Geometry Nets

Volume and Surface area of Cone, Pyramid and Triangular pris

Surface Area and Volume of a Pyramid

Volume of a Pyramid

Volume of a Cube

Surface Area

Surface Area and Volume of a Sphere

Volume of a Sphere

Volume of a Cylinder

Volume of a Cone

Shakuntala Devi

Calculating prodigy Shakuntala Devi on RT

Numbers link us all together

Revisiting Numbers with Shakuntala Devi

Indian Mathematicians BHASKARACHARYA

Bhaskaracharya

• He was born in a village of Mysore district.
• He was the first to give that any number divided by 0 gives infinity (00).
• He has written a lot about zero, surds, permutation and combination.
• He wrote, “The hundredth part of the circumference of a circle seems to be straight. Our earth is a big sphere and that’s why it appears to be flat.”
• He gave the formulae like sin(A ± B) = sinA.cosB ± cosA.sinB

Indian Mathematicians SHAKUNTALA DEVI

Shakuntala Devi

• She was born in 1939
• In 1980, she gave the product of two, thirteen digit numbers within 28 seconds, many countries have invited her to demonstrate her extraordinary talent.
• In Dallas she competed with a computer to see who give the cube root of 188138517 faster, she won. At university of USA she was asked to give the 23^rd root of 91674867692003915809866092758538016248310668014430862240712651642793465704086709659 32792057674808067900227830163549248523803357453169351119035965775473400756818688305 620821016129132845564895780158806771.

She answered in 50seconds. The answer is 546372891. It took a UNIVAC 1108 computer, full one minute (10 seconds more) to confirm that she was right after it was fed with 13000 instructions.

• Now she is known to be Human Computer.

Indian Mathematicians BRAHMAGUPTA

Brahmagupta

• Brahma Gupta was born in 598A.D in Pakistan.
• He gave four methods of multiplication.

• He gave the following formula, used in G.P series

a + ar + ar² + ar³ +……….. + ar^n-1 = (ar^n-1) ÷ (r – 1)

• He gave the following formulae :

Area of a cyclic quadrilateral with side a, b, c, d= √(s -a)(s- b)(s -c)(s- d) where 2s = a + b + c + d Length of its diagonals =

Indian Mathematicians ARYABHATA

Aryabhatta

• Aryabhatta was born in 476A.D in Kusumpur, India.
• He was the first person to say that Earth is spherical and it revolves around the sun.
• He gave the formula (a + b)² = a² + b² + 2ab

• He taught the method of solving the following problems:

Indian Mathematicians RAMANUJAN

Ramanujan

He was born on 22na of December 1887 in a small village of Tanjore district, Madras. He failed in English in Intermediate, so his formal studies were stopped but his self-study of mathematics continued.

He sent a set of 120 theorems to Professor Hardy of Cambridge. As a result he invited Ramanujan to England.

Ramanujan showed that any big number can be written as sum of not more than four prime numbers.

He showed that how to divide the number into two or more squares or cubes.

when Mr Litlewood came to see Ramanujan in taxi number 1729, Ramanujan said that 1729 is the smallest number which can be written in the form of sum of cubes of two numbers in two ways, i.e. 1729 = 93 + 103 = 13 + 123 since then the number 1729 is called Ramanujan’s number.

In the third century B.C, Archimedes noted that the ratio of circumference of a circle to its diameter is constant. The ratio is now called ‘pi ( Π )’ (the 16th letter in the Greek alphabet series)

The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers as big as 1053 with specific names as early as 5000 B.C. during the Vedic period.

Prime Numbers

What are prime numbers?
Prime numbers are natural numbers which have only two divisors , i.e 1 and the number itself.Identifying prime numbers is very important. Prime numbers comes under the study of elementary math. Learn about prime numbers and distinguish it with composite numbers with our expert tutors.

List of prime numbers
Till now it has been explained about the concept of prime numbers and below is listed prime numbers for your better knowledge.
The list of prime numbers from 1 to 200 are as follows:
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
127
131
137
139
149
151
157
163
167
173
179
181
191
193
197
199




Is 1 a prime number?
According to the definition, we could say 1 is not a prime number. Below is given the Explanation:
The definition says that a prime number should have exactly two divisors. But 1 has only one divisor. So, 1 is not a prime number.Students can also get to learn about the concept of First prime number and working first prime number problems.
Prime numbers and Composite numbers
The numbers which are exactly divisible by a number except 1 and the same number are termed as composite numbers.
Example: 4 is a composite number since it is exactly divisible by 2 other than 1 and 4. Set of prime numbers with composite numbers and 1 forms the set of positive integers.Students can learn more about the Prime numbers by observing the examples of Prime numbers.
Prime number theorem states that if we select a random number nearby some larger number N, the chance of the random number to be prime is 1/ln N. ln N represents natural logarithm of N. Below are few facts of prime numbers which will help you to get math answers:
2 is the only even prime number as all other even numbers are divided by 2.
Zero and 1 are not considered as prime numbers
If the sum of a number's digits is a multiple of 3, that number can be divided by 3.
No prime number greater than 5 ends in a 5 as any number greater than 5 that ends in a 5 is divided by 5

Exponential and Logarithmic Series

The Number e

The sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...¥ is called the exponential number and is denoted by 'e'.

Exponential Series

If x is any complex number then the series is called the exponential series. It can be proved mathematically that this exponential series has a sum and we denote it by e^x.

Exponential Theorem

If a > 0, then prove that

Graph of Exponential Function

We see that as x increases, the value of e^x also increases indefinitely. Also, as x decreases, the value of e^x tends toward zero. The function e^x is one-one.

Exponential and Logarithmic Function

The function e^x (where e is the number approximately 2.718) is called exponential function.
and the functions of the form y= log_bx where b > 0 and b ≠ 1 is known as logarithmic functions.

Some Particular Exponential Series

The following are the Some Particular Exponential Series:

Graph of Logarithmic Series

We see that as x increases from 0 to ¥, the value of log x also increases indefinitely. The function log x is one-one.

Logarithmic Series

If x is a real number such that |x|<1,>

Pie Graph or Pie Chart

To find the angle of each sector

Total of data corresponds to 360^o.

Let x^o = the angle at the centre for item A, then

The data given in example 1 can be used to draw a pie graph.

Calculation of Angles

Food:

Angle at centre

= 150^o

Rent:

Angle at centre

= 40^o

Similarly we can calculate the remaining angles, and the total of angles column should always come to 360^o.

A survey was conducted to find out the consumption of various brands of soap. The results of the survey are given below:

(i) Soap Cake:

(ii) Soap Powder:

(i) Using the information given in the table for soap cake, draw a pie chart.

(ii) Using the angles in the pie chart for soap powder, complete the table that follows:

(i) Draw Pie Chart

Total of the items = 100

A. Angle at centre

= 216^o

B. Angle at centre

= 72^o

C. Angle at centre

= 54^o

Other Angle at centre

= 18^o

(ii) Complete the table

For P,

Similarly

Same way we can calculate the values of R, S and others.

R = 25%, S = Others =

Graph of y = cosx

Graph:

From the graph it is clear that the curve repeats itself every 360^o (2prad). This fact is expressed by the statement that the function has a period of 360^o (2p radians). In symbols we write cos(x + 360^o.n) = cos(x + 2pn) = cos x where n is a positive or negative integer.

From the graph we also observe that cosx does not pass through the origin. The maximum and minimum values of cosx are +1 and -1 respectively. As x increases from 0^o to 90^o cosx decreases from 1 to 0, as x increases from 90^o to 180^o cosx decreases from 0 to -1, as x increases from 180^o to 270^ocosx increases from -1 to 0, as x increases from 270^o to 360^o cosx increases from 0 to 1. Cosx is period and has a period 2p.

Graph of y = sin x

Now let us construct the graph of y = sinx from x = 0^o to 360^o. The following table is readily constructed for intervals of 30^o.

Plotting the points and drawing a smooth curve through them we have the curve as shown in figure.

Table:

Graph:

From the figure it is evident that the curve repeats itself every 360^o or 2p. This fact is expressed by saying that the function has a period of 360^o or 2p.

In symbols we write sin (x + n.360^o) or sin (x + 2np), sinx = sin (x + n.360^o) = sin (x + 2np), where n is any positive or negative integer. This infers that sinx varies and takes a complete ordered range of values once and that sinx is periodic has the period 2p. From the figure we observe that as x increases from 0^o to 90^o, sinx increases from 0 to 1 and as x increases from 90^o to 180^o, sinx decreases from 1 to 0.

[A function f(x) is periodic with period T if f(x+T) = f(x) for all values of x]

As x increases from 180^o to 270^o, sinx decreases from 0 to -1 and as x increases from 270^o to 360^o, sinx increases from -1 to 0. The maximum absolute value of sin x = 1.

Angles

Fig.(i) Fig.(ii) Fig.(iii)

Let OA and OB two half lines with common end point O. The half lines OA and OB are the sides of an angle and the point O is the vertex of the angle. An angle is an amount of rotation of a half-line (or ray) in a plane about its end point from an initial position to a terminal position.

Measurement of angle

The amount of rotation from initial side to terminal is called the measure of an angle.

Positive and Negative angles

Angles that are formed by counter clockwise (anti clockwise) rotation, such as the one shown in fig (ii) are said to be positive or to have positive measure.

Angles that are formed by a clockwise rotation, like the one in fig(iii) are said to be negative or to have negative measure.

Lines at right angles

The lines are said to be at right angles if the rotating half line (or ray) from starting from initial position to the final position describes one quarter of a circle.

Quadrants

Let X'OX and Y'OY perpendicular coplanar lines intersecting each other at O. We refer X'OX as x-axis and Y'OY as y-axis. It is clear from the adjoining figure, that these two lines divide the plane into four equal parts, each part is called a Quadrant.

The four Quadrants are:

XOY - first Quadrant

YOX' - second Quadrant

X'OY' - third Quadrant

Y'OX - fourth Quadrant

Angle in standard position

If an angle of any measure be given, one can always construct (or draw) a cartesian co-ordinate reference frame in such a way that the origin is at the vertex of the angle, and positive half of the x axis coincides with the initial side of the angle. When this has been achieved, the angle is said to be in standard position. An angle in standard position is said to be in the Quadrant in which its terminal side lies.

vi) An angle is called Quadrant angle if it is in standard position and its terminal side coincides with one of the co-ordinate axis.

Trignometric Ratios