BHASKARA I – REPRESENTATION OF      

                               NUMBERS

                Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations had been known to Indian astronomers approximately 500 years prior to this work. However, these numbers, prior to Bhaskara, were written not in figures but in words or allegories, and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were right from the lower ones.

His system is truly positional, since the same words representing, can also be used to represent the values 40 or 400.^[5] Quite remarkably, he often explains a number given in this system, using the formula ankair api ("in figures this reads"), by repeating it written with the first nine Brahmi numerals, using a small circle for the zero . Contrary to his word system, however, the figures are written in descending valuedness from left to right, exactly as we do it today. Therefore, at least since 629 the decimal system is definitely known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit.

      BRAHMAGUPTA’S  CONTRIBUTION

Pythagorean triples

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta provides a formula useful for generating Pythagorean triples:

         The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.

Or, in other words, if d = mx/x + 2, then a traveller who "leaps" vertically upwards a distance d from the top of a mountain of height m, and then travels in a straight line to a city at a horizontal distance mx from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city. Stated geometrically, this says that if a right-angled triangle has a base of length a = mx and altitude of length b = m + d, then the length, c, of its hypotenuse is given by c = m(1 + x) − d. And, indeed, elementary algebraic manipulation shows that a² + b² = c² whenever d has the value stated. Also, if m and x are rational, so are d, a, b and c. A Pythagorean triple can therefore be obtained from a, b and c by multiplying each of them by the least common multiple of their denominators.

                    VALUE OF PIE

Pi is defined as the ratio of the circumference of a circle to the diameter that is approximately equal to 3.14159. In a circle, if you divide the circumference (is the total distance around the circle) by the diameter, you will get exactly the same number. Whether the circle is big or small, the value of pi remains the same. The symbol of Pi is denoted by π and pronounced as “pie”. It is the 16th letter of the Greek alphabet and used to represent a mathematical constant.

Pi is also known as an irrational number which means that the digits after decimal point never end being a non-terminating value. Therefore, 22/7 is used for everyday calculations. π is not equal to the ratio of any two number which makes it an irrational number.

How to calculate value of Pi?

π=circumferenceDiameter

π = 3.14159

Let us understand the concept of Pi with the following a few examples.

Let’s have a look at the first 100 decimal places of Pi

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 7067

Examples

Question – A boy walks around a circle which has a diameter of 100 m, how far the boy has walked?

Solution:

Distance walked = Circumference

Circumference = π × 100m = 314.159 m

Therefore, circumference = 3.14m

length, weight, sizes and percentages as well.

            RATIO AND PROPORTION

Ratio and Proportion are mathematical entities that help in comparison of numbers. In other words a ratio can be defined as a relationship between two numbers that helps in defining the quantity of the first number with the second number. You can learn more about it by downloading the Ratio and Proportion PDF below.

In solving any kind of problems it is important to learn ratio and proportion formulas, as this helps in solving any kind of mathematical problems one may encounter. If we consider two numbers as a and b, then the ratio of a to b is written as a : b = a / b. Here a is antecedent and b is the consequent. This is what as known as a ratio.

Proportions, on the other hand can simply be defined when two ratios are equal to each other. These ratios can be of definable quantities such as shapes and sizes or similar numbers such as fractions. Proportions can be used to calculate metrics such as length, weight, sizes and percentages as well.

#Surface Area Of Sphere 

Have you ever wondered how the Surface Area of a Sphere was derived?

Well here is a great visualization to alter your perception.

Step 1: Cut the sphere in the following way.

Step 2: Spread the cut out part across the paper.

Step 3: Collate the pieces together in the following way

Step 4: Spread the areas out separately to form a sine curve

Step 5: The area of the sine curve is the surface area of the sphere.

                          CUBOID

In everyday life, objects like a wooden box, a matchbox, a tea packet, a chalk box, a dice, a book etc are encountered. All these objects have a similar shape. In fact, all these objects are made of six rectangular planes. The shape of these objects is a cuboid.

Cuboid:

A cuboid is a closed 3-dimensional geometrical figure bounded by six rectangular plane regions.

Figure 1

Face – A Cuboid is made up of six rectangles, each of the rectangle is called the face. In the figure above, ABFE, DAEH, DCGH, CBFG, ABCD and EFGH are the 6-faces of cuboid. The top face ABCD and bottom face EFGH form a pair of opposite faces. Similarly, ABFE, DCGH, and DAEH, CBFG are pairs of opposite faces.

Any two faces other than the opposite faces are called adjacent faces.

Consider a face ABCD, the adjacent face to this are ABFE, BCGF, CDHG, and ADHE.

Base and lateral faces: Any face of a cuboid may be called as the base of the cuboid. The four faces which are adjacent to the base are called the lateral faces of the cuboid.

Usually the surface on which a cube rest on is known to be the base of the cube.

In Figure (1) above, EFGH represents the base of a cuboid.

Edges – The edge of the cuboid is a line segment between any two adjacent vertices.

There are 12 edges, they are AB,AD,AE,HD,HE,HG,GF,GC,FE,FB,EF and CD and the opposite sides of a rectangle are equal.

Hence, AB=CD=GH=EF, AE=DH=BF=CG and EH=FG=AD=BC.

Vertex – The point of intersection of the 3 edges of a cuboid is called vertex of a cuboid.

A cuboid has 8 vertices A,B,C,D,E,F, G and H represents vertices of cuboid in fig 1.

By observation, the twelve edges of a cuboid can be grouped into three groups such that all edges in one group are equal in length, so there are three distinct groups and the groups are named as length, breadth and height.

                      DIFFERENTIAL EQUATION

A differential equation can simply be termed as an equation with a function and one or more of its derivatives. You can read more about it from the differential equations PDF below. The functions usually represent physical quantities. The simplest ways to calculate quantities is by using differential equations formulas.

Differential Equations are used to solve practical problems like Elmer Pump Heat Equation

Differential Equations first came into existence by Newton and Leibniz who also invented Calculus. The three kinds of equations Newton initially conceptualized were:

MATH TRICKS THAT WILL BLOW YOUR MIND

1) Six Digits Become Three
Take any three-digit number and write it twice to make a six digit number. Examples include 371371 or 552552.
Divide the number by 7.
Divide it by 11.
Divide it by 13. (The order in which you do the division is unimportant.)
The answer is the three digit number

Examples: 371371 gives you 371 or 552552 gives you 552.

A related trick is to take any three-digit number.
Multiply it by 7, 11, and 13.
The result will be a six digit number that repeats the three-digit number.

Example: 456 becomes 456456.

2)Multiplying by 6
If you multiply 6 by an even number, the answer will end with the same digit. The number in the tens place will be half of the number in the ones place.

Example: 6 x 4 = 24

3)Memorizing Pi
To remember the first seven digits of pi, count the number of letters in each word of the sentence:

"How I wish I could calculate pi."

This gives 3.141592