Title: AJ
Tags: mff_vocabulary math_notation geometrie

Q: úhel
A: angle

Q: ostrý (úhel)
A: acute

Q: pravý (úhel)
A: right

Q: tupý (úhel)
A: obtuse

Q: přímý (úhel)
A: straight

Q: nekonvexní (úhel)
A: reflex

Q: přilehlý
A: adjacent

Q: výška
A: altitude

Q: obsah / plocha
A: area

Q: osa – osy (mn. č.)
A: axis – axes (pl.)

Q: základna / podstava
A: base

Q: střed
A: centre (center in NAmE)

Q: kruh / kružnice
A: circle

Q: obvod (kruhu)
A: circumference

Q: kužel – kuželovitý (příd. jm.)
A: cone – conical (adj.)

Q: kuželosečka
A: conic section

Q: parabola

Q: hyperbola

Q: elipsa
A: ellipse

Q: krychle
A: cube

Q: válec
A: cylinder

Q: průměr
A: diameter

Q: hrana
A: edge

Q: rovnostranný
A: equilateral

Q: stěna
A: face

Q: ohnisko – ohniska (mn. č.)
A: focus – foci (pl.)

Q: přepona
A: hypotenuse

Q: protnout / protínat se (slov.)
A: intersect (v.)

Q: úsečka
A: line segment

Q: přímka
A: line

Q: rovnoběžky / rovnoběžné přímky
A: parallel lines

Q: různoběžky / protínající se přímky
A: intersecting lines

Q: mimoběžky
A: skew lines

Q: protilehlý / protější
A: opposite

Q: rovnoběžný / rovnoběžka
A: parallel

Q: obvod
A: perimeter

Q: kolmý / kolmice
A: perpendicular

Q: rovina
A: plane

Q: bod
A: point

Q: mnohoúhelník
A: polygon

Q: mnohostěn
A: polyhedron

Q: poloměr
A: radius

Q: obdélník
A: rectangle

Q: strana
A: side

Q: prostorový útvar / těleso
A: solid figure

Q: koule / sféra
A: sphere

Q: čtverec
A: square

Q: tečna – dotýkat se / být tečnou k (slov.)
A: tangent line – (v.) to be tangent to

Q: trojúhelník
A: triangle

Q: rovnostranný trojúhelník
A: equilateral t.

Q: rovnoramenný trojúhelník
A: isosceles t.

Q: ostroúhlý / tupoúhlý / pravoúhlý trojúhelník
A: acute / obtuse / right t.

Q: vrchol – vrcholy (mn. č.)
A: vertex – vertices (pl.)

Q: \(A_{m\times n}\)
A: an m-by-n matrix A
Note the position of the dimensions before the matrix itself.

Q: \[\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\]
A: a two-by-two matrix, (where) the first row is 1, (and) 2, the second row is 3, (and) 4
a two-by-two matrix, (where) the first column is 1, (and) 3, the second column is 2, (and) 4
For larger matrices simply add more columns/rows.

Q: \[\begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix}\]
A: an m-by-n matrix, the first row is a sub one one, a sub one two, dot dot dot, a sub one n, second row a sub two one, a sub two two, dot dot dot, a sub two n, third row dot dot dot, mth (or last) row a sub m one, a sub m two, dot dot dot, a sub m n

Q: \(A^T\)
A: A transpose
the transpose of A
It is not usual to say *A transposed which would imply the action that happened to the matrix.

Q: \(\det(A)\) or \(|A|\)
A: the determinant of A
It is always the determinant, as the number is uniquely associated with each matrix.

Q: \(A^{-1}\)
A: A inverse
the inverse of A
the inverse matrix A

Q: \(a_{ij}\)
A: a sub i j
the ij-th entry/element of matrix A

Q: \(a + b = c\)
A: a plus b is equal to/ equals c
the sum of a and b is/ is equal to/equals c
Note Let a + b = c be … is read as “Let a plus b equal to c be …”, i.e. dropping “is” when used as a way of rewriting a given quantity as a sum or vice versa (the equality is a fait accompli rather than an active assertion). A similar remark applies to a – b = c, ab = c, a/b = c.

Q: \(2 + 3 = 5\)
A: two plus three equals/makes five
two added to three makes five
the sum of two and three equals five
And the reverse order, e.g. five is the sum of two and three.
Two plus three is five is less formal but acceptable;
two and three is/are five is less formal still and not sufficiently clear without context.

Q: \(a - b = c\)
A: a minus b is equal to/ equals/makes c
a take b is equal to/ equals c
the difference of a and b is/ is equal to/ equals c
The first is most common; the second less formal.
“the difference between” two numbers is also possible.

Q: \(5 - 3 = 2\)
A: five minus three equals/makes two
three from five leaves two
five diminished by three is equal to two
two is/equals the difference of five and three
Less formal but acceptable is five take three is two (or five take three equals/is equal to two).
The “diminished by” formulation is uncommon.

Q: \(ab = c\)
A: a times b equals c
a multiplied by b equals c
the product of a and b is/ is equal to/ equals c
ab equals c

Q: \(2 \cdot 3 = 6\)
A: two times three equals/makes six
two multiplied by three equals six
the product of two and three is/ is equal to/ equals six

Q: \(a \div b = c\)
A: a divided by b equals c

Q: \(6 \div 3 = 2\)
A: six divided by three equals/makes two

Q: \(\frac{a}{b}\)
A: (the fraction) a over b;
a divided by b
fraction of … is incorrect

Q: \(\frac{a+b}{c+d}\)
A: (the fraction equal to) (the sum/ the quantity) a plus b over/ divided by (the sum/ the quantity) c plus d;
the fraction whose numerator is ... and whose denominator is ...
If the numerator A in A/B is a lengthy expression, read as “A, all (that) over B” or “A, that quantity (then) divided by B”.
The correct terms numerator and denominator are sometimes erroneously called the *nominator (=a person who nominates) and *denumerator (=nonsense).

Q: \(0.\overline{3}\)
A: zero point three recurring/repeating
We do not say *periodic.

Q: \(a^n\)
A: a to the n;
a to the n-th power;
a (raised) to the power (of) n;
the n-th power of a
a to the n-th is incorrect, and might in any case be confused with the n-th root of a (“one nth” being 1/n);
a to n is incorrect.

Q: \(a^0\)
A: a to the zero;
a to the zeroth power;
a (raised) to the power (of) zero;
the zeroth power of a

Q: \(a^2\)
A: a squared,
the square of a;
a to the two,
a (raised) to the power (of) two;
a (raised) to the second power;
the second power of a

Q: \(a^3\)
A: a cubed,
the cube of a;
a to the three,
a (raised) to the power (of) three;
a (raised) to the third power;
the third power of a

Q: \(\sqrt[n]{a}\)
A: the n-th root of a
Equals a^(1/n) read as “a to the one over n”, “a to the one nth” etc.

Q: \(\sqrt{a}\)
A: root a;
the square root of a
the second root of … - this phrasing is not used with this meaning.

Q: \(\sqrt[3]{a}\)
A: the cube root of a;
the third root of a
the third root of is uncommon but understood in the required sense of denoting the cube root. For higher roots, the fourth root of a, the fifth root of a, etc. are used, in line with “the n-th root of a”.

Q: \(a \in A\)
A: a is an element of A
a belongs to A
Written after an equation this could be read where a belongs to A, or for a from A.
Example: 0^n = 0, n ∈ ℕ
…where n is a natural number
…where n is natural
…for natural n
…for n from the set of natural numbers

Q: \(A \subset B\)
A: A is a subset of B
A is contained in B
The article a is very important here.

Q: \(\forall\)
A: for all
for every
In various contexts it is also possible to use for any, for each.

Q: \(\forall a \in \mathbb{R}\)
A: for all real a
for all a from the set of real numbers

Q: \(\exists\)
A: there exist(s)
there is/are
Not enough to say just *exist(s), the grammar requires some subject.

Q: \(p \wedge q\)
A: p and q
the conjunction of p and q
The conjunction/disjunction option is quite infrequent.

Q: \(p \vee q\)
A: p or q
the disjunction of p and q

Q: \(p \Rightarrow q\)
A: if p, then q
p implies q
Make sure you use the if…then… structure, which is the most common way of reading implications (for example in theorems), e.g. if a is greater than 2, then a is greater than 3.

Q: \(p \Leftrightarrow q\)
A: p if and only if q
p is equivalent to q
In writing, this might be informally shortened to iff.

Q: \(\neg p\)
A: not p
the negation of p

Q: \(A \cap B\)
A: the intersection of A and B
A intersect B

Q: \(A \cup B\)
A: the union of A and B
A union B

Q: \(C_U(A)\)
A: the complement of A with respect to U
the complement of A in U
It is usual that the set with respect to which we take the complement is only understood and not explicitly stated, see below.

Q: \(C(A)\) or \(A'\)
A: the complement of A

Q: \(A \setminus B\)
A: A minus B
the difference A minus B
the difference of A and B

Q: \(:\)
A: such that („takové, že“)
or
it holds that („platí, že“)
The meaning of this symbol depends on the context. Two examples below.

Q: \(\forall a \exists b : a \cdot b = 0\)
A: for all a there exists b such that a times b equals 0

Q: \(\forall A : A \cap \emptyset = \emptyset\)
A: for all A it holds that A intersect empty set is an empty set
In this context it is perfectly fine to not read the symbol at all, i.e. for all A, A intersect the empty set is the empty set.

Q: \(\{a_n\}\)
A: the sequence a sub n
The n-th element of the sequence would be a sub n.

Q: \((a, b)\)
A: the open interval a b
The notation can refer to other objects than intervals, e.g. ordered pair.

Q: \([a, b]\)
A: the closed interval a b
For all intervals, a and b are called the endpoints. It is the interval with endpoints a, b.

Q: \([a, b)\) or \((a, b]\)
A: the half-closed interval, open on the left/right and closed on the right/left

Q: \(f(x)\)
A: f of x
function f of x
This structure works for all functions:
the [function] of [argument]
just insert any specific function, e.g. trigonometric functions (the sine of x), logarithms, roots, derivative.

Q: \(f^{-1}(x)\)
A: the inverse of f of x
the inverse of the function f of x

Q: \(\int f(x) dx\)
A: the (indefinite) integral of f of x with respect to x
The word definite/indefinite can be left out, it is clear from the context.
with respect to x can be replaced with d x at any time.

Q: \(\int_a^b f(x) dx\)
A: the (definite) integral from a to b of f of x with respect to x
the (definite) integral of f of x from a to b with respect to x

Q: \(+ c\)
A: plus c
plus constant
this is the constant of integration.

Q: \(\lim_{x \to \infty} f(x)\)
A: the limit of f of x as x approaches infinity
the limit of f of x as x goes to infinity
the limit of f of x as x tends to infinity
The order of the function (of f of x) and where it approaches (as x goes to …) can be switched.
The three verbs are interchangeable.

Q: \(\lim_{x \to 0^-} f(x)\)
A: the limit of f of x as x approaches zero form the left
the limit of f of x as x approaches zero form below
the left-handed limit of f of x as x approaches zero

Q: \(\lim_{x \to 0^+} f(x)\)
A: the limit of f of x as x approaches zero form the right
the limit of f of x as x approaches zero form above
the right-handed limit of f of x as x approaches zero

Q: \(\sum_{n=1}^\infty a_n\)
A: the sum of a sub n from n equals 1 to infinity
The order of the sequence (of a sub n) and the bounds (from … to …) can be switched.
equals can be also is equal to.

Q: \(f'(x)\)
A: the (first) derivative of f of x
f prime of x
The first can be omitted, it is redundant unless you want to stress this fact.

Q: \(f'(a)\)
A: the derivative of f at a
This is the derivative at a point – mind the preposition.

Q: \(\frac{df}{dx}\) or \(\frac{d}{dx}f(x)\)
A: the (first) derivative of f of x (with respect to x)
df by dx
d by dx of f of x
This means the same as the above; so called ‘Leibnizian notation’, still used in the Anglo-Saxon context. Expresses explicitly what you differentiate with respect to.
extra example: d/dx(sin x) “d by dx of sine x”

Q: \(f''(x)\)
A: the second derivative of f of x
f double prime of x
For derivatives of higher order:
the third/fourth… derivative …
f triple prime …

Q: \(\sin x\)
A: the sine of x
sine x
Do not say *sin x.
Note that the names of the trigonometric functions in English do not derive from Latin (as they do in Czech), so no *sinus, *cosinus, *tangens.

Q: \(\cos x\)
A: the cosine of x
cosine x
cos x

Q: \(\tan x\)
A: the tangent of x
tangent x
tan x

Q: \(\cot x\)
A: the cotangent of x
cotangent x
cot x

Q: \(\sin^{-1} x\)
\(\cos^{-1} x\)
\(\tan^{-1} x\)
\(\cot^{-1} x\)
A: the arc sine/cosine/… of x
the inverse sine/cosine/… of x
It is not a power, so do not say *sine to the minus one.
It is preferred to avoid *arcus (as it comes from Latin).
The spelling arcsine and arc-sine is also correct.

Q: \(\log_a b\)
A: the logarithm to (the) base a of b
the logarithm of b to/with (the) base a
log-a of b
the base-a logarithm of b

Q: \(\log x\)
A: the (common/decimal/decadic) logarithm of x
the logarithm to (the) base 10 of x
the logarithm of x to/with (the) base 10
log-ten of x
the base-10 logarithm of x
It is useful to remember that it is called the common logarithm.

Q: \(\ln x\)
A: the natural logarithm of x
the logarithm to the base e of x

Q: Grupa je komutativní, jestliže ... / Grupa se nazývá komutativní, jestliže ...
A: A group is (called) commutative if ...
A group is said to be commutative if ...

Q: Nechť / Budiž f = kx, kde k je ...
A: Define f = kx, where k is ...
Let f = kx, where k is ...

Q: Získáváme / Dostáváme ..., což budeme dále označovat jako ...
A: We obtain what will be referred to as ...

Q: Získáváme / Dostáváme ..., což budeme dále nazývat jako ...
A: We obtain what we shall call ...

Q: Získáváme / Dostáváme ..., známé jako ...
A: We obtain what is known as ...

Q: Rozšířená matice soustavy je matice, kterou získáme ...
A: An augmented matrix is a matrix obtained by ...

Q: Označme f zobrazení ...
A: Let us denote by f the map ...

Q: Nechť f označuje zobrazení ...
A: Let f denote the map ...

Q: Rozšířenou matici soustavy budeme značit Ar
A: The augmented matrix will be denoted by Ar

Q: Ar zde označuje matici ...
A: Here Ar denotes the matrix ...
Here Ar stands for the matrix ...

Q: Exp budeme krátce psát jako e.
A: We abbreviate exp to e.

Q: Jednoduše budeme označovat / zapisovat písmenem e.
A: We denote it briefly by e.

Q: Krátce označujeme / zapisujeme písmenem e.
A: We write it e for short.
We write it e for brevity.

Q: Prvek / element k takový, že ...
A: The element k such that ...

Q: Prvek / element k s vlastností / jehož vlastností je ...
A: The element k with the property that ...

Q: Prvek k, který splňuje / splňující ...
A: The element k satisfying ...

Q: Prvek k, který je tak malý, že ...
A: The element k so small that ...

Q: Konstanta k, která je nezávislá na / nezávislá na ...
A: The constant k being independent of ...

Q: Naším základním předpokladem je ...
A: Our basic assumption is the following.

Q: Nyní vyslovíme / uvedeme následující předpoklady: ...
A: We will make the following assumptions: ...

Q: Předpokládáme, že k je ...
A: We assume k to be ...

Q: Vlastnosti k je nutno omezit.
A: It is necessary to put some restrictions on k.

Q: Předpokládáme, že ... / Předpokládá se, že ...
A: It is assumed that ...

Q: Je nutné, aby ... / Požaduje se, aby ...
A: It is required that ...

Q: F splňuje podmínku F(x) = 0.
A: F satisfies the condition that F(x) = 0.

Q: Tento postup lze použít bez újmy na obecnosti.
A: This involves no loss of generality.

Q: Existuje právě jedno ...
A: There exists a unique ... / ... one and only one ...

Q: Předpokládejme / Mějme kladné x ...
A: Given a positive x ...

Q: Věta říká, že ...
A: The theorem states that ...
The theorem asserts that ...

Q: Věta ukazuje, že ...
A: The theorem shows that ...

Q: Věta je rozšířením ...
A: The theorem is an extension of ...

Q: Věta je zobecněním ...
A: The theorem is a generalization of ...

Q: Věta je upřesněním ...
A: The theorem is a refinement of ...

Q: Věta je jiným vyjádřením ...
A: The theorem is a reformulation of ...

Q: Ekvivalentní formulací (a) je: ...
A: An equivalent formulation of (a) is: ...

Q: Jestliže ..., pak ...
A: If ..., then ...

Q: Nechť k je ... Potom ..., za předpokladu ...
A: Let k be ... Then ..., provided ...

Q: Nechť k splňuje ... Potom ...
A: Let k satisfy ... Then ...

Q: Předpokládejme, že ... Potom ..., pokud ...
A: Suppose that ... Then ..., unless ...

Q: Předpokládejme, že ... Potom ...
A: Assume that ... Then ...

Q: ... tehdy a jen tehdy ... / právě tehdy, když ...
A: ... if and only if ...

Q: přímý důkaz
A: direct proof

Q: nepřímý důkaz
A: indirect proof

Q: důkaz indukcí
A: proof by induction

Q: důkaz sporem
A: proof by contradiction

Q: Nejprve dokážeme, že ...
A: We first prove that ...

Q: To dokážeme následujícím způsobem.
A: We prove this as follows.

Q: Předpokládejme, že toto tvrzení je chybné.
A: Suppose the assertion is false.

Q: Předpokládáme, že vzorec platí pro k a jeho platnost dokážeme pro k + 1.
A: Assume the formula holds for k; we will prove it for k + 1.

Q: Z toho vyplývá, že a = b.
A: ... it follows that a = b.

Q: To nám dává (rovnost) a = b.
A: This gives a = b.

Q: Výsledkem je (rovnost) a = b.
A: The result is a = b.

Q: Tímto dostáváme (rovnost) a = b.
A: We thus get a = b.

Q: ... a tedy a = b.
A: ... and so a = b.

Q: ... v důsledku čehož platí a = b.
A: ... and consequently a = b.

Q: ... což nám dává a = b.
A: ... which gives a = b.
... which yields a = b.

Q: ... což implikuje a = b.
A: ... which implies a = b.

Q: Rovnost implikuje, že ...
A: The equality implies that ...

Q: Protože x je kladné, dostáváme ax > 0.
A: As x is positive, we have ax > 0.

Q: Ale ax > 0, protože x je kladné.
A: But ax > 0 since x is positive.

Q: Z (a) vyplývá, že ... a nakonec také ...
A: We conclude from (a) that ..., and finally that ...

Q: Stejný závěr lze učinit pro ...
A: The same conclusion can be drawn for ...

Q: Stejným způsobem lze ukázat, že ...
A: In the same manner we can see that ...

Q: Uvažujte ...
A: Consider ...

Q: Zvolte ...
A: Choose ...

Q: Definujte ...
A: Define ...

Q: Nechť / Budiž ...
A: Let ...

Q: Položte / Stanovte ...
A: Set ...

Q: Předpokládejme ...
A: Let us suppose ...
Let us assume ...

Q: Uvažujme k takové, že ...
A: Let us regard k as ...

Q: Vypočítejme ...
A: Let us compute ...

Q: Připočtením a k pravé straně dostáváme ...
A: Adding a to the right-hand side yields ...
Adding a to the right-hand side gives ...

Q: Odečtením (2) od (1) získáváme ...
A: Subtracting (2) from (1), we obtain ...
Subtracting (2) from (1), we get ...
Subtracting (2) from (1), we have ...

Q: Postačí dokázat, že ...
A: It suffices to show that ...
It is sufficient to show that ...

Q: Musíme zvážit dva případy: ...
A: We need only consider two cases: ...

Q: Musíme ukázat, že ...
A: We only need to show that ...

Q: Ukážeme, že ... a tím bude důkaz proveden.
A: The proof is completed by showing that ...

Q: Je zřejmé, že ...
A: It is clear that ...
It is evident that ...

Q: Snadno vidíme / nahlédneme, že ...
A: It is easily seen that ...

Q: Jednoduché ověření ukáže, že ...
A: A trivial verification shows that ...
A trivial verification makes it obvious that ...

Q: ..., a tím je důkaz proveden.
A: ..., which completes the proof.
..., which proves the theorem.

Q: ..., což jsme požadovali.
A: ..., which is our claim.

Q: ..., což odpovídá našemu tvrzení.
A: ..., which is our assertion.

Q: ..., což je požadovaný závěr.
A: ..., which is the desired conclusion.

Q: ..., a tím je důkaz dokončen.
A: ..., and the proof is complete.

Q: To je v rozporu s naším předpokladem.
A: This contradicts our assumption.

Q: ..., což je v rozporu s (a). / na rozdíl od (a).
A: ..., contrary to (a).

Q: ..., což je v rozporu s našimi předpoklady.
A: ..., which contradicts our assumptions.

Q: ..., což není možné. / ..., což je spor.
A: ..., which is impossible.

Q: Proto / tedy / z tohoto důvodu ...
A: Therefore ...

Q: A tak / tak / takto / tedy ...
A: Thus ...

Q: Proto / z tohoto důvodu / tudíž ...
A: Hence ...

Q: Níže / dále ...
A: Here and subsequently, ...

Q: Během důkazu ...
A: Throughout the proof, ...

Q: V následující části ...
A: In what follows, ...

Q: Od nynějška / níže / dále ...
A: From now on, ...

Q: Takto / tímto způsobem ...
A: In this way, ...

Q: Stručně / jednoduše (budeme zapisovat) ...
A: For simplicity of notation, ...
For abbreviation, ...
... for brevity.
... for short.

Q: Jak množina X tak množina Y jsou spočitatelné.
A: Both X and Y are countable.

Q: Ani množina X ani množina Y nejsou spočitatelné.
A: Neither X nor Y is countable.

Q: Ani jedna z nich (ze dvou) není spočitatelná.
A: Neither of them is countable.

Q: Žádná z množin Ai (více než dvou) není spočetelná.
A: None of the sets Ai is countable.

Q: Bez újmy na obecnosti ...
A: Without loss of generality ...

Q: Na základě výše uvedených předpokladů ...
A: Under the above assumptions, ....

Q: Na základě podmínek uvedených výše ...
A: Under the conditions stated above, ...

Q: Na základě předpokladů uvedených ve větě 5 ...
A: Under the assumptions of Theorem 5, ...

Q: Na základě hypotéz uvedených ve větě 5 ...
A: Under the hypotheses of Theorem 5, ...

Q: Z definice ...
A: By definition, ...

Q: Na základě výše uvedeného ...
A: By the above, ...

Q: Dle Cauchyho věty ...
A: By Cauchy's theorem, ...

Q: Dle předpokladů ...
A: By assumption, ...

Q: Lze zvolit libovolné k. / Můžeme zvolit libovolné k.
A: It is immaterial which k we choose.

Q: Nyní se budeme zabývat ...
A: We now turn to ...

Q: Abychom získali ..., budeme pokračovat stejným způsobem.
A: We continue in this fashion to obtain ...

Q: Zhruba řečeno ...
A: Roughly speaking, ...

Q: Zjednodušeně řečeno ...
A: Loosely speaking, ...

Q: Samozřejmě / Jistě ...
A: Of course, ...

Q: (Zcela) jasně / zřejmě / zřetelně ...
A: Clearly, ...
Obviously, ...