chord

names of polygons (3- trigon, triangle, 4- tetragon, quadrilateral, 5- pentagon, 6- hexagon, 8- octagon)

to draw marks or a line

plot a graph to show….

sketch, shade, graph, mark

Sketch, by shading the appropriate area, the set … on the coordinate diagram of RxR.

plot on a plane

such … will plot as a straight line

contour

at a point

rectangular coordinate system

coordinates, point with coordinates (a,b)

origin

three-dimensional space, 3-space

intersect

left/right half plane

COMPLEX NUMBERS

complex number,

conjugate of … ('kän-ji-g&t)absolute value, modulus

real, imaginary part,

polar form, trigonometric representation

radius, argument,

determined only up to multiples of 30*

de Moivre's Formula

points on a coordinate diagram

multiple-valued functions, many valued, multiform

single valued, uniform function

POLYNOMIALS

polynomial of degree n in x over field F

coefficient, x variable,
constant,
leading element,

if
, the polynomial is called monic

(a+x)ⁿ binomial,
binomial coefficient

quadratic equation, quadratic trinomial

discriminant Δ

factor a polynomial into (…)(…)….(….)

monomial - axⁿ

polynomial solvable by radicals

root, distinct roots of a polynomial,

zero of P(z) of the order h

completing the square, i.e. translate the eq.
into

solve equation using the quadratic formula

terms

reduce

decompose

eliminate

Fundamental Theorem of Algebra

MATRICES

matrix, table with assigned mathematical operations

In general, a matrix (plural matrices) is something that provides support or structure, especially in the sense of surrounding and/or shaping. It comes from the Latin word for "womb", which itself derived from the Latin word for "mother", which is mater. Various disciplines use the term "matrix" with differing meanings.

order, matrix of order m x n

size of matrix, square matrix, m by n

lower triangular matrix

upper triangular matrix

diagonal m.

scalar

identity matrix

rows

columns,

diagonal,

coefficient matrix

transpose of a matrix

Gaussian elimination

(reduced) row echelon form, equivalence of matrices

elementary row operations (transformations)

leading element

system of m equations in n unknowns, coefficients

solve the system by means of …,

consistent, inconsistent, dependent,

unique

sub-matrix - obtained by removing elements of whole rows and/or columns

constants

determinant,

minor - determinant of sub-matrix

cofactor - signed minor (-1)^i+j |A_ij|

adjoint matrix - the transpose of matrix of cofactors

expand along a row (column)

inverse of a matrix, invertible (Forward)

rank of matrix

solution by Cramer's Rule (familiar solution by determinants)

rank

Find the relationship between the value of parameter "a" and the number of solutions of the system:

Determine the number of solutions without solving the system:

Determine how many solutions the system … has.

Find an equivalent Cramer system (square system) of eq.

pivot /'pi-v&t / (non zero element in reduced row echelon form)

LINEAR VECTOR FIELDS

linear space over a field

m-vector or m-dimensional vector (ordered set of m elements)

components of a vector

origin and terminus of a vector, head and tail

n-tuple, ordered triple of numbers

closed under scalar multiplication

closed under addition

set of "vectors" on which you define operations….

linear dependence, (for 2- (3-) space not collinear (coplanar) vectors

basis of ….

basis of a subspace, pl. bases - the minimum number of vectors required to span a space

unit, elementary vectors, [1,0,0,..],[0,1,0,…]

dimension (dI-men(t)-sh&n)

coordinates relative to the basis

let v, w be two vectors of V, their coordinates being relative to the same basis of the space

coordinates of vector with respect to given basis

linear combination

span a lin. space, spanned by vectors

generate a …, generated by vectors

syst. of homogeneous equation

[0,0,…] trivial solution

solution space,

For Ax=0 the solution vectors constitute a vector sp. called the null space of A

The dimension of this space is called the nullity of A

Complete solution = particular solution + complete solution of hom.syst

sprain (skrecenie w stawie, med)

linear transformation, mapping, the matrix of the transformation

kernel

image = f(domain)

congruence Objects which are exactly the same size and shape are said to be congruent

similarity - contraction, amplification

homothety (houm'otheti)

Two figures are homothetic if they are related by an expansion or geometric contraction This means that they lie in the same plane and corresponding sides are parallel; such figures have connectors of corresponding points which are concurrent at a point known as the homothetic center.

homothetic transformation (jednokladnosc wg.Kopalinskiego i Terlikowskiej)

homothety with center

homothety with coefficient

concurrent (existing or happening at the same time)

reflection in a line

line reflection

reflecting over a line

Exactly one line that passes through two points;

The line provides the shortest connection between the points.

Two different lines can intersect in at most one point; two different planes can intersect in at most one line.

Remember, a reflection is often called a flip. 
 Under a reflection, the figure does not change size.
 It is simply flipped over the line of reflection.

reflection in a point e.g. reflection in the origin

point reflection

This diagram shows points A and C reflected through point P

rotation - turns a figure around (about) a point (with respect to a point) eg. 45 degrees

rotate 45 degrees

rotate by (through) an angle of theta

line is inclined at angle of 45 degrees

symmetry

axis of symmetry (about a line), axial symmetry

center of symmetry

line of symmetry

…. is symmetric about the x-axis, about the 'L' line, about the point 'P'

symmetry with respect to (about) v1 along v2

to mirror about the x-axis:

mirror image

translation

to translate by v

dilatation

A dilatation produces an image which is the same shape as the original, but is a different size.  The description of a dilation includes the scale factor and the center of the dilation.

A dilation of scalar factor k whose center of dilation is the origin may be written:  D_k(x,y) = (kx,ky).

projection of space V on v1 along v2

image of … under transformation

the transformation … carries vectors into …

Jordan normal form

block square matrix (nilpotent)

Matrix A has the form ….

the matrix of a transformation L: Fⁿ ->F^m

transition matrix from one basis to another

Gramian determinant

Euclidean space

orthogonal decomposition

dot product (inner, scalar )

vector product

diagonable matrices

approximate - adjective

approximate - verb

CALCULUS

f(x) (eff of ex)

function, F: Domain -> Co-domain (Range - Kuratowski)

Image = F(domain)

one-to-one mapping : F(x1) =F(x2) => x1=x2 - injection

onto mapping : every element of the codomain is the image under f of at least one element of the domain- surjection

one-to-one correspondence: one-to-one mapping and onto- bijection

x is in the domain

df/dx (dee f over/by dx) the derivative of x with respect to x

iterated integral

double integral

"region perpendicular to", divide the region into slices parallel to the y-axis and add up the …

slice the region perpendicular to….

inverse function, forward function

mirror image, to mirror about the bisector, flip over line…

partial fraction decomposition

trigonometric functions

period

sine cosine tangent cotangent,

arc sine (symb. arcsin)..

sine, cosine, tangent- tan

exponential function

logarithm, base of log

limit

sequence, terms

increasing, decreasing

convergent

squeeze principle, sandwich theorem, pinch test

tends to, approaches

as … approaches infinity

piecewise defined function

point at which we piecing two functions

slope

product rule

chain rule:

The derivative of a composite function is the derivative of the

outside function times the derivative of the inside function.

extremum (pl.extrema)

minimum, maximum

line tangent

suspect

attain value, assume value,

take on a value

'assign a value'

concave (down)

convex (up)

inflection point

Taylor series- power series expansion for f(x) about the point x=x_0,

expansion with respect to x, of order n,

truncated series

to truncate a series

coefficients

convergence

approximation

partial fraction, ratios of polynomials, where the degree of each numerator polynomial is less than that of the corresponding denominator polynomial.

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