ASSIGNMENT 3 (REPORT ON RAMANUJAN MUSEUM)

ASSIGNMENT 2 (DIFFERENT METHODOLOGY OF TEACHING)

TEACHER CENTERED METHOD
STUDENT CENTERED METHOD
RECENT TRENDS METHOD

ASSIGNMENT 1 (QUESTION PAPER, ANSWER PAPER,WEIGHTAGE,BLUE PRINT)

MATHS QUOTES

1.        As for everything else, so for mathematical theory: beauty can be perceived but not explained.—Arthur Cayley

2.        It isn’t that they cannot see the solution.  It is that they cannot see the problem.—GK Chesterton

3.        The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems.—C.W. Churchmann

4.        The mathematician has reached the highest rung on the ladder of human thought.—Havelock Ellis

5.        The heart has its reasons which reason knows not of.—Blaise Pascal

6.        The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.—Ernest Renan

7.        An expert problem solver must be endowed with two incompatible quantities: a restless imagination and a patient pertinacity.--Howard W. Even

8.        Technical skill is mastery of complexity.  Creativity is mastery of simplicity.—E.C. Zeeman

9.        Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition.—John Arbuthnot

10.     I must study politics and war that my sons may have liberty to study mathematics and philosophy.—John Adams

11.     A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.—Howard W. Eves

12.     The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality.—C.O Oakley

13.     You have to run as fast as you can just to stay where you are.  If you want to get anywhere, you'll have to run much faster.—Lewis Carroll

14.     Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.—John William Navin Sullivan (1886-1937)

15.     I have had my results for a long time: but I do not yet know how I am to arrive at them.—Karl Friedrich Gauss (1777-1855)

16.     How could youths better learn to live than by once trying the experiment of living? Methinks this would educate their minds as much as mathematics.— Henry David Thoreau

17.     The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.—William James (1842 - 1910)

18.     Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.—Blaise Pascal (1623-1662)

19.     Give me a place to stand and I will move the earth.—On the lever in Pappus Synagoge

20.     Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.—Newton

21.     Perfect clarity would profit the intellect but damage the will.—Blaise Pascal

22.     It is not enough to have a good mind. The main thing is to use it well.—Rene Descartes

23.     Each problem that I solved became a rule which served afterwards to solve other problems.—Rene Descartes

24.     If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.—John Louis von Neumann

25.     Pure mathematics is, in its way, the poetry of logical ideas.—Albert Einstein

26.     So if a man's wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again.—Francis Bacon, "Of Studies"

27.     Go down deep enough into anything and you will find mathematics.—Dean Schlicter

28.     Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.  Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.—W.S. Anglin

29.     Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.—Alfred North Whitehead

30.     Mathematics is the supreme judge; from its decisions there is no appeal.—Tobias Dantzig

31.     It is clear that the chief end of mathematical study must be to make the students think.—John Wesley Young

32.     Mathematics is no more computation than typing is literature.—John Allen Paulos

33.     The moving power of mathematical invention is not reasoning but imagination.—Augustus de Morgan

34.     Questions are creative acts of intelligence.—Frank Kingdon

35.     Two and two the mathematician continues to make four, in spite of the whine of the amateur for three, or the cry of the critic for five.—James McNeill Whistler

36.  The work of a teacher -- exhausting, complex, idiosyncratic, never twice the same -- is at its heart, an intellectual and

      ethical enterprise.  Teaching is the vocation of vocations...—William Ayres

37.     To learn, you must want to be taught.–Proverbs 12:1

38.     Failure is the opportunity to begin again, more intelligently.—Henry Ford

39.     You live your life between your ears.—Bebe Moore Campbell

40.     The study of mathematics is apt to commence in disappointment.—Alfred North Whitehead

41.     If there's no struggle, there's no progress.—Frederick Douglass

42.     The only way to learn mathematics is to do mathematics.-- Paul Halmos

43.     The purpose of computing is insight, not numbers!—R. W. Hamming

44.     Do not falter or shrink.  Just think out your work, and work out your think.—Nixon Waterman

45.     There is a difference between not knowing and not knowing yet.—Shelia Tobias

46.     Millions saw the apple fall, but Newton asked why.—Bernard Baruch

47.     Problems cannot be solved at the same level of awareness that created them.—Albert Einstein

ARYABHATA- BIOGRAPHY

Born: 476, probably in Ashmaka

Died: 550 (at age 74), location unknown

Nationality: Indian

Famous For: Early mathematician who calculated the value of pi

Aryabhata (476-550) was an Indian mathematician and astronomer. He is generally considered to have begun the line of great Indian astronomer-mathematicians that flourished during the country’s classical period. Several of his calculations showed remarkable accuracy for the era, with some remaining the best available for many centuries. He is sometimes referred to as Aryabhata I, since several later scientists of the same name also produced notable works.

Aryabhata’s Early Life

Aryabhata came from southern India, but his precise place of birth is not known. Some authorities suggest that Kerala is the most likely location, while others believe that Dhaka or Maharashtra are more probable. It is, however, generally accepted that he studied at an advanced level in Kusumapura in modern-day Patna, where he remained for some years.

A contemporary poem places Aryabhata as the manager of a scientific institution; the precise nature of the body is not given, but there are grounds for suspecting that it may have been linked to the astronomical observatory that was maintained there by the University of Nalanda.

The Aryabhatiya

While studying at the university, Aryabhata produced the Aryabhatiya, his major work. Written at the age of just 23, it ranges widely across mathematics and astronomy, but is particularly notable for its calculations regarding planetary periods. The value given for the length of the Earth’s astronomical day differs from the true value by only a matter of minutes.

Aryabhata also worked out a value for pi that equates to 3.1416, very close to the approximations still used today. Using this value, he was able to calculate that the Earth had a circumference of 24,835 miles. This is correct to within 0.2%, and remained the best figure available well into medieval times.

While working on the calculation of pi, it is possible that Aryabhata may also have discovered that number’s irrationality. The relevant text is inconclusive on this point, but if he did establish the irrational nature of pi, he beat the first European mathematicians to do this by many hundreds of years.

The Aryabhatiya also contains solid work regarding the solar system. It states correctly that the light cast by planets and the moon is caused by sunlight reflecting off their surfaces, and that all planets follow elliptical orbits. Aryabhata was also able to describe accurately the processes that lead to both solar and lunar eclipses.

Aryabhata’s Legacy

For several hundred years after its author’s death, the Aryabhatiya was unknown in the West, although its Arabic translation in the 9th century was of great use to the scientists of the Islamic Golden Age. The book was eventually translated into Latin shortly after 1200. The mathematical ideas contained within it were quickly adopted by Europeans, especially those dealing with areas and volumes, and with finding cube and square roots.

However, Aryabhata’s astronomical findings had less impact, and it was left to later men such as Copernicus and Galileo to bring about the Western astronomical revolution. The first Indian artificial was named Aryabhata in his honor, as was a new university in the state of Bihar.

BASIC MATH SYMBOLS

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

Calculus & Analysis Symbols

Symbol	Symbol Name	Meaning / definition	Example
ε	epsilon	represents a very small number, near zero	ε → 0
$\underset{x\to a}{lim}$	limit	limit value of a function	$\underset{x\to a}{lim}(3x+1)=3\times a+1=3a+1$
y ‘	derivative	derivative – Lagrange’s notation	${\left(5x^3\right)}^{\prime }=15x^2$
e	e constant / Euler’s number	e = 2.718281828…	e = lim (1+1/x)x , x→∞
y(n)	nth derivative	n times derivation	nth derivative of $3x^n=3n(n-1)(n-2)\dots .\left(2\right)\left(1\right)=3n!$
y ”	second derivative	derivative of derivative	$\left(4x^3\right)”=24x$
$\frac{d^{2}y}{d{x}^2}$	second derivative	derivative of derivative	$\frac{d^2}{d{x}^2}(6x^3+x^2+3x+1)=36x+1$
dy/dx	derivative	derivative – Leibniz’s notation	$\frac{d}{dx}\left(5x\right)=5$
$\frac{d^{n}y}{d{x}^n}$	nth derivative	n times derivation
$\ddot{y}=\frac{d^{2}y}{d{t}^2}$	Second derivative of time	derivative of derivative
$\dot{y}$	Single derivative of time	derivative by time – Newton’s notation
$D^{2}x$	second derivative	derivative of derivative
Dx	derivative	derivative – Euler’s notation
∫	integral	opposite to derivation
$\frac{af(x,y)}{ax}$	partial derivative		∂(x2+y2)/∂x = 2x
∭	triple integral	integration of function of 3 variables
∬	double integral	integration of function of 2 variables
∯	closed surface integral
∮	closed contour / line integral
[a,b]	closed interval	[a,b] = {x | a ≤ x ≤ b}
∰	closed volume integral
(a,b)	open interval	(a,b) = {x | a < x < b}
z*	complex conjugate	z = a+bi → z*=a-bi	z* = 3 + 2i
i	imaginary unit	i ≡ √-1	z = 3 + 2i
∇	nabla / del	gradient / divergence operator	∇f (x,y,z)
z	complex conjugate	z = a+bi → z = a–bi	z = 3 + 2i
$\vec{x}$	vector	$\vec{V}=x\hat{i}+y\hat{j}+z\hat{k}$
x * y	convolution	y(t) = x(t) * h(t)
∞	lemniscate	infinity symbol
δ	delta function