Category Archives: Learning Center

Who Invented the Equal sign?

 

MathsBaron_EqualEqual sign commonly used in mathematics and computing around the world, was first recorded by welsh mathematician Robert Recorde in 1557 in his book “The Whetstone of Witte”. Recorde composed the symbol consisting of two parallel lines to avoid having to write over again and again “is equal to”. Recorde’s representation of the equal sign consisted of two long parallel lines. It took around 150 years before the cocept became popular. In 1700s some people used the symbol || , while others widely used the symbols-  æ or œ (derived from latin word aequalis / equal).

Below is the snippet from the book where Robert Recorde introduced the equal sign for the first time.

The_Whetstone_of_Witte_Euqual_Sign“To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can be moare equalle.”

 

Pi Day

MathsBaron_PiMarch 14 (03/14) is celebrated as Pi (π) day by mathematician scholars and enthusiasts around the world because 3, 1 and 4 are the first three significant digits in the value of Pi.

The number Pi is defined as ratio of circumference of circle to its diameter and is a constant number.  In other words you measure the circumference of the circle and divide it by its diameter to get the value of Pi and this holds true for circles of any radius or diameter.

For centuries the number π aka Pi has been the center of attention and fascination by mathematician and scholars. Ancient Greek mathematician and scientists, Archimedes (287 – 312 BC) is credited with first to approximate value of Pi.  He used the 96 sided polygon to come up with approximate value of Pi lay between 317 (approximately 3.1429) and 31071 (approximately 3.1408), consistent with its actual value of approximately 3.1416. Archimedes also proved that area of circle was Pi multiplied by the square of radius ( A = πr²). In 18th century Greeks gave the Pi its famous symbol (π). 

Ramanujan’s Number

Ramanujan1729 is called Ramanujan’s number. This number became famous through the conversation bwteen two great mathematicians – Srinivas Ramanujan and Godfrey Harold Hardy. English mathematician G.H. Harold narrated

” I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.

The number 1729  can be expressed as sum of cube of 1 & cube of 12 and cuber of 9 & cuber of 10.

1729 = (1)3 + (12)3 = (9)3 + (10)3

Sutra 10 – Yaavadunam

The Sutra means “By the Deficiency”. Here find the deficiency of the number to its nearest base. The difference between the number and the base is termed as deviation or reference base or complement which may be positive or negative.

Example:

Number

Base

Deficiency

8

10

10 – 8 = 2

14

10

10   – 14 = -4

88

100

100 – 88 = 12

109

100

100   – 109 = -9

999

1000

1000 – 999 =    1

99995

100000

100000   – 99995 = 5

Exercise:

1)      Find the deficiency of the following numbers to its nearest base

a)      98

b)      7

c)      13

d)      98

e)      105

f)       989

g)      99999

h)      1000001

Square of a number:

Square of a number can be determined by implementing the corollary of this Sutra “Yavadunam Tavadunikrtya Varganca Yojayet

This means” Whatever  is deficiency,  subtract  it from the number and write the square of that deficiency” This Sutra can be applied to find the square of any number closer to the base of  powers of 10. Below are the easy steps.

1)      Find the deficiency with the nearest base.

2)      Square the deficiency and place at the right side.

3)      Subtract the Deficiency from the number

4)      Result = [Number – Deficiency + carry over][Square of Deficiency]

Example 1 – Find the Square of 9

1)      Base = 10, Deficiency = 1

2)      Square of the Deficiency = 1

3)      Subtract the Deficiency from Number -> 9 – 1 = 8, Carry over = 0

4)      Square of 9 is [Number – Deficiency + Carry over][Square of Deficiency] = 81

 Example 2 – Find the Square of 14

1)      Base = 10, Deficiency =  – 4

2)      Square of the Deficiency = 16

3)      Subtract the Deficiency from Number -> 14 – (-4) =  18, Carry over =1

4)      Square of 9 is [Number – Deficiency + carry over][Square of Deficiency] = 196

Example 3 – Find the Square of 98

1)      Base = 100, Deficiency =  2

2)      Square of the Deficiency = 04

3)      Subtract the Deficiency from Number -> 98 –  2 =  96, Carry over =0

4)      Square of 9 is [Number – Deficiency + carry over][Square of Deficiency] = 9604

Example 4 – Find the Square of 104

1)      Base = 100, Deficiency =  – 4

2)      Square of the Deficiency = 16

3)      Subtract the Deficiency from Number -> 104 –  (-4) =  108, Carry over = 0

4)      Square of 9 is [Number – Deficiency + carry over][Square of Deficiency] = 10816

Example 5 – Find the Square of 9997

1)      Base = 10000, Deficiency =  3

2)      Square of the Deficiency = 0009

3)      Subtract the Deficiency from Number -> 9997 –  3 =  9994, Carry over =0

4)      Square of 9 is [Number – Deficiency + carry over][Square of Deficiency] = 99940009

Exercise

Find the Square of following Numbers

1)      96

2)      6

3)      990

4)      111

5)      102

6)      10003

7)      999995

8)      10000008

9)      12

10)  89

Answers:

Sutra 2 – Nikhilam navatascaramam Dasatah

The Sutra means “All from 9 and the last from 10”.  In mathematical terms, this means find complement of a number.  Subtract the number from the nearest power of 10 such as 10, 100, 1000 etc. The power of 10 from which the difference is calculated is called the Base.

The difference between the number and the base is termed as deviation or reference base or complement which may be positive or negative.

Example:

 Number Base Base – Number Complement / Nikhilam
7 10 7 – 10 -3
14 10 14   – 10 4
88 100 88 – 100 -122
109 100 109   – 100 9
5200 10000 5200 – 10000 -4800

Exercise:

1)      Find the compliment and the nearest base of the following numbers

a)      45

b)      8

c)      13

d)      8910

e)      98

f)       105

g)      989

h)      99999

i)        1000001