Maths Science: ફેબ્રુઆરી 2015

ગુરુવાર, 12 ફેબ્રુઆરી, 2015

વેગ અને પ્રવેગ (Velocity and Acceleration)

વેગ અને પ્રવેગ (Velocity and Acceleration)

વેગ(Velocity) એ બીજું કંઈ નહીં પણ દિશા ધરાવતી ઝડપ(Speed) છે એવું કહી શકાય. ઉદા.તરીકે, કોઈ કાર ઉત્તરથી દક્ષિણ તરફ 40 કિમી/ કલાક ની ઝડપે ગતિ કરી રહી હોય તો તેનો વેગ 40 કિમી/ કલાક ઉત્તરથી દક્ષિણ તરફ છે એમ કહેવાય.

આપણે સૌ જાણીએ છીએ કે વ્યવહારમાં ટ્રાફિક, ખરાબ રોડ વગેરે કારણોને લીધે કોઈપણ વાહન કે વસ્તુનો વેગ સતત બદલાતો રહે છે. જેમ કે કોઈ કાર ખુલ્લા હાઈવે પર 120 કિમી/કલાકની ઝડપે પ્રવાસ કરતી હશે, પણ એ જ કારની ઝડપ શહેરી વિસ્તારમાં 25કિમી/કલાક જેટલી જ હશે. આ કારણથી કોઈપણ વસ્તુના વેગમાં થતા ફેરફારની જાણકારી મેળવવા માટે એક નવી રાશિ વ્યાખ્યાયિત કરવામાં આવી -પ્રવેગ (Acceleration).

પ્રવેગની વ્યાખ્યા આ પ્રમાણે છે :- "ગતિમાન પદાર્થના વેગમાં થતા ફેરફારના દરને પ્રવેગ કહે છે."

તેથી પ્રવેગ = વેગમાં થતો ફેરફાર / તે માટે લાગતો સમય

= (અંતિમ વેગ - પ્રારંભિક વેગ) / સમય

કેમ કે વેગનો એકમ એટલે કે ઝડપનો એકમ મીટર/સેકન્ડ છે અને સમયનો એકમ સેકન્ડ છે, તો ઊપરના સૂત્ર પ્રમાણે અથવા પ્રવેગની વ્યાખ્યા મુજબ તેનો એકમ (મીટર/સેકન્ડ)/સેકન્ડ મતલબ મીટર/સેકન્ડ^{2 થાય.}

જો વેગમાં વધારો થતો હોય તો વેગ્માં થતા ફેરફારના દરને પ્રવેગ(Acceleration) કહેવાય, અને જો વેગમાં ઘટાડો થતો હોય તો ફેરફારના દરને પ્રતિપ્રવેગ(Retardation) કહેવાય છે.

પ્રવેગી ગતિને સમજવા માટે આકૃતિમાં બતાવ્યા મુજબ કોઈ ઊંચી ઈમારત પરથી મુક્ત પતન(free-fall) કરાવેલા દડાની ગતિને ચકાસો.

આકૃતિમાં લખેલા ગતિના આંકડા પરથી કહી શકાય કે દડાની ગતિ દર સેકન્ડે 9.8 મીટર/સેકન્ડ જેટલી વધે છે. મતલબ કે તેનો પ્રવેગ 9.8 મીટર/સેકન્ડ2 જેટલો છે.

દડાનું મુક્તપતન પૃથ્વીના ગુરુત્વાકર્ષણબળ(Gravitational Force)ના કારણે થયુ હોવાથી આ9.8 મીટર/સેકન્ડ2 ને ગુરુત્વ પ્રવેગ(Gravitational Acceleration) "g" કહે છે.

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Tricks of Mathematics

(1)

(2)

કાટકોણ ત્રિકોણની બાજૂઓના માપ n ગણા કરવા છતાં તેમની વચ્ચેનો સંબંધ જળવાઈ રહે છે.

(3)

કોઈ સંખ્યાના વર્ગનો તાળો મેળવવા માટે-

ન્યૂટનનો ગતિનો પહેલો નિયમ અથવા જડત્વનો નિયમ

ન્યૂટનનો ગતિનો પહેલો નિયમ કે જેને જડત્વનો નિયમ પણ કહે છે. તેને સમજતાં પહેલા આપણે કેટલાક સાદા પ્રયોગો કરી લઇએ, જેના દ્વારા આ નિયમ સમજવામાં સરળતા રહેશે.

તમે સૌએ આ પ્રયોગ તો કરેલો જ હશે. કાચના ખાલી ગ્લાસ ઊપર પોસ્ટકાર્ડ રાખી તેની ઉપર રૂપિયાનો સિક્કો રાખો. આકૃતિમાં બતાવ્યા પ્રમાણે ગ્લાસ પરના કાર્ડને આંગળી વડે એવી રીતે ધક્કો મારો કે તે સીધે સીધુ આગળ નીકળી જાય અને જૂઓ સિક્કાનું શું થાય છે? સિક્કો ગ્લાસની અંદર પડી ગયો ખરૂ ને ! આમ થવાનું કારણ પછી સમજીશું.

તમારામાંથી કોઈએ ચાલુ બસે ઊતરવાનો પ્રયત્ન કર્યો છે? જો કર્યો હોય તો તમને અનુભવ હશે કે તમારા પગ જમીન પર પડતાંની સાથે જ સ્થિર થઈ જાય છે, જ્યારે તમારૂ ધડ સહિતનું બાકીનું શરીર આગળની તરફ ઝૂકી જાય છે. અને તમે બરાબર ઊભા પણ નથી રહી શકતા.

ટેબલ પર 1 KGનું વજનિયું મૂકો. હવે આ વજનિયાને નમારી ટચલી આંગળી વડે ધક્કો મારવાનો પ્રયત્ન કરી જૂઓ. કેમ શું થયું? વજનિયું માંડ માંડ ખસ્યુ કે ન ખસ્યુ હોય તેવું પણ બને. સાચુ ને!
હવે એક ફૂટબોલ કે વોલિબોલના દડાને લો. તમારા મિત્રને કહો કે તે દડાને જમીન પર એવી રીતે ગબડાવે કે જેથી કરીને દડો ટપ્પા ખાધા વિના સરકીને તમારી તરફ આવે. આ વખતે પણ દડાને તમારે કોઈ એક આંગળી વડે જ રોકવાનો છે. શું થાય છે?

ઉપર જણાવેલા દરેક પ્રયોગ કરતી વેળાએ તમે અનુભવ્યુ હશે કે કોઈ સ્થિર પડેલી વસ્તુને ખસેડવા માટે(ગતિમાં લાવવા માટે) કે પછી ગતિમાં રહેલી વસ્તુને રોકવા માટે(અથવા તેની ગતિમાં ફેરફાર કરવા માટે) બળ લગાડવું પડે છે.

તો ન્યુટનનો ગતિનો પહેલો નિયમ આવા જ તારણો પરથી આ પ્રમાણે લખી શકાય-

" જ્યાં સુધી કોઈ પદાર્થ પર અસંતુલિત બાહ્ય બળ ન લગાડવામાં આવે ત્યાં સુધી તે પદાર્થ સ્થિર અવસ્થામાં હોય તો સ્થિર અવસ્થામાં રહેશે અને અચળવેગી ગતિ કરતો હોય તો તેની અચળવેગી ગતિ ચાલુ રહેશે."

(ગેલેલિયોએ કરેલા પ્રયોગો પરથી ગતિ વિશેના જે તારણો આપેલા તે ન્યૂટનના ગતિના પહેલા નિયમ જેવા જ હતા. માટે એમ કહી શકાય કે ન્યૂટને ગેલિલિયોના તારણોને વ્યવસ્થિત રીતે નિયમના સ્વરૂપે રજૂ કર્યા.)(જડત્વ મતલબ પદાર્થનો પોતાની અવસ્થા- સ્થિર કે ગતિ જાળવી રાખવાનો ગુણધર્મ.)

અહીં 'અસંતુલિત બાહ્યબળ' શબ્દ વાપરવાનું કારણ નીચે મુજબ સમજાવી શકાય-

તમે સૌ શાળામાં 'રસ્સા ખેંચ' નામની રમત તો રમ્યા જ હશો. અથવા તમારા મિત્રોને આ રમત રમતા જોયા હશે. આકૃતિમાં બતાવ્યા મુજબ તેમાં એક લાંબા મજબુત દોરડાંની બંને બાજુ બેય ટીમના ખેલાડી રસ્સાને પોતપોતાની બાજુ ખેંચવાનો પ્રયત્ન કરે છે. બંને ટીમની વચ્ચે એક મધ્યરેખા દોરેલી હોય છે. હવે જરાક વિચાર કરો કે જો બંને ટીમ સરખું બળ લગાવે તો શું થાય? બેય ટીમે લગાડેલું બળ સરખું અને વિરૂદ્ધદિશાનું હોવાથી પરિણામ શુન્ય આવશે. મતલબ કે દોરડું મધ્યરેખાની એકેય બાજુ નહીં ખસે. પણ જો ડાબી બાજુની ટીમે લગાડેલું બળ વધારે હોય તો? સ્વભાવિક રીતે જ જમણી બાજુનો પ્રથમ ખેલાડી મધ્યરેખાની ડાબી બાજુ ખેંચાઈ આવશે અને ડાબી ટીમ વિજેતા બનશે.
અહીં દોરડાને મધ્યરેખાની કોઈ એક બાજુ પર લાવવા માટે અસંતુલિત બળ લગાવવું જરૂરી છે કારણ કે સમાન મૂલ્યના વિરૂદ્ધ દિશાના બળ એકબીજાની અસર નાબૂદ કરી નાખે છે. આ જ કારણથી નિયમની અંદર 'અસંતુલિત બળ' શબ્દ વાપરવો જરૂરી બન્યો.

કદાચ તમારામાંથી કોઈને એવો વિચાર આવ્યો હશે કે ગતિમાન પદાર્થ પર બાહ્ય બળ ન લગાવવામાં આવે ત્યાં સુધી તે ગતિમાં રહેવો જોઈએ. તો પછી જમીન પર ગબડતો દડો કેમ અમુક અંતર કાપ્યા બાદ ધીમો પડીને છેવટે સ્થિર કેમ થઈ જાય છે? આ તો ન્યૂટનના ગતિના પ્રથમ નિયમનો ભંગ કહેવાયને?

જો તમને આ વિચાર આવ્યો હોય તો તમને અભિનંદન. અહીં નિયમનો ભંગ નથી થયો. નિયમ એવું કહે છે કે બાહ્ય બળ લાગુ ન પડે ત્યાં સુધી કોઈ ફેરફાર ન થાય, પરંતુ તમે જણાવેલા (વિચારેલા) કિસ્સામાં દડાને હવાનો અવરોધ તેમજ જમીન દ્વારા લાગતું ઘર્ષણબળ દડાની ગતિને ધીમો પાડીને છેવટે સ્થિર કરી દે છે. અહીં હવાનો અવરોધ અને જમીન દ્વારા લાગતું ઘર્ષણબળ પણ બાહ્યબળ જ કહેવાય ભલેને તે કોઈ વ્યક્તિ દ્વારા લગાવવામાં ન આવ્યા હોય.

તો હવે ન્યૂટનનો ગતિનો પહેલો નિયમ કે જેને જડત્વનો નિયમ પણ કહે છે, તે તમારા ભેજામાં બરાબર ઊતરી ગયો હશે. જો હજુ કોઈ શંકા હોય તો ફરીવાર પહેલેથી વાંચી લો. છેલ્લે બીજા એવા ઊદાહરણ જોઈ લઈએ જ્યાં જડત્વનો નિયમ વપરાયો હોય.

જ્યારે તમે ટોમેટો સોસની બોટલમાંથી સોસને બહાર કાઢો છો ત્યારે શું કરો છો? બોટલને ઝડપથી ખાલી પ્લેટ તરફ લાવી અચાનક જ રોકી પાડશો અને બોટલમાંનો બિચારો સોસ પ્લેટમાં પડી જશે.
જ્યારે તમે સોસની બોટલને ગતિમાં લાવો છો ત્યારે અંદરનો સોસ પણ ગતિમાં આવે છે. પણ જ્યારે બોટલને રોકી લો ત્યારે સોસની ગતિને રોકે તેવું કોઈ બાહ્ય બળ હોતું નથી. (સોસ અને બોટલ વચ્ચેનું ઘર્ષણબળ ખૂબ જ ઓછું હોવાથી સોસની ગતિને રોકવા માટે પૂરતું નથી હોતું.) પરિણામે તમારો મનગમતો સ્વાદિષ્ટ સોસ તમારી પ્લેટમાં આવી જાય છે.

જ્યારે બસનો ડ્રાઈવર અચાનક બ્રેક મારે ત્યારે તેના મુસાફરોને આગળ તરફ ઝૂકી જતાં જોયાં હશે. અહીં પણ જડત્વનો નિયમ જ કામ કરે છે.

Newton's Law of Motion

Newton's First Law of Motion:

I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

This we recognize as essentially Galileo's concept of inertia, and this is often termed simply the "Law of Inertia".

Newton's Second Law of Motion:

II. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors (as indicated by their symbols being displayed in slant bold font); in this law the direction of the force vector is the same as the direction of the acceleration vector.

This is the most powerful of Newton's three Laws, because it allows quantitative calculations of dynamics: how do velocities change when forces are applied. Notice the fundamental difference between Newton's 2nd Law and the dynamics of Aristotle: according to Newton, a force causes only a change in velocity (an acceleration); it does not maintain the velocity as Aristotle held.
This is sometimes summarized by saying that under Newton, F = ma, but under Aristotle F = mv, where v is the velocity. Thus, according to Aristotle there is only a velocity if there is a force, but according to Newton an object with a certain velocity maintains that velocity unless a force acts on it to cause an acceleration (that is, a change in the velocity). As we have noted earlier in conjunction with the discussion of Galileo, Aristotle's view seems to be more in accord with common sense, but that is because of a failure to appreciate the role played by frictional forces. Once account is taken of all forces acting in a given situation it is the dynamics of Galileo and Newton, not of Aristotle, that are found to be in accord with the observations.

Newton's Third Law of Motion:

III. For every action there is an equal and opposite reaction.

This law is exemplified by what happens if we step off a boat onto the bank of a lake: as we move in the direction of the shore, the boat tends to move in the opposite direction (leaving us facedown in the water, if we aren't careful!).

Mathematical Symbols

List of all mathematical symbols and signs - meaning and examples.

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	$\frac{6}{2}=3$
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√a · ³√a · ³√a = a	³√8 = 2
⁴√a	fourth root	⁴√a · ⁴√a · ⁴√a · ⁴√a = a	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	1ppt = 10^-12	10ppt × 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''
	line	infinite line
AB	line segment	line from point A to point B
	ray	line that start from point A
	arc	arc from point A to point B	= 60º
\|	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! = 123*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	∑ x_i= x₁+x₂+...+x_n
∑∑	sigma	double summation
∏	capital pi	product - product of all values in range of series	∏ x_i=x₁∙x₂∙...∙x_n
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
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle	c = π·d = 2·π·r

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
$\langle x,y \rangle$	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	(A^T)_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
σ_X	standard deviation	standard deviation value of random variable X	σ_X= 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) = 0.6
ρ_X,Y	correlation	correlation of random variables X and Y	ρ_X,Y = 0.6
∑	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 = (x_max+x_min)/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
s²	sample variance	population samples variance estimator	s² = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
z_x	standard score	z_x = (x-x) / s_x
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 x^c-1e^-λx / Γ(c), x≥0
χ²(k)	chi-square distribution	f (x) = x^k^/2-1e^-x/2 / ( 2^k/2Γ(k/2) )
F (k₁, k₂)	F distribution
Bin(n,p)	binomial distribution	f (k) = _nC_k p^k(1-p)^n-k
Poisson(λ)	Poisson distribution	f (k) = λ^ke^-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
_nP_k	permutation	$_{n}P_{k}=\frac{n!}{(n-k)!}$	₅P₃ = 5! / (5-3)! = 60
_nC_k	combination	$_{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!}$	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 fewer elements or equal to the set	{9,14,28} ⊆ {9,14,28}
A ⊂ B	proper subset / strict subset	subset has fewer 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}
2^A	power set	all subsets of A
$\mathcal{P}(A)$	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
A^c	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
	aleph-null	infinite cardinality of natural numbers set
	aleph-one	cardinality of countable ordinal numbers set
Ø	empty set	Ø = { }	C = {Ø}
$\mathbb{U}$	universal set	set of all possible values
$\mathbb{N}$ ₀	natural numbers / whole numbers set (with zero)	$\mathbb{N}$ ₀ = {0,1,2,3,4,...}	0 ∈ $\mathbb{N}$ ₀
$\mathbb{N}$ ₁	natural numbers / whole numbers set (without zero)	$\mathbb{N}$ ₁ = {1,2,3,4,5,...}	6 ∈ $\mathbb{N}$ ₁
$\mathbb{Z}$	integer numbers set	$\mathbb{Z}$ = {...-3,-2,-1,0,1,2,3,...}	-6 ∈ $\mathbb{Z}$
$\mathbb{Q}$	rational numbers set	$\mathbb{Q}$ = {x \| x=a/b, a,b∈ $\mathbb{Z}$ }	2/6 ∈ $\mathbb{Q}$
$\mathbb{R}$	real numbers set	$\mathbb{R}$ = {x \| -∞ < x <∞}	6.343434 ∈ $\mathbb{R}$
$\mathbb{C}$	complex numbers set	$\mathbb{C}$ = {z \| z=a+bi, -∞<a<∞, -∞<b<∞}	6+2i ∈ $\mathbb{C}$

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 (iff)
↔	equivalent	if and only if (iff)
∀	for all
∃	there exists
∄	there does not exists
∴	therefore
∵	because / since

Calculus & analysis symbols

Symbol	Symbol Name	Meaning / definition	Example
$\lim_{x\to x0}f(x)$	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 - Lagrange's notation	(3x³)' = 9x²
y ''	second derivative	derivative of derivative	(3x³)'' = 18x
y⁽ⁿ⁾	nth derivative	n times derivation	(3x³)⁽³⁾ = 18
$\frac{dy}{dx}$	derivative	derivative - Leibniz's notation	d(3x³)/dx = 9x²
$\frac{d^2y}{dx^2}$	second derivative	derivative of derivative	d²(3x³)/dx² = 18x
$\frac{d^ny}{dx^n}$	nth derivative	n times derivation
$\dot{y}$	time derivative	derivative by time - Newton's notation
	time second derivative	derivative of derivative
D_xy	derivative	derivative - Euler's notation
D_x²y	second derivative	derivative of derivative
$\frac{\partial f(x,y)}{\partial x}$	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)
	Laplace transform	F(s) = {f (t)}
	Fourier transform	X(ω) = {f (t)}
δ	delta function
∞	lemniscate	infinity symbol

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	Greek Letter Name	English Equivalent	Pronunciation
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
0	not defined
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

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! = 123*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	∑ x_i= x₁+x₂+...+x_n
∑∑	sigma	double summation
∏	capital pi	product - product of all values in range of series	∏ x_i=x₁∙x₂∙...∙x_n
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
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle	c = π·d = 2·π·r

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
$\langle x,y \rangle$	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	(A^T)_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
σ_X	standard deviation	standard deviation value of random variable X	σ_X= 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) = 0.6
ρ_X,Y	correlation	correlation of random variables X and Y	ρ_X,Y = 0.6
∑	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 = (x_max+x_min)/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
s²	sample variance	population samples variance estimator	s² = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
z_x	standard score	z_x = (x-x) / s_x
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 x^c-1e^-λx / Γ(c), x≥0
χ²(k)	chi-square distribution	f (x) = x^k^/2-1e^-x/2 / ( 2^k/2Γ(k/2) )
F (k₁, k₂)	F distribution
Bin(n,p)	binomial distribution	f (k) = _nC_k p^k(1-p)^n-k
Poisson(λ)	Poisson distribution	f (k) = λ^ke^-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
_nP_k	permutation	$_{n}P_{k}=\frac{n!}{(n-k)!}$	₅P₃ = 5! / (5-3)! = 60
_nC_k	combination	$_{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!}$	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 fewer elements or equal to the set	{9,14,28} ⊆ {9,14,28}
A ⊂ B	proper subset / strict subset	subset has fewer 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}
2^A	power set	all subsets of A
$\mathcal{P}(A)$	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
A^c	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
	aleph-null	infinite cardinality of natural numbers set
	aleph-one	cardinality of countable ordinal numbers set
Ø	empty set	Ø = { }	C = {Ø}
$\mathbb{U}$	universal set	set of all possible values
$\mathbb{N}$ ₀	natural numbers / whole numbers set (with zero)	$\mathbb{N}$ ₀ = {0,1,2,3,4,...}	0 ∈ $\mathbb{N}$ ₀
$\mathbb{N}$ ₁	natural numbers / whole numbers set (without zero)	$\mathbb{N}$ ₁ = {1,2,3,4,5,...}	6 ∈ $\mathbb{N}$ ₁
$\mathbb{Z}$	integer numbers set	$\mathbb{Z}$ = {...-3,-2,-1,0,1,2,3,...}	-6 ∈ $\mathbb{Z}$
$\mathbb{Q}$	rational numbers set	$\mathbb{Q}$ = {x \| x=a/b, a,b∈ $\mathbb{Z}$ }	2/6 ∈ $\mathbb{Q}$
$\mathbb{R}$	real numbers set	$\mathbb{R}$ = {x \| -∞ < x <∞}	6.343434 ∈ $\mathbb{R}$
$\mathbb{C}$	complex numbers set	$\mathbb{C}$ = {z \| z=a+bi, -∞<a<∞, -∞<b<∞}	6+2i ∈ $\mathbb{C}$

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 (iff)
↔	equivalent	if and only if (iff)
∀	for all
∃	there exists
∄	there does not exists
∴	therefore
∵	because / since

Calculus & analysis symbols

Symbol	Symbol Name	Meaning / definition	Example
$\lim_{x\to x0}f(x)$	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 - Lagrange's notation	(3x³)' = 9x²
y ''	second derivative	derivative of derivative	(3x³)'' = 18x
y⁽ⁿ⁾	nth derivative	n times derivation	(3x³)⁽³⁾ = 18
$\frac{dy}{dx}$	derivative	derivative - Leibniz's notation	d(3x³)/dx = 9x²
$\frac{d^2y}{dx^2}$	second derivative	derivative of derivative	d²(3x³)/dx² = 18x
$\frac{d^ny}{dx^n}$	nth derivative	n times derivation
$\dot{y}$	time derivative	derivative by time - Newton's notation
	time second derivative	derivative of derivative
D_xy	derivative	derivative - Euler's notation
D_x²y	second derivative	derivative of derivative
$\frac{\partial f(x,y)}{\partial x}$	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)
	Laplace transform	F(s) = {f (t)}
	Fourier transform	X(ω) = {f (t)}
δ	delta function
∞	lemniscate	infinity symbol

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	Greek Letter Name	English Equivalent	Pronunciation
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
0	not defined
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

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! = 123*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	∑ x_i= x₁+x₂+...+x_n
∑∑	sigma	double summation
∏	capital pi	product - product of all values in range of series	∏ x_i=x₁∙x₂∙...∙x_n
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
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle	c = π·d = 2·π·r

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
$\langle x,y \rangle$	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	(A^T)_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
σ_X	standard deviation	standard deviation value of random variable X	σ_X= 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) = 0.6
ρ_X,Y	correlation	correlation of random variables X and Y	ρ_X,Y = 0.6
∑	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 = (x_max+x_min)/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
s²	sample variance	population samples variance estimator	s² = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
z_x	standard score	z_x = (x-x) / s_x
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 x^c-1e^-λx / Γ(c), x≥0
χ²(k)	chi-square distribution	f (x) = x^k^/2-1e^-x/2 / ( 2^k/2Γ(k/2) )
F (k₁, k₂)	F distribution
Bin(n,p)	binomial distribution	f (k) = _nC_k p^k(1-p)^n-k
Poisson(λ)	Poisson distribution	f (k) = λ^ke^-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
_nP_k	permutation	$_{n}P_{k}=\frac{n!}{(n-k)!}$	₅P₃ = 5! / (5-3)! = 60
_nC_k	combination	$_{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!}$	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 fewer elements or equal to the set	{9,14,28} ⊆ {9,14,28}
A ⊂ B	proper subset / strict subset	subset has fewer 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}
2^A	power set	all subsets of A
$\mathcal{P}(A)$	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
A^c	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
	aleph-null	infinite cardinality of natural numbers set
	aleph-one	cardinality of countable ordinal numbers set
Ø	empty set	Ø = { }	C = {Ø}
$\mathbb{U}$	universal set	set of all possible values
$\mathbb{N}$ ₀	natural numbers / whole numbers set (with zero)	$\mathbb{N}$ ₀ = {0,1,2,3,4,...}	0 ∈ $\mathbb{N}$ ₀
$\mathbb{N}$ ₁	natural numbers / whole numbers set (without zero)	$\mathbb{N}$ ₁ = {1,2,3,4,5,...}	6 ∈ $\mathbb{N}$ ₁
$\mathbb{Z}$	integer numbers set	$\mathbb{Z}$ = {...-3,-2,-1,0,1,2,3,...}	-6 ∈ $\mathbb{Z}$
$\mathbb{Q}$	rational numbers set	$\mathbb{Q}$ = {x \| x=a/b, a,b∈ $\mathbb{Z}$ }	2/6 ∈ $\mathbb{Q}$
$\mathbb{R}$	real numbers set	$\mathbb{R}$ = {x \| -∞ < x <∞}	6.343434 ∈ $\mathbb{R}$
$\mathbb{C}$	complex numbers set	$\mathbb{C}$ = {z \| z=a+bi, -∞<a<∞, -∞<b<∞}	6+2i ∈ $\mathbb{C}$

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 (iff)
↔	equivalent	if and only if (iff)
∀	for all
∃	there exists
∄	there does not exists
∴	therefore
∵	because / since

Calculus & analysis symbols

Symbol	Symbol Name	Meaning / definition	Example
$\lim_{x\to x0}f(x)$	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 - Lagrange's notation	(3x³)' = 9x²
y ''	second derivative	derivative of derivative	(3x³)'' = 18x
y⁽ⁿ⁾	nth derivative	n times derivation	(3x³)⁽³⁾ = 18
$\frac{dy}{dx}$	derivative	derivative - Leibniz's notation	d(3x³)/dx = 9x²
$\frac{d^2y}{dx^2}$	second derivative	derivative of derivative	d²(3x³)/dx² = 18x
$\frac{d^ny}{dx^n}$	nth derivative	n times derivation
$\dot{y}$	time derivative	derivative by time - Newton's notation
	time second derivative	derivative of derivative
D_xy	derivative	derivative - Euler's notation
D_x²y	second derivative	derivative of derivative
$\frac{\partial f(x,y)}{\partial x}$	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)
	Laplace transform	F(s) = {f (t)}
	Fourier transform	X(ω) = {f (t)}
δ	delta function
∞	lemniscate	infinity symbol

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	Greek Letter Name	English Equivalent	Pronunciation
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
0	not defined
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

images

ગુરુવાર, 12 ફેબ્રુઆરી, 2015

ન્યૂટનનો ગતિનો પહેલો નિયમ અથવા જડત્વનો નિયમ

Newton's First Law of Motion:

Newton's Second Law of Motion:

Newton's Third Law of Motion:

Mathematical Symbols

Geometry symbols

Algebra symbols

Linear Algebra Symbols

Probability and statistics symbols

Combinatorics Symbols

Set theory symbols

Logic symbols

Calculus & analysis symbols

Numeral symbols

Greek alphabet letters

Roman numerals

Algebra symbols

Linear Algebra Symbols

Probability and statistics symbols

Combinatorics Symbols

Set theory symbols

Logic symbols

Calculus & analysis symbols

Numeral symbols

Greek alphabet letters

Roman numerals

Algebra symbols

Linear Algebra Symbols

Probability and statistics symbols

Combinatorics Symbols

Set theory symbols

Logic symbols

Calculus & analysis symbols

Numeral symbols

Greek alphabet letters

Roman numerals