the moon

Then He decorated the heavens with the stars and the light of the meteors and set the shining sun and the bright moon in an orbit that rolls around.

(Sermon 1)

Percentage (%)

Percentage is per-cent which means parts per hundred.
One percent is equal to 1/100 fraction:

1% = 1/100 = 0.01

Ten percent is equal to 10/100 fraction:

10% = 10/100 = 0.1

Fifty percent is equal to 50/100 fraction:

50% = 50/100 = 0.5

One hundred percent is equal to 100/100 fraction:

100% = 100/100 = 1

One hundred and ten percent is equal to 110/100 fraction:

110% = 110/100 = 1.1

Percent sign

The percent sign is the symbol: %
It is written to the right side of the number: 50%

Percentage Definition

Percentage is a value that represents the proportion of one number to another number.
1 percent represents 1/100 fraction.
100 percent (100%) of a number is the same number:

100% × 80 = 100/100×80 = 80

50 percent (50%) of a number is half of the number:

50% × 80 = 50/100×80 = 40

So 40 is 50% of 80.

Percentage of a Value Calculation

x% of y is calculated by the formula:

percentage value = x% × y = x × y / 100

Example:

Find 40% of 200.

40% × 200 = 40 × 200 / 100 = 80

Percentage Calculation

The percentage of x from y, is calculated by the formula:

percentage% = (x / y) × 100

Example:

The percentage of 30 out of 60.

30 / 60 × 100 = 50%

Percentage Change (increase/decrease)

Percentage change from x1 to x2 is calculated by the formula:

percentage change = 100 × (x2 - x1) / x1

When the result is positive, we have percentage growth or increase.

Example:

Percentage change from 60 to 80.

100 × (80 - 60) / 60 = 33.33% increase

When the result is negative, we have percentage decrease.

Example:

Percentage change from 80 to 60.

100 × (60 - 80) / 80 = -25% decrease

the winds

He brought out the creation by His power and made the winds blow With His compassion.

(Sermon 1)

English Idioms & Idiomatic Expressions (Beginning with B)

Babe in arms

A babe in arms is a very young child, or a person who is very young to be holding a position.

Babe in the woods

A babe in the woods is a naive, defenceless, young person.

Baby boomer

(USA) A baby boomer is someone born in the years after the end of the Second World War, a period when the population was growing very fast.

Back burner

If an issue is on the back burner, it is being given low priority.

Back foot

(UK) If you are on your back foot, you are at a disadvantage and forced to be defensive of your position.

Back number

Something that's a back number is dated or out of fashion.

Back the wrong horse

If you back the wrong horse, you give your support to the losing side in something.

Back to back

If things happen back to back, they are directly one after another.

Back to square one

If you are back to square one, you have to start from the beginning again.

Back to the drawing board

If you have to go back to the drawing board, you have to go back to the beginning and start something again.

Back to the salt mines

If someone says they have to go back to the salt mines, they have to return, possibly unwillingly, to work.

Back to the wall

If you have your back to the wall, you are in a difficult situation with very little room for manoeuvre.

Backseat driver

A backseat driver is an annoying person who is fond of giving advice to the person performing a task or doing something, especially when the advice is either wrong or unwelcome.

Bad Apple

A person who is bad and makes other bad is a bad apple.

Bad blood

If people feel hate because of things that happened in the past, there is bad blood between them.

Bad egg

A person who cannot be trusted is a bad egg. Good egg is the opposite.

Bad hair day

If you're having a bad hair day, things are not going the way you would like or had planned.

Bad mouth

(UK) When you are bad mouthing,you are saying negative things about someone or something.('Bad-mouth' and 'badmouth' are also used.)

Bad shape

If something's in bad shape, it's in bad condition. If a person's in bad shape, they are unfit or unhealthy.

Bad taste in your mouth

If something leaves you with a bad taste in your mouth, you feel there is something wrong or bad about it.

Bad workers always blame their tools

"A bad worker always blames their tools" - If somebody does a job badly or loses in a game and claims that they were let down by their equipment, you can use this to imply that this was not the case.

Bag and baggage

Bag and baggage means all your possessions, especially if you are moving them or leaving a place.

Bag of bones

If someone is a bag of bones, they are very underweight.

Bag of nerves

If someone is a bag of nerves, they are very worried or nervous.

Baker's dozen

A Baker's dozen is 13 rather than 12.

Bald as a coot

A person who is completely bald is as bald as a coot.

Ball is in your court

If the ball is in your court, it is up to you to make the next decision or step.

Balloon goes up

When the balloon goes up, a situation turns unpleasant or serious.

Ballpark figure

A ballpark figure is a rough or approximate number (guesstimate) to give a general idea of something, like a rough estimate for a cost, etc.

Balls to the walls

(USA) If you do something balls to the wall, you apply full acceleration or exertion.

Banana republic

Banana republic is a term used for small countries that are dependent on a single crop or resource and governed badly by a corrupt elite.

Banana skin

(UK) A banana skin is something that is an embarrassment or causes problems.

Bandit territory

An area or an industry, profession, etc, where rules and laws are ignored or flouted is bandit territory.

Baptism of fire

A baptism of fire was a soldier's first experience of shooting. Any unpleasant experience undergone, usually where it is also a learning experience, is a baptism of fire.

Bar fly

A bar fly is a person who spends a lot of time drinking in different bars and pubs.

Bare your heart

If you bare your heart to someone, you tell them your personal and private feelings. ('Bare your soul' is an alternative form of the idiom.)

Barefaced liar

A barefaced liar is one who displays no shame about lying even if they are exposed.

Bark is worse than their bite

Someone who's bark is worse than their bite may well get angry and shout, but doesn't take action.

Barking up the wrong tree

If you are barking up the wrong tree, it means that you have completely misunderstood something or are totally wrong.

Barkus is willing

This idiom means that someone is willing to get married.

Barrack-room lawyer

(UK) A barrack-room lawyer is a person who gives opinions on things they are not qualified to speak about.

Barrel of laughs

If someone's a barrel of laughs, they are always joking and you find them funny.

Basket case

If something is a basket case, it is so bad that it cannot be helped.

Bat an eyelid

If someone doesn't bat an eyelid, they don't react or show any emotion when surprised, shocked, etc.

Bated breath

If someone says they're waiting with bated breath, they're very excited and find it difficult to be patient.('Baited breath' is a common mistake.)

Bats in the belfry

Someone with bats in the belfry is crazy or eccentric.

Batten down the hatches

If you batten down the hatches, you prepare for the worst that could happen to you.

Batting a thousand

(USA) (from baseball) It means to do something perfectly.

Battle of nerves

A battle of nerves is a situation where neither side in a conflict or dispute is willing to back down and is waiting for the other side to weaken. ('A war of nerves' is an alternative form.)

Be all ears

If you are all ears, you are very eager to hear what someone has to say.

Multiplication Crossword (to x12)

Complete the crossword puzzle by answering the multiplication questions below.

1	2			3			4
		5				6
	7		8		9
10			11
12
		13		14		15	16
	17

Across		Down
1.    11 x 12 =		2.    3 x 12 =	14.    8 x 9 =
3.    12 x 7 =		3.    9 x 9 =	16.    5 x 5 =
6.    4 x 11 =		4.    12 x 12 =
7.    11 x 11 =		5.    2 x 11 =
11.    12 x 10 =		8.    10 x 11 =
12.    12 x 5 =		9.    12 x 9 =
15.    6 x 12 =		10.    8 x 12 =
17.    4 x 12 =		13.    8 x 11 =

the earth

He created the earth and suspended it, retained it without support, made it stand without legs, and raised it without pillars.

(Sermon 185)

CROOKED JACKAL

Long ago, there lived a crooked jackal named 'Sam' in a forest. He had no friends as he behaved like a crooked animal. Even animal of his gene like fox, dog, wolf did not mingle with him. They all avoided the jackal.
During the night, he terrorized other animals in the forest by howling loudly. In the calm, serene atmosphere the howling of 'Sam' became a nuisance. The little animals used to carve into their burrows as soon as the jackal produces the ugly sound. It was pure sadism, which followed the wicked animal. 'No, other animal in this jungle can howl louder than me, my mere voice will horrify all other animals', he added.
The harassment of the jackal continued un-abated. Unfortunately, none of the animals in the forest dared to ask 'Sam' to stop this menace. The pressure tactics of the jackal, to bewilder the animals, took a quiver turn one day. He got a mask of the giant animal dinosaur, left out by a group of tourists. After adorning the mask, the jackal shouted at other animals, 'Look, now onwards I am a dinosaur, with the strength and courage, I can sabotage the entire universe'. I will walk always wearing this mask, he announced.
The little animals shivered and frightened. They could not withstand the extent of damage to be caused by a dinosaur. The 'Sam' ruled the jungle like a king adorning the mask of a dinosaur. Days passed, One day the jackal in an inebriated condition traveled to the remote corner of the forest. He was alone. He could not howl, suddenly a huge rock like creature came in front of him. The jackal could not believe his eyes. It was a dinosaur. He was not at all a match for the real dinosaur. He had heard about these giant animals, which lived on the earth, centuries ago. Sam tried to howl loudly for help, but was in vain. The real dinosaur crushed the jackal under his feet and walked away.

Father in Many Languages Word Search Puzzle

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DAD		PAPA
DADA		PAPPA
DADDY		PATER
FADER		PATRIS
FAR		PERE
FATHER		POP
ISA		POPPA
OJCIEC		VADER
PADRE		VATER
PAI		VATI

Can Two People have Identical Fingerprints?

By Ritu Asthana; Illustration by Anup Singh

 

I was once watching a detective serial on T.V. where the fingerprints of the suspect are required. The hero invites the villain to his home and offers him a glass of water. The bad guy takes the glass and drinks the water from it. After he leaves the hero dramatically whips out a handkerchief and picks up the glass. His expressions showed that satisfaction at a job well done.

At that time, I found it amazing - how can smudges on a glass identify people? How can my hand be different from yours - apart from the size that is? It was only after my dad explained me the reason that the mystery behind that highly dramatic scene became clear.

My dad explained that if you were to press your thumb on an inkpad and then on a sheet of white paper you will leave a smudge or print, which no one else in the world can make. The same would be true for each of your fingers. The Chinese were the first to use fingerprints to affix their signature on important documents though they had no way of identifying or matching it with the owner.

If you look closely at the inside of your palm you can see tiny lines all over. These are absolutely unique and they remain practically unchanged from birth to death.

Each print is one-of-a-kind and no two people have the same characteristic. Scientists and criminologists (those who study criminal characteristics) determine the individuality of a fingerprint by a careful study of its ridge characteristics (minutiae) and not by its general shape or pattern.

The surface of the skin has been designed to provide our bodies with a firm grasp and to prevent objects grasped from slipping. Skin is composed of layers of cells. The outer portion of skin is called the epidermis while the inner skin is the dermis. Separating the two layers is a boundary of cells called the papillae.

Each skin ridge has a single row of pores that open out for the sweat glands. Once the finger touches a surface, sweat from these pores, along with other body oils layer the ridge of the skin and are thus transferred to that surface. The result is an impression of the finger's ridge pattern. Such prints are referred to as hidden fingerprints because they are invisible to the naked eye.

How this fact was discovered and now used by the police force worldwide is very interesting to trace. In 1823, John Evangelist Purkinji, a professor of anatomy at the University of Breslau in Czechoslovakia, published the first document on the nature of fingerprints.

In 1856, Sir William Herschel, who was then working for the Indian Civil Service in West Bengal, used thumbprints on documents as a substitute for signatures. In 1880, a British physician Henry Faulds, who was then working in Tokyo, published a paper suggesting that fingerprints left at the scene of a crime could identify the offender. However, Faulds never got the credit he deserved!

In 1892, an English scientist, Sir Francis Galton, published a comprehensive book on using fingerprints to solve crimes. At the same time in Argentina, a police researcher Juan Vucetich was also working towards a fingerprint classification system.

Math symbols

Basic Math Symbols

Symbol	Symbol Name	Meaning / definition	Example
=	equals sign	equality	5 = 2+3
≠	not equal sign	inequality	5 ≠ 4
>	strict inequality	greater than	5 > 4
<	strict inequality	less than	4 < 5
≥	inequality	greater than or equal to	5 ≥ 4
≤	inequality	less than or equal to	4 ≤ 5
( )	parentheses	calculate expression inside first	2 × (3+5) = 16
[ ]	brackets	calculate expression inside first	[(1+2)*(1+5)] = 18
+	plus sign	addition	1 + 1 = 2
−	minus sign	subtraction	2 − 1 = 1
±	plus - minus	both plus and minus operations	3 ± 5 = 8 and -2
∓	minus - plus	both minus and plus operations	3 ∓ 5 = -2 and 8
*	asterisk	multiplication	2 * 3 = 6
×	times sign	multiplication	2 × 3 = 6
∙	multiplication dot	multiplication	2 ∙ 3 = 6
÷	division sign / obelus	division	6 ÷ 2 = 3
/	division slash	division	6 / 2 = 3
–	horizontal line	division / fraction
mod	modulo	remainder calculation	7 mod 2 = 1
.	period	decimal point, decimal separator	2.56 = 2+56/100
a b	power	exponent	23 = 8
a^b	caret	exponent	2 ^ 3 = 8
√a	square root	√a · √a  = a	√9 = ±3
3√a	cube root		3√8 = 2
4√a	forth root		4√16 = ±2
n√a	n-th root (radical)		for n=3, n√8 = 2
%	percent	1% = 1/100	10% × 30 = 3
‰	per-mille	1‰ = 1/1000 = 0.1%	10‰ × 30 = 0.3
ppm	per-million	1ppm = 1/1000000	10ppm × 30 = 0.0003
ppb	per-billion	1ppb = 1/1000000000	10ppb × 30 = 3×10-7
ppt	per-trillion	1ppb = 10-12	10ppb × 30 = 3×10-10

Geometry symbols

Symbol	Symbol Name	Meaning / definition	Example
∠	angle	formed by two rays	∠ABC = 30º
∡	measured angle		∡ABC = 30º
∢	spherical angle		∢AOB = 30º
∟	right angle	= 90º	α = 90º
º	degree	1 turn = 360º	α = 60º
´	arcminute	1º = 60´	α = 60º59'
´´	arcsecond	1´ = 60´´	α = 60º59'59''
AB	line	line from point A to point B
	ray	line that start from point A
|	perpendicular	perpendicular lines (90º angle)	AC | BC
||	parallel	parallel lines	AB || CD
≅	congruent to	equivalence of geometric shapes and size	∆ABC ≅ ∆XYZ
~	similarity	same shapes, not same size	∆ABC ~ ∆XYZ
Δ	triangle	triangle shape	ΔABC ≅ ΔBCD
| x-y |	distance	distance between points x and y	| x-y | = 5
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle	c = π·d = 2·π·r
rad	radians	radians angle unit	360º = 2π rad
grad	grads	grads angle unit	360º = 400 grad

Algebra symbols

Symbol	Symbol Name	Meaning / definition	Example
x	x variable	unknown value to find	when 2x = 4, then x = 2
≡	equivalence	identical to
≜	equal by definition	equal by definition
:=	equal by definition	equal by definition
~	approximately equal	weak approximation	11 ~ 10
≈	approximately equal	approximation	sin(0.01) ≈ 0.01
∝	proportional to	proportional to	f(x) ∝ g(x)
∞	lemniscate	infinity symbol
≪	much less than	much less than	1 ≪ 1000000
≫	much greater than	much greater than	1000000 ≫ 1
( )	parentheses	calculate expression inside first	2 * (3+5) = 16
[ ]	brackets	calculate expression inside first	[(1+2)*(1+5)] = 18
{ }	braces	set
⌊x⌋	floor brackets	rounds number to lower integer	⌊4.3⌋= 4
⌈x⌉	ceiling brackets	rounds number to upper integer	⌈4.3⌉= 5
x!	exclamation mark	factorial	4! = 1*2*3*4 = 24
| x |	single vertical bar	absolute value	| -5 | = 5
f (x)	function of x	maps values of x to f(x)	f (x) = 3x+5
(f ∘g)	function composition	(f ∘g) (x) = f (g(x))	f (x)=3x, g(x)=x-1 ⇒(f ∘g)(x)=3(x-1)
(a,b)	open interval	(a,b) ≜ {x | a < x < b}	x ∈ (2,6)
[a,b]	closed interval	[a,b] ≜ {x | a ≤ x ≤ b}	x ∈ [2,6]
∆	delta	change / difference	∆t = t1 - t0
∆	discriminant	Δ = b2 - 4ac
∑	sigma	summation - sum of all values in range of series	∑ xi= x1+x2+...+xn
∑∑	sigma	double summation
∏	capital pi	product - product of all values in range of series	∏ xi=x1∙x2∙...∙xn
e	e constant / Euler's number	e = 2.718281828...	e = lim (1+1/x)x , x→∞
γ	Euler-Mascheroni  constant	γ = 0.527721566...
φ	golden ratio	golden ratio constant

Linear Algebra Symbols

Symbol	Symbol Name	Meaning / definition	Example
∙	dot	scalar product	a ∙ b
×	cross	vector product	a × b
A⊗B	tensor product	tensor product of A and B	A ⊗ B
	inner product
[ ]	brackets	matrix of numbers
( )	parentheses	matrix of numbers
| A |	determinant	determinant of matrix A
det(A)	determinant	determinant of matrix A
|| x ||	double vertical bars	norm
A T	transpose	matrix transpose	(AT)ij = (A)ji
A †	Hermitian matrix	matrix conjugate transpose	(A†)ij = (A)ji
A *	Hermitian matrix	matrix conjugate transpose	(A*)ij = (A)ji
A -1	inverse matrix	A A-1 = I
rank(A)	matrix rank	rank of matrix A	rank(A) = 3
dim(U)	dimension	dimension of matrix A	rank(U) = 3

Probability and statistics symbols

Symbol	Symbol Name	Meaning / definition	Example
P(A)	probability function	probability of event A	P(A) = 0.5
P(A ∩ B)	probability of events intersection	probability that of events A and B	P(A∩B) = 0.5
P(A ∪ B)	probability of events union	probability that of events A or B	P(A∪B) = 0.5
P(A | B)	conditional probability function	probability of event A given event B occured	P(A | B) = 0.3
f (x)	probability density function (pdf)	P(a ≤ x ≤ b) = ∫ f (x) dx
F(x)	cumulative distribution function (cdf)	F(x) = P(X ≤ x)
μ	population mean	mean of population values	μ = 10
E(X)	expectation value	expected value of random variable X	E(X) = 10
E(X | Y)	conditional expectation	expected value of random variable X given Y	E(X | Y=2) = 5
var(X)	variance	variance of random variable X	var(X) = 4
σ2	variance	variance of population values	σ2 = 4
std(X)	standard deviation	standard deviation of random variable X	std(X) = 2
σX	standard deviation	standard deviation value of random variable X	σX  = 2
	median	middle value of random variable x
cov(X,Y)	covariance	covariance of random variables X and Y	cov(X,Y) = 4
corr(X,Y)	correlation	correlation of random variables X and Y	corr(X,Y) = 3
ρX,Y	correlation	correlation of random variables X and Y	ρX,Y = 3
∑	summation	summation - sum of all values in range of series
∑∑	double summation	double summation
Mo	mode	value that occurs most frequently in population
MR	mid-range	MR = (xmax+xmin)/2
Md	sample median	half the population is below this value
Q1	lower / first quartile	25% of population are below this value
Q2	median / second quartile	50% of population are below this value = median of samples
Q3	upper / third quartile	75% of population are below this value
x	sample mean	average / arithmetic mean	x = (2+5+9) / 3 = 5.333
s 2	sample variance	population samples variance estimator	s 2 = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
zx	standard score	zx = (x-x) / sx
X ~	distribution of X	distribution of random variable X	X ~ N(0,3)
N(μ,σ2)	normal distribution	gaussian distribution	X ~ N(0,3)
U(a,b)	uniform distribution	equal probability in range a,b	X ~ U(0,3)
exp(λ)	exponential distribution	f (x) = λe-λx , x≥0
gamma(c, λ)	gamma distribution	f (x) = λ c xc-1e-λx / Γ(c), x≥0
χ 2(k)	chi-square distribution	f (x) = xk/2-1e-x/2 / ( 2k/2 Γ(k/2) )
F (k1, k2)	F distribution
Bin(n,p)	binomial distribution	f (k) = nCk pk(1-p)n-k
Poisson(λ)	Poisson distribution	f (k) = λke-λ / k!
Geom(p)	geometric distribution	f (k) =  p (1-p) k
HG(N,K,n)	hyper-geometric distribution
Bern(p)	Bernoulli distribution

Combinatorics Symbols

Symbol	Symbol Name	Meaning / definition	Example
n!	factorial	n! = 1·2·3·...·n	5! = 1·2·3·4·5 = 120
nPk	permutation		5P3 = 5! / (5-3)! = 60
nCk	combination		5C3 = 5!/[3!(5-3)!]=10

Set theory symbols

Symbol	Symbol Name	Meaning / definition	Example
{ }	set	a collection of elements	A={3,7,9,14}, B={9,14,28}
A ∩ B	intersection	objects that belong to set A and set B	A ∩ B = {9,14}
A ∪ B	union	objects that belong to set A or set B	A ∪ B = {3,7,9,14,28}
A ⊆ B	subset	subset has less elements or equal to the set	{9,14,28} ⊆ {9,14,28}
A ⊂ B	proper subset / strict subset	subset has less elements than the set	{9,14} ⊂ {9,14,28}
A ⊄ B	not subset	left set not a subset of right set	{9,66} ⊄ {9,14,28}
A ⊇ B	superset	set A has more elements or equal to the set B	{9,14,28} ⊇ {9,14,28}
A ⊃ B	proper superset / strict superset	set A has more elements than set B	{9,14,28} ⊃ {9,14}
A ⊅ B	not superset	set A is not a superset of set B	{9,14,28} ⊅ {9,66}
2A	power set	all subsets of A
Ƥ (A)	power set	all subsets of A
A = B	equality	both sets have the same members	A={3,9,14}, B={3,9,14}, A=B
Ac	complement	all the objects that do not belong to set A
A \ B	relative complement	objects that belong to A and not to B	A={3,9,14},     B={1,2,3}, A-B={9,14}
A - B	relative complement	objects that belong to A and not to B	A={3,9,14},     B={1,2,3}, A-B={9,14}
A ∆ B	symmetric difference	objects that belong to A or B but not to their intersection	A={3,9,14},     B={1,2,3}, A ∆ B={1,2,9,14}
A ⊖ B	symmetric difference	objects that belong to A or B but not to their intersection	A={3,9,14},     B={1,2,3}, A ⊖ B={1,2,9,14}
a∈A	element of	set membership	A={3,9,14}, 3 ∈ A
x∉A	not element of	no set membership	A={3,9,14}, 1 ∉ A
(a,b)	ordered pair	collection of 2 elements
A×B	cartesian product	set of all ordered pairs from A and B
|A|	cardinality	the number of elements of set A	A={3,9,14}, |A|=3
#A	cardinality	the number of elements of set A	A={3,9,14}, #A=3
א	aleph	infinite cardinality
Ø	empty set	Ø = { }	C = {Ø}
U	universal set	set of all possible values
ℕ0	natural numbers set (with zero)	ℕ0 = {0,1,2,3,4,...}	0 ∈ ℕ0
ℕ1	natural numbers set (without zero)	ℕ1 = {1,2,3,4,5,...}	6 ∈ ℕ1
ℤ	integer numbers set	ℤ = {...-3,-2,-1,0,1,2,3,...}	-6 ∈ ℤ
ℚ	rational numbers set	ℚ = {x | x=a/b, a,b∈ℕ}	2/6 ∈ ℚ
ℝ	real numbers set	ℝ = {x | -∞ < x <∞}	6.343434 ∈ ℝ
ℂ	complex numbers set	ℂ = {z | z=a+bi, -∞<a<∞,      -∞<b<∞}	6+2i ∈ ℂ

Logic symbols

Symbol	Symbol Name	Meaning / definition	Example
·	and	and	x · y
^	caret / circumflex	and	x ^ y
&	ampersand	and	x & y
+	plus	or	x + y
∨	reversed caret	or	x ∨ y
|	vertical line	or	x | y
x'	single quote	not - negation	x'
x	bar	not - negation	x
¬	not	not - negation	¬ x
!	exclamation mark	not - negation	! x
⊕	circled plus / oplus	exclusive or - xor	x ⊕ y
~	tilde	negation	~ x
⇒	implies
⇔	equivalent	if and only if
∀	for all
∃	there exists
∄	there does not exists
∴	therefore
∵	because / since

Calculus & analysis symbols

Symbol	Symbol Name	Meaning / definition	Example
	limit	limit value of a function
ε	epsilon	represents a very small number, near zero	ε → 0
e	e constant / Euler's number	e = 2.718281828...	e = lim (1+1/x)x , x→∞
y '	derivative	derivative - Leibniz's notation	(3x3)' = 9x2
y ''	second derivative	derivative of derivative	(3x3)'' = 18x
y(n)	nth derivative	n times derivation	(3x3)(3) = 18
	derivative	derivative - Lagrange's notation	d(3x3)/dx = 9x2
	second derivative	derivative of derivative	d2(3x3)/dx2 = 18x
	nth derivative	n times derivation
	time derivative	derivative by time - Newton notation
	time second derivative	derivative of derivative
	partial derivative		∂(x2+y2)/∂x = 2x
∫	integral	opposite to derivation
∬	double integral	integration of function of 2 variables
∭	triple integral	integration of function of 3 variables
∮	closed contour / line integral
∯	closed surface integral
∰	closed volume integral
[a,b]	closed interval	[a,b] = {x | a ≤ x ≤ b}
(a,b)	open interval	(a,b) = {x | a < x < b}
i	imaginary unit	i ≡ √-1	z = 3 + 2i
z*	complex conjugate	z = a+bi → z*=a-bi	z* = 3 + 2i
z	complex conjugate	z = a+bi → z = a-bi	z = 3 + 2i
∇	nabla / del	gradient / divergence operator	∇f (x,y,z)
	vector
	unit vector
x * y	convolution	y(t) = x(t) * h(t)
ℒ	Laplace transform	F(s) = ℒ{f (t)}
ℱ	Fourier transform	X(ω) = ℱ{f (t)}
δ	delta function

Numeral symbols

Name	European	Roman	Hindu Arabic	Hebrew
zero	0		٠
one	1	I	١	א
two	2	II	٢	ב
three	3	III	٣	ג
four	4	IV	٤	ד
five	5	V	٥	ה
six	6	VI	٦	ו
seven	7	VII	٧	ז
eight	8	VIII	٨	ח
nine	9	IX	٩	ט
ten	10	X	١٠	י
eleven	11	XI	١١	יא
twelve	12	XII	١٢	יב
thirteen	13	XIII	١٣	יג
fourteen	14	XIV	١٤	יד
fifteen	15	XV	١٥	טו
sixteen	16	XVI	١٦	טז
seventeen	17	XVII	١٧	יז
eighteen	18	XVIII	١٨	יח
nineteen	19	XIX	١٩	יט
twenty	20	XX	٢٠	כ
thirty	30	XXX	٣٠	ל
fourty	40	XL	٤٠	מ
fifty	50	L	٥٠	נ
sixty	60	LX	٦٠	ס
seventy	70	LXX	٧٠	ע
eighty	80	LXXX	٨٠	פ
ninety	90	XC	٩٠	צ
one hundred	100	C	١٠٠	ק

Greek alphabet letters

Greek Symbol		Greek Letter Name	English Equivalent	Pronunciation
Upper Case	Lower Case
Α	α	Alpha	a	al-fa
Β	β	Beta	b	be-ta
Γ	γ	Gamma	g	ga-ma
Δ	δ	Delta	d	del-ta
Ε	ε	Epsilon	e	ep-si-lon
Ζ	ζ	Zeta	z	ze-ta
Η	η	Eta	h	eh-ta
Θ	θ	Theta	th	te-ta
Ι	ι	Iota	i	io-ta
Κ	κ	Kappa	k	ka-pa
Λ	λ	Lambda	l	lam-da
Μ	μ	Mu	m	m-yoo
Ν	ν	Nu	n	noo
Ξ	ξ	Xi	x	x-ee
Ο	ο	Omicron	o	o-mee-c-ron
Π	π	Pi	p	pa-yee
Ρ	ρ	Rho	r	row
Σ	σ	Sigma	s	sig-ma
Τ	τ	Tau	t	ta-oo
Υ	υ	Upsilon	u	oo-psi-lon
Φ	φ	Phi	ph	f-ee
Χ	χ	Chi	ch	kh-ee
Ψ	ψ	Psi	ps	p-see
Ω	ω	Omega	o	o-me-ga

Roman numerals

Number	Roman numeral
1	I
2	II
3	III
4	IV
5	V
6	VI
7	VII
8	VIII
9	IX
10	X
11	XI
12	XII
13	XIII
14	XIV
15	XV
16	XVI
17	XVII
18	XVIII
19	XIX
20	XX
30	XXX
40	XL
50	L
60	LX
70	LXX
80	LXXX
90	XC
100	C
200	CC
300	CCC
400	CD
500	D
600	DC
700	DCC
800	DCCC
900	CM
1000	M
5000	V
10000	X
50000	L
100000	C
500000	D
1000000	M