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Number Systems
Multiplication table
of
(maximum 24)
1 x
2 x
3 x
4 x
5 x
6 x
7 x
8 x
9 x
10 x
11 x
12 x
13 x
14 x
15 x
16 x
17 x
18 x
19 x
20 x
21 x
22 x
23 x
24 x
The most common number system types are:
Decimal number system (Base- 10)Input Decimal number
Binary number system (Base- 2)
Octal number system (Base-8)
Duodecimal base 12:
Hexadecimal number system (Base-16)
Our number system is a decimal system, or base ten system, meaning it's built powers of ten. Most number systems around the world use base ten, the origin of this system is likely based on hand counting: most people have ten fingers, and so we have ten digits. Aside from the number of fingers belonging to the median human, there's nothing particularly special about ten. We can use any whole number larger than one as a base, and get a perfectly good number system.

Computers are built around binary, or base two. Processors and operating systems are 32 bit or 64 bit since these are powers of two. Since binary is hard for humans to directly read, computer scientists and engineers often look at raw computer code represented in base eight octal, or base sixteen hexadecimal. Since eight and sixteen are powers of two (8 = 2 × 2 × 2, and 16 = 2 × 2 × 2 × 2), blocks of binary code are nicely translatable into these bases.

Suppose we lived in a world in which humans evolved to generally have four fingers on each hand, instead of five. It's pretty likely that we would have adopted an octal, or base eight, system, using powers of eight instead of powers of ten. In octal, we have eight digits — 0, 1, 2, 3, 4, 5, 6, and 7 — and now place values are based on powers of eight. The octal number 10 represents the number 8 in decimal. The octal 100 represents 8 × 8, or 64. 1000 is 8 × 8 × 8, or 512.

The hexadecimal system (also known as base 16) is a positional notation numeral system using twelve as its base. Any group of four bits translates into a hexadecimal digit: 0011 becomes a 3, and 1010 becomes 10, which is usually represented in hexadecimal as a capital A.

The Septimal system (also known as base 7) is a positional notation numeral system using 7 as its base. Examples are Weeks time keeping and music letter notation.

The duodecimal system (also known as base 12) is a positional notation numeral system using twelve as its base. The origin of the duodecimal system is typically traced back to a system of finger counting based on the knuckle bones of the four larger fingers. Using the thumb as a pointer, it is possible to count to 12 by touching each finger bone, starting with the farthest bone on the fifth finger, and counting on. In this system, one hand counts repeatedly to 12, while the other displays the number of iterations, until five dozens, i.e. the 60, are full. This system is still in use in many regions of Asia. Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day

Zero to Infinity

zero is a complete absence and infinity is a complete lack of absence. That means Zero & infinity are two sides of the same coin. Everything (∞) is nothing (o), with a twist.
  1. Mathematics was developed to count definite things, starting with one. zero represents no things or nothing.
  2. Zero is a start number and Infinity is end number, while both defy measurement. Are Zero and Infinity just concepts?
  3. Between complexity of zero and infinity, number system is straightforward.
  4. Before God, nothing existed and God is defined as the starting point. God has also been defined as infinitely more powerful, good, knowing and perfection.
  5. Out of zero comes the infinite universe (or Big Bang). Nothing is a probability wave, of nothing and everything (nothing=everything)
  6. Zero and Infinity are the limits of human thinking.
  7. The Infinite is one of the most intriguing ideas in which the human mind has ever engaged.

More about Zero

  1. Some of the properties of zero:
  2. Zero is even (not odd, not neutral)
  3. Zero is not a prime number (smallest prime number is 2)
  4. Zero is neither positive nor negative
  5. Zero is an integer
  6. If you add or subtract zero from any other number, the answer will always result in the other number.
  7. The concept of dividing by zero is fraught with illogical consequences and does not work.
  8. Numbers to the zero power always equal one.

Zero Vector - Zero dimension

  1. Zero vector or null vector is a vector which has zero magnitude and an arbitrary direction. If a vector is multiplied by zero, the result is a zero vector. The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector.
  2. An illustration of a nildimensional space is a point. Zero-dimensional Polish spaces include the Cantor space and Baire space.The physical meaning of can be understood from the following examples.
  3. The displacement of a ball thrown up and received back by the thrower is a zero vector.
  4. The velocity vector of a stationary body is a zero vector.
  5. The acceleration vector of a body in uniform motion is a zero vector.
  6. Zero dimension also includes absence of time dimension. As time does not exist, past, present and future coexist simultaneously.
  7. Three dimensions is our "normal" space. Two dimensions is a plane "Flatland". One dimension is a "line-world". Zero dimension is "point-world", where everything happens in the same place.
  8. Zero dimensions is like a rip in spacetime, where there is no spacetime

More about Infinity

Ancient Indians and Greeks approached infinity as a philosophical concept. Many ancient cultures had various ideas about the nature of infinity. Indians used the number line with zero and infinity. Two Indian words for infinity are asaṃkhyata ("countless, innumerable") and ananta ("endless, unlimited"). The Indian mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
Enumerable: lowest, intermediate, and highest
Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
Infinite: nearly infinite, truly infinite, infinitely infinite
The universe is infinite, by Thomas Digges in 1576. In 1584, Giordano Bruno proposed an unbounded universe, Infinite Universe, Innumerable suns and planets around them
The Mahavaipulya Buddhavatamsaka Sutra or Avatamsaka Sutra, contains a description of an "incalculable" number, known as the asamkhyeya (lie outside the bounds of human calculability). Asamkhyeya is defined as a particular power of a laksha.
  1. Infinity does not do anything.
  2. The sequence of natural numbers never ends, and is infinite. Infinity is not a real number.
  3. Infinity is only an idea of something that has no end.
  4. Infinity does not grow and it is already and always fully formed. "endless", or "boundless"
  5. A Googol is 1 followed by one hundred zeros (less than infinity)
  6. Infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.
  7. Three types of infinities are mathematical infinity; physical infinity; and a philosophical infinity.
  8. The infinite is an elusive concept, full of paradoxes and controversies. People look for precision, certainty, objectivity, and effectiveness in modeling things around us. Our bodies and our experiences are finite. We have never experienced or seen infinity.
  9. Philosophers have speculated about the nature of the infinite, such as Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea also is used in physics and the other sciences.
  10. In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.
  11. A pre-Socratic Greek philosopher who lived in Miletus, used the word apeiron which means infinite or limitless. Mathematical infinity also comes from Zeno of Elea who is known for his paradoxes.

Some more on numbers

The concept of zero is essential to a system that’s at the basis of modern computing: binary numbers. Modern-day digital computers operate on the principle of two possible states, ‘on’ and ‘off’. The ‘on’ state is assigned the value ‘1’, while the ‘off’ state is assigned the value ‘0’. Or, zero. In the 2nd or 3rd Centuries BCE by a musicologist named Pingala, used this.

Two

In binary system, 2 = 10. So from single state to 2 states and generating large varieties by repeating the process. The number two is a symbolic representation of the state of duality which we experience objectively through our minds and the senses. It also symbolizes Purusha and Prakriti (God and Nature), Brahman and Atman, the knower and the known, the subject and the object, the doer and the deed, cause and effect, the right and wrong, knowledge and ignorance, life and death, etc

Three or Trinity

The number 3 is a prime number, and the first odd prime number.
The Trinity based on roles: Brahma is the creator. Vishnu is the preserver. Siva is the destroyer.
We are body - mind - spirit/soul
It was also the number of time – past, present, future;
Three worlds of heaven, earth and the hell;
Triple qualities of sattva, rajas and tamas;
Three layers of nature, light, fire and darkness;
Three states of being immanent, transcendent and the absolute;
Three worlds of heaven, earth and the hell;
Three phases of life, young age, middle age and old age;
Three paths to self-realization, the path of knowledge, the path of love and the path of action;
The triple control of control of body, mind and action or thought, word and deed;

Four

Four is the smallest composite number.
4**2 = 2**4
A circle divided by 4 makes right angles and four quadrants. Because of it, four (4) is the base number of the plane (mathematics).
A solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces (as well as vertices) of a polyhedron.
The god Brahma has four faces, may be indicating ability to see all 4 directions.
The aims of life (purusharthas) are four, dharma (righteousness), artha (wealth), kama (desire) and moksha.(salvation).
The stages (ashrama) in life are four: the age of studentship (brahmacharya), the age of householder (grihasta), the age of retirement (vanaprastha) and the age of ascetic (renunciation).
There are four ages or yugas: Golden, Silver, Bronze, and Iron (Satya, Dvapara, Treta and Kali)
The symbolic meanings of the number four are linked to those of the cross and the square.

Five

5 is the first safe prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. 5 is the smallest Pythagorean triple (3, 4, 5)
Humans, and many other animals, have 5 digits on their limbs. Five elements: Ether (akasa); earth; Air (vayu); Agni (fire); and Water (jalam)
The senses are five in number: skin, eyes, nose, ears and tongue.

Six

6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number; its proper divisors are 1, 2 and 3. Since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and -perfect number.
6 represents 6 points forming hexagon, by intersection of two equal triangles
The mind is the sixth sense organ, with six faculties: reason, emotion, thought, awareness, knowledge and intelligence, and six enemies or impurities: lust, anger, greed, pride, delusion and envy.

Seven

Highest single digit prime number, which cannot be divided by any other number other than 1 and themselves.
7 is the only Number "that can divide "Infinitely" - infinitely, or "Infinite number of times"
1/7 = 0.142857 142857 142857 ....
2/7 = 0.285714 285714 285714 ....
3/7 = 0.428571 428571 428571 ....
4/7 = 0.571428 571428 571428 ....
5/7 = 0.714285 714285 714285 ....
6/7 = 0.857142 857142 857142 ....

Eight

8 can be halved successively till it reduces to 1.

number 9

9 = 3*3 (3 squre, smallest magic square)

The human body is considered as a city of nine gates which correspond with the nine openings (two eyes, two ears, two nostrils, navel and two excretory openings).

Ten

First 4 numbers totals 10 (1+2+3+4=10)
Ten is the first double digit number formed by the coming together of one and zero.

Twelve 12

12 is a composite number with four divisors (2, 3, 4 and 6).
12 Knuckles in a hand
12 Gods, 12 Disciples, 12 Tribes and the Zodiac of 12 months, rashis and so on.

Eighteen 18

18 a manifestation of 9, is a composite number, with large number of divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semi perfect number. 18 is the only number that equals twice the sum of its decimal digits.

Twenty-seven

27 is the value of the expression n^n, where n=3.
27 54 108 216 432 Harshad number series
27 is also a decagonal number.

60 and Sexagesimal system

60 is convenient for division by 2,3,4,5 and 6.
That way one can express fraction of 60 in whole numbers.
60 = 12 (Knuckles in a hand) * 5 (fingers in a hand)
The sexagesimal system of measurement is based on the number sixty. There are sixty seconds in a minute, sixty minutes in a hour. When we measure angles, we use the sexagesimal to express units in degrees, minutes and seconds. This method of measurement familiar to both the Indian and Mesopotamian cultures.
Sexagesimal system may be based in large part upon the observation of the planets, specifically Jupiter and Saturn. After every sixty years, Jupiter and Saturn will return to the same relative place in the zodiac.
The sixty years cycle of Jupiter and Saturn gives rise to another interesting number. In a sixty year period, Jupiter will complete five circuits of the zodiac and Saturn will complete two circuits. The combined individual cycles equal seven which is also the total number of visible planets plus the two luminaries.

Number 108 is LCM of 27 and 12

108 is a Harshad (means "great joy") number, which is an integer divisible by the sum of its digits (1+0+8 = 9)
The number 108 appears often in Hindu and Buddhist philosophy. 27 constallations and each one has 4 directions or 4 parts, and 27 * 4 = 108
9 corresponds to grahas and 12 to rassi and 9 * 12 = 108
Sun is approx 108 times the diameter of the Earth. The distance from the Sun to the Earth is approx 108 times the diameter of the Sun. Distance of the Moon from the Earth is approx 108 times the diameter of the Moon.
Silver represent the moon and atomic weight is 108.
There are a 108 marmas in the body.

Number 360

360 is a highly composite number
The smallest number divisible by every natural number from 1 to 10 except 7.
A circle is divided into 360 degrees for the purpose of angular measurement.
This choice of unit allows round angle to be divided into equal sectors measured in integer degrees rather than fractional degrees. 360 is the sum of two twin primes (179 + 181).
one year is close to 360 days, making it equal to 12 months (30*12).

Number 432

The number 432 = (2^4)*(3^3).
Equals 2 to the power of 4 multiplied by 3 to the power of 3.
432 is a Harshad number
It was consifered as magic number for astronomical calculations.
Kalacakra calendar in use, 60 year cycle starting with Prabhava, is by Pandita Somanatha of Kashmir/Himalayas, in 367 CE. Vernal equinox of 367 CE is Prabava Varsham. He further developed sexa decimal system. The earth's axis wobble that causes the precession of the equinoxes is approximately 25,920 years or 432 sixty year cycles.

Number 252 1729 and 2520

252 is a Harshad number in base 10. Divisible by 1, 2, 3, 4, 6, 7, 9.
2520 is the smallest number divisible by all integers from 1 to 10, i.e., it is their least common multiple.
It is the product of five consecutive numbers, namely 3×4×5×6×7
2520=7*30*12 - 7 day week 30 days a month and 12 month
It is a Harshad number in all popular bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16.

1729 is the smallest number expressible as the sum of two cubes in two different ways. 1729 is the sum of the cubes of 10 and 9. Cube of 10 is 1000 and the cube of 9 is 729. Both the cubes, therefore, add up to 1729. 1729 is now known as the Ramanujan-Hardy number.

Infinity

Infinite and Zero are two boundaries.
The first unenumerable number was constructed using effectively a recursive construction. Still the highest enumerable number has not been attained.
Five different types of infinity: infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite.
The set of real numbers (or even complex numbers) does not include infinity. It’s not a number, but a symbol representing a particular kind of non-number. It’s either the “end” of a process that never ends (limits), or it’s a marker indicating something didn’t “converge” (series).
As a concept “infinity” is pre-historic. Indian scriptures reserve infinity, as attributes of almighty. It represents eternal time & space. Infinity make us accept of our limitations and feel humble in a good way. The Ouroboros or Lemniscate, meaning ‘ribbon’, by ancient Greeks, is an ancient symbol depicting a serpent biting its own tail. It represents the infinite immortality, continuity, and eternal return.

The Indian mathematical text Surya Prajnapti classifies all numbers into three sets: enumerable, innumerable, and infinite.
In real analysis, the symbol, called "infinity", denotes an unbounded limit.
sum of the infinite series converges to some real value a.
sum of the infinite series diverges in the specific sense that the partial sums grow without bound.
Infinity is often used not only to define a limit
Physical infinity-no measurable quantity could have an infinite value.

Different Number systems

DecimalBinaryOctalHexagonal
1111
21022
31133
410044
510155
611066
711177
81000108
91001119
10101012A
11101113B
12110014C
13110115D
14111016E
15111117F
16100002010
17100012111
18100102212
19100112313
20101002414
21101012515
22101102616
23101112717
24110003018
25110013119
2611010321A
2711011331B
2811100341C
2911101351D
3011110361E
3111111371F
321000004020
331000014121
341000104222
351000114323
361001004424
371001014525
381001104626
391001114727
401010005028
411010015129
42101010522A
43101011532B
44101100542C
45101101552D
46101110562E
47101111572F
481100006030
491100016131
501100106232
511100116333
521101006434
531101016535
541101106636
551101116737
561110007038
571110017139
58111010723A
59111011733B
60111100743C
61111101753D
62111110763E
63111111773F
64100000010040
65100000110141
66100001010242
67100001110343
68100010010444
69100010110545
70100011010646
71100011110747
72100100011048
73100100111149
7410010101124A
7510010111134B
7610011001144C
7710011011154D
7810011101164E
7910011111174F
80101000012050
81101000112151
82101001012252
83101001112353
84101010012454
85101010112555
86101011012656
87101011112757
88101100013058
89101100113159
9010110101325A
9110110111335B
9210111001345C
9310111011355D
9410111101365E
9510111111375F
96110000014060
97110000114161
98110001014262
99110001114363
100110010014464
101110010114565
102110011014666
103110011114767
104110100015068
105110100115169
10611010101526A
10711010111536B
10811011001546C
10911011011556D
11011011101566E
11111011111576F
112111000016070
113111000116171
114111001016272
115111001116373
116111010016474
117111010116575
118111011016676
119111011116777
120111100017078
121111100117179
12211110101727A
12311110111737B
12411111001747C
12511111011757D
12611111101767E
12711111111777F
1281000000020080
1291000000120181
1301000001020282
1311000001120383
1321000010020484
1331000010120585
1341000011020686
1351000011120787
1361000100021088
1371000100121189
138100010102128A
139100010112138B
140100011002148C
141100011012158D
142100011102168E
143100011112178F
1441001000022090
1451001000122191
1461001001022292
1471001001122393
1481001010022494
1491001010122595
1501001011022696
1511001011122797
1521001100023098
1531001100123199
154100110102329A
155100110112339B
156100111002349C
157100111012359D
158100111102369E
159100111112379F
16010100000240A0
16110100001241A1
16210100010242A2
16310100011243A3
16410100100244A4
16510100101245A5
16610100110246A6
16710100111247A7
16810101000250A8
16910101001251A9
17010101010252AA
17110101011253AB
17210101100254AC
17310101101255AD
17410101110256AE
17510101111257AF
17610110000260B0
17710110001261B1
17810110010262B2
17910110011263B3
18010110100264B4
18110110101265B5
18210110110266B6
18310110111267B7
18410111000270B8
18510111001271B9
18610111010272BA
18710111011273BB
18810111100274BC
18910111101275BD
19010111110276BE
19110111111277BF
19211000000300C0
19311000001301C1
19411000010302C2
19511000011303C3
19611000100304C4
19711000101305C5
19811000110306C6
19911000111307C7
20011001000310C8
20111001001311C9
20211001010312CA
20311001011313CB
20411001100314CC
20511001101315CD
20611001110316CE
20711001111317CF
20811010000320D0
20911010001321D1
21011010010322D2
21111010011323D3
21211010100324D4
21311010101325D5
21411010110326D6
21511010111327D7
21611011000330D8
21711011001331D9
21811011010332DA
21911011011333DB
22011011100334DC
22111011101335DD
22211011110336DE
22311011111337DF
22411100000340E0
22511100001341E1
22611100010342E2
22711100011343E3
22811100100344E4
22911100101345E5
23011100110346E6
23111100111347E7
23211101000350E8
23311101001351E9
23411101010352EA
23511101011353EB
23611101100354EC
23711101101355ED
23811101110356EE
23911101111357EF
24011110000360F0
24111110001361F1
24211110010362F2
24311110011363F3
24411110100364F4
24511110101365F5
24611110110366F6
24711110111367F7
24811111000370F8
24911111001371F9
25011111010372FA
25111111011373FB
25211111100374FC
25311111101375FD
25411111110376FE
25511111111377FF
256100000000400100

Mystery of 3, 6, and 9

In the decimal system, these numbers behave uniquely:
Multiples of 3 form repeating patterns.
9 always results in digits that sum to 9.
6 completes the cycle between them.
Tesla was intrigued by these patterns and their relation to vortex mathematics — a theory where 9 represents completion, and 3 & 6 mark critical points within an energetic cycle.

Ramanujan - brief on numbers

Ramanujan–Soldner constant is a mathematical constant defined as the unique positive zero of the logarithmic integral function.
It is named after Srinivasa Ramanujan and Johann Georg von Soldner. Its value is approximately 1.45136923488338105028396848589202744949303228
1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic positive integers in two different ways. It is known as the Hardy–Ramanujan number.
Ramanujan's most famous formula for calculating π is:
1/π = (2√2/9801) * Σ [ (4k)! / (k!) ⁴ * (1103 + 26390k) / (396^(4k))] from k=0 to ∞.

Almost integers relating to e and π
(e^π) called Gelfond's constant = 23.14069263277926900572
Like both e and π, this constant is both irrational and transcendental.
(e^π)-π = 19.99909997918947576726
e^{pi*sqrt(43)} = 8847 36743.99977 7466
e^{pi*sqrt(67)} = 14 71979 52743.99999 86624 54
e^{pi*sqrt(163)} = 262 53741 26407 68743.99999 99999 99250 07

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