Thursday 28 September 2017

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.

Thursday 21 September 2017

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
4a · 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
limxa
limit
limit value of a function
limxa(3x+1)=3×a+1=3a+1
y
derivative
derivative – Lagrange’s notation
(5x3)=15x2
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 3xn=3n(n1)(n2).(2)(1)=3n!
y
second derivative
derivative of derivative
(4x3)=24x
d2ydx2
second derivative
derivative of derivative
d2dx2(6x3+x2+3x+1)=36x+1
dy/dx
derivative
derivative – Leibniz’s notation
ddx(5x)=5
dnydxn
nth derivative
n times derivation
 
¨y=d2ydt2
Second derivative of time
derivative of derivative
 
˙y
Single derivative of time
derivative by time – Newton’s notation
 
D2x
second derivative
derivative of derivative
 
Dx
derivative
derivative – Euler’s notation
 
integral
opposite to derivation
 
 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+biz = abi
z = 3 + 2i
x
vector
V=x^i+y^j+z^k
 
x * y
convolution
y(t) = x(t) * h(t)
 
lemniscate
infinity symbol
 
δ
delta function