Additional Math Concepts

These basics are not presented in as rigorous a manner as the axioms, but they serve to clarify and build upon the axioms.

 (1) A unit is anything that can be grasped as an entity.

In mathematics, the most fundamental idea is that of a unit. A unit is commonly represented by the number “1” (one).

(2) A unit is counted one at a time.

Counting provides natural numbers that are commonly represented by the numbers, 1, 2, 3, 4, 5, and so on.

(3) All numbers are referenced from 0 (zero).

Any quantity present is relative to no quantity. Therefore, all numbers are referenced from the idea of “no units.” This concept is commonly represented by the numeral “0” (zero).

(4)  Therefore, the number n is fully defined by “0 + n”.

Therefore, the number 1 is fully defined by “0+1”; the number 2 is fully defined by “0+2”; the number 3 is fully defined by “0+3”, and so on.

(5) The number “0 + n” is abbreviated as n, or as the positive integer +n.

The number “0+1” is abbreviated as +1; the number “0+2” is abbreviated as +2; the number “0+3” is abbreviated as +3, and so on.  The numbers +1, +2, +3, +4, +5, etc. are called positive integers.

(6) If a number is n, then the next number is “n + 1”.

The next number is obtained by counting one more. This gives us the basic function of adding. Addition is represented by the sign “+”. Thus, the next number after 1 is “1+1” written as 2; the next number is “2+1” written as 3; the next number is “3+1” written as 4, and so on. One may keep on counting forward without limit.

 (7) If a number is n, then the previous number is “n – 1”. 

The previous number is obtained by counting one less. This function of taking away (subtracting) is the opposite of adding. Subtraction is represented by the sign “–”. Thus, the number previous to 3 is “3–1” or 2; the number previous to 2 is “2–1” or 1; the number previous to 1 is “1–1” or 0.

(8) The counts previous to 0 (zero) account for units that are missing.

As mentioned in (3) above, 0 (zero) represents the reference point of “no units”. The number previous to 0 is, 0–1; the number previous to 0–1 is 0–2; the number previous to 0–2 is 0–3, and so on. These counts define units that are missing. One may thus keep on counting backward without limit.

(9) A missing number is fully defined by “0 – n”.

“0–1”, “0–2”, “0–3”, etc., provide a count of units that are missing. An example would be a count of the money that one owes.

(10) The number “0 – n” is abbreviated as the negative integer –n.

The number “0–1” is abbreviated as –1; the number “0–2” is abbreviated as –2; the number “0–3” is abbreviated as –3, and so on. The numbers “–1, –2, –3, –4, –5, etc.” are called negative integers.

(11) The reference point zero (0) is neither positive nor negative.

Zero (0) is simply the reference point for quantities that are present, as well as for the quantities that are missing.

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November 24, 2011 – 11:50 AM Categories: Mathematics | Comments (12)

False Predictions

FALSE PREDICTIONS from http://www.mhealthtalk.com/2011/09/false-barriers/

These quotes show that it’s risky to say that something can’t or won’t be done, especially when technology is concerned and it’s the right thing to do.

  • PHONOGRAPH – “The phonograph has no commercial value at all.” (Thomas Edison)

  • TELEGRAPH – “I watched his countenance closely, to see if he was not deranged … and I was assured by other senators after he left the room that they had no confidence in it.” (U.S. Senator Smith of Indiana, after witnessing a demonstration of Samuel Morse’s telegraph, 1842)

  • TELEPHONE – “Well-informed people know it is impossible to transmit the voice over wires and that were it possible to do so, the thing would be of no practical value.” (Boston Post, on the telephone, 1865)

  • TELEPHONE – “This telephone has too many shortcomings to be seriously considered as a means of communication. The device is inherently of no value to us.” (Western Union internal memo, 1876)

  • TELEPHONE – “The Americans think we need of the telephone, but we do not. We have plenty of messenger boys.” (Sir William Preece, chief engineer of Britain’s Post Office, 1876)

  • ELECTRICITY – “Fooling around with alternating current is just a waste of time. Nobody will use it, ever.” (Thomas Edison, 1889)

  • CARS – “The horse is here to stay, but the automobile is only a novelty–a fad.” (President of the Michigan Savings Bank, speaking to Henry Ford’s lawyer, Horace Rackham. Rackham ignored the advice, invested $5000 in Ford stock, and sold it later for $12.5 million.)

  • PLANES – “Heavier-than-air flying machines are fantasy. Simple laws of physics make them impossible.” (Lord Kelvin, president, British Royal Society, 1895)

  • INVENTION – “Everything that can be invented has been invented.” (Charles H. Duell, commissioner of the US Patent Office, recommending that his office should be abolished, 1899)

  • PLANES – “Man will not fly for 50 years.” (Wilbur Wright, to brother Orville after a disappointing flying experiment in 1901. Their first successful flight was in 1903.)

  • PLANES – “There will never be a bigger plane built.” (A Boeing engineer, after the first flight of the 247, a twin-engine plane that holds ten people)

  • RADIO – “The wireless music box has no imaginable commercial value. Who would pay for a message sent to nobody in particular?” (David Sarnoff’s associates responding to his urgings for investment in radio, 1912)

  • TANKS – “Caterpillar land ships are idiotic and useless. Those officers and men are wasting their time and are not pulling their proper weight in the war.” (Fourth Lord of the British Admiralty, regarding the introduction of tanks in war, 1915)

  • TANKS – “The idea that cavalry will be replaced by these iron coaches is absurd. It is little short of treasonous.” (ADC to Field Marshal Haig, at tank demonstration, 1916)

  • MOVIES – “Who the hell wants to hear actors talk?” (H. M. Warner, Warner Brothers, 1927)

  • NUCLEAR – “There is not the slightest indication that nuclear energy will ever be obtainable. It would mean that the atom would have to be shattered at will.” (Albert Einstein, 1932)

  • NUCLEAR – “That is the biggest fool thing we have ever done. The bomb will never go off, and I speak as an expert in explosives.” (Admiral William Leahy, when President Truman asked for his opinion on the project to build an atomic bomb)

  • SPACE – “A rocket will never be able to leave the earth’s atmosphere.” (New York Times, 1936)

  • COMPUTERS – “I think there is a world market for about five computers.” (Thomas J. Watson Jr., chairman of IBM, 1943)

  • TELEVISION – “Television won’t last because people will soon get tired of staring at a plywood box every night.” (Darryl Zanuck, Movie Producer, 20th Century Fox, 1946)

  • TELEVISION – “The problem with television is that the people must sit and keep their eyes glued on a screen; The average American family hasn’t time for it.” (New York Times, 1949)

  • SPACE – “Space travel is bunk.” (Sir Harold Spencer Jones, Astronomer Royal of the UK, 1957, two weeks before Sputnik orbited the Earth)

  • COPIERS – “The world potential market for copying machines is 5000 at most.” (IBM to the founders of Xerox as it turned down their proposal, 1959)

  • SPACE – “There is practically no chance communications space satellites will be used to provide better telephone, telegraph, television, or radio service inside the United States.” (T. Craven, FCC Commissioner, 1961)

  • MUSIC – “Guitar music is on the way out.” (Decca Records, declining to record a new group called The Beatles, 1962)

  • COMPUTERS – “There is no reason for any individual to have a computer in their home.” (Kenneth Olson, founder of Digital Equipment Corporation, 1977)

  • COMPUTERS – “So we went to Atari and said, ‘Hey, we’ve got this amazing thing, even built with some of your parts, and what do you think about funding us? Or we’ll give it to you. We just want to do it. Pay our salary, we’ll come work for you.’ And they said, ‘No.’ So then we went to Hewlett-Packard, and they said, ‘Hey, we don’t need you. You haven’t gone through college yet.‘” (Steve Jobs, founder of Apple)

  • COMPUTERS – “640 K [of computer memory] ought to be enough for anybody.” (Bill Gates, founder and CEO of Microsoft, 1981)

  • COMPUTERS – “We see a corporate market of maybe 15,000 PCs a year by 1990.” (DataQuest, 1984)

  • COMPUTERS – “By 1990 75-80 percent of IBM compatible computers will be sold with OS/2.” (Bill Gates, founder and CEO of Microsoft, 1988)

  • COMPUTERS – “I predict that the last mainframe will be unplugged on March 15, 1996.” (Stewart Alsop, InfoWorld columnist, 1991)

  • INTERNET – “I predict the Internet will soon go spectacularly supernova and in 1996 catastrophically collapse.” (Robert Metcalfe, founder of 3Com and inventor of Ethernet, 1995)

  • NEW BUSINESSES – “A cookie store is a bad idea. Besides, market research and focus groups confirm that America wants soft, not chewy, cookies.” (Investor rejection letter to Debby Fields, founder of Mrs. Fields’ Cookies)

  • NEW BUSINESSES – “The concept is interesting and well-informed, but in order to earn better than a ‘C’ the idea must be feasible.” (Yale professor’s comments on a term paper submitted by Fred Smith for an overnight delivery system. Two years later, Smith founded Federal Express.)

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November 24, 2011 – 8:08 AM Categories: General | Comments (3)

Fundamentals, Consistency and Breakthroughs

Breakthroughs occur when deeper consistencies are found in the fundamentals of a subject. A breakthrough is always preceded by the discovery of some inconsistency. It then leads to the breakthrough and a deeper consistency.

For a long time, mathematics was based on numerals that did not include the zero. Look at the Roman numerals. There is no zero there. This is because a number was supposed to represent “something” only. For a long time it was inconceivable that the idea of “nothing” had any use in mathematics.

We know that the next number is arrived at by adding one to the number. There is no limit to the next number. It keeps on going for ever. We also know that the previous number is determined by subtracting one from the number. However, at one time the previous number stopped at one, because counting started from one. Nobody seemed to see any utility in seeking a number previous to one.

It was not until the 4th century BC that somebody took a hard look at the inconsistency of not going beyond one for the previous number. This new number was perceived as “void” or “nothing.” If a person spent all his money, he had no money left. This was consistent with the idea of previous number. But this was resisted simply because counting started from one, and “nothing” could not be counted. The moment that resistance was overcome, breakthroughs started to occur in the subject of mathematics. Zero could now be used as a reference point from which to measure. The biggest breakthrough came with the positional notation for the numbering system. Suddenly, it became much simpler to write numbers (compare current numbering system to the Roman system), and mathematics started to progress by leaps and bounds.

Inconsistencies at the fundamental level are hard to accept as inconsistencies because the popular consensus endorses it as the norm. Thus, it took several centuries before the possibility of numbers previous to zero could even be considered. The idea of negative numbers was resisted in Europe as late as the 17th century AD, even though they were known to facilitate the representation of debt. Finally, with the formal introduction of negative numbers later in 17th century, the “previous” numbers could also be extended without limit. A much deeper consistency was achieved at the fundamental level of mathematics. This led to incredible breakthroughs.

There are many such examples of overcoming mental resistance at the fundamental level for the sake of greater consistency. And each time it happened it was followed by wonderful breakthroughs. Here is one more example. We know that larger and larger magnitudes may be represented by the idea of a unit, such as, a planet, a galaxy, a cluster of galaxies, and a universe. Is there a limit in magnitude to which a unit is bound? The expanding universe seems to indicate that the answer is no. Now looking in the opposite direction of smaller and smaller magnitudes, is their another limit to which a unit is bound? The discovery of the irrational number seems to indicate that the answer is no again. So, the mathematical logic points to the possibility of discovering smaller and smaller particles endlessly. We don’t know for certain yet. We are simply looking at a deeper consistency.

In summary, breakthroughs occur when one seeks a deeper consistency in the fundamentals of a subject, even when this requires going against the general consensus.

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November 21, 2011 – 11:22 PM Categories: KHTK | Comments (11)

The Basics of Math

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Mathematics presents “logical tools” for learning.

Arithmetic forms the first part of mathematics that presents the “number skill”. Arithmetic starts with counting.

Counting is a tool for learning how many things are there. Counting starts with one. The next count is one more.

A unit is the thing being counted one at a time. If one is counting houses, then each house is a unit. If one is counting inches of a length, then each inch is a unit.

The digits are the ten symbols – 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 – that are used to write numbers.

Numbers provide a system to represent the counts. A number is made up of one or more digits, just like words are made up of one or more letters.

Addition is counting together of numbers. Subtraction is opposite of addition.

Multiplication is repeated addition of a number. Division is opposite of multiplication.

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THE FIELD AXIOMS

Real numbers are undefined objects that satisfy certain properties.

AXIOM #1: CLOSURE PROPERTY OF ADDITION

If x and y are real numbers, then x+y is a unique real number.

Addition is an operation such that for every pair of real numbers x and y we can form the sum of x and y, which is another real number denoted by x+y. The sum x+y is uniquely determined by x and y.

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AXIOM #2: CLOSURE PROPERTY OF MULTIPLICATION

If x and y are real numbers, then xy is a unique real number.

Multiplication is an operation such that for every pair of real numbers x and y we can form the product of x and y, which is another real number denoted by xy or by x.y. The product xy is uniquely determined by x and y.

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AXIOM # 3: COMMUTATIVE PROPERTIES OF ADDITION & MULTIPLICATION

If x and y are real numbers, then x+y = y+x.

If x and y are real numbers, then xy = yx.

ADDITION: Two numbers may be added in any order. For example,

5 + 8       =       8 + 5

One may visualize the numbers as items of a one-dimensional array. For example,

$ $ $ $ $ $ $ $ $ $ $ $ $

One may count the items in this array as “5 first and 8 next”; or “8 first and 5 next”. The result is the same.

$ $ $ $ $      $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $      $ $ $ $ $

“Subtraction” is accounted by this law by treating the number being added as a negative integer. The sign moves with the following number. The first unsigned number is treated as having a positive sign. For example,

[8 – 5]     is           +8 –5    =     –5 +8

MULTIPLICATION: Two numbers may be multiplied in any order. For example,

5 x 8       =       8 x 5

One may visualize the numbers as items of a two-dimensional array. For example,

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

One may count the items in this array, as “5 rows of 8 each”, or “8 columns of 5 each”. The result is the same.

8+8+8+8+8     =     5+5+5+5+5+5+5+5

“Division” is accounted by this law by using the reciprocal (multiplicative inverse) of the divisor as the multiplicand.

[8 ÷ 2]     is           8 x ½    =     ½ x 8

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AXIOM # 4: ASSOCIATIVE PROPERTIES OF ADDITION & MULTIPLICATION

If x, y and z are real numbers, then (x+y)+z = x+(y+z).

If x, y and z are real numbers, then (xy)z = x(yz).

In addition, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

3   +  (5 + 8)      =        (3 + 5)   +  8

Similarly, in multiplication, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

3   x   (5 x 8)      =        (3 x 5)   x   8

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AXIOM 5: DISTRIBUTIVE PROPERTY

If x, y and z are real numbers, then x(y+z) = xy + xz.

Multiplication distributes over addition. For example, a factor may be multiplied by the other factor as a sum of two parts with the same outcome as follows.

5 x 17   =   5 x (10 + 7)   =   5 x 10 + 5 x 7   =   50 + 35   =   85

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AXIOM 6: IDENTITY ELEMENTS

The identity element for addition is 0, i.e., for any real number x, 

x+0 = x.

The identity element for multiplication is 1, i.e., for any real number x, 

x.1 or 1x = x.

There exist two real numbers, which we denote by 0 and 1, such that for every real x we have

0 + x  =  x + 0  =  x                    (the idea of adding nothing)

1 . x  =  x . 1  =  x                      (the idea of a single occurrence)

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AXIOM 7: INVERSES

A unique ADDITIVE INVERSE exists for every real number, i.e., for every x the additive inverse is -x such that

x     +     (-x)         =           0.

A unique MULTIPLICATIVE INVERSE exists for every real number, i.e., for every non-zero x the multiplicative inverse is 1/x such that

x      .       (1/x)      =           1.

For every real number x there is a real number y such that

x + y   =   y + x   =   0       (the idea of negating something into nothing)

For every real number x (except 0) there is a real number y such that

x . y   =   y . x   =   1         (the idea of reducing something to its unit)

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Note 1: 0 is an exception because it denotes nothing, whereas all other numbers denote something.

Note 2: 1 is unique because it denotes a unit, whereas all other non-zero numbers denote multiple units.

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November 19, 2011 – 10:33 AM Categories: Mathematics | Post a comment

The Factors of Unknowable

This is an effort to go as deep as possible into the Unknowable using the Vedic process “neti, neti.”

The most fundamental starting factor:

(1) There is observation (because this is an observation being registered).

Phenomena:

(2) There are things to be observed.

(3) These things are being manifested and unmanifested.

(4) A manifestation begins, continues, and then ends.

Manifestations:

(5) The universe is a manifestation that is continuing.

(6) There are sub-manifestations to this universe that begin, continue, end.

(7)  Manifestations may come and go, but the manifesting continues.

Existence and Time:

(8) It is the property of manifesting, which is known as existence.

(9) It means that something or other continues to be manifested.

(10) Thus, existence provides the idea of Time.

Form:

(11) A manifestation can be observed only because of its form.

(12) The form of a manifestation can be physical, such as, rocks, trees, chair, molecules, atoms, light and electromagnetic waves.

(13) The form of a manifestation can also be mental, such as, visualizations, thoughts, evaluations, conclusions, considerations and speculations.

Dimensions:

(14) The form of a manifestation has finite extents.

(15) It is the property of extents, which is perceived as dimensions.

(16) It means that something or other continues to be extended.

Space:

(17) These dimensions provide the idea of space.

(18) Physical dimensions provide the idea of physical space.

(19) Mental dimensions provide the idea of mental space.

Motion:

(20) As these extents reach out and withdraw there is change.

(21) It is the property of change, which is perceived as motion.

(22) It means that something or other continues to move.

Energy:

(23) This motion provides the idea of energy.

(24) Physical motion provides the idea of physical energy.

(25) Mental motion provides the idea of mental energy.

Patterns:

(26) Motion of extents may follow certain fixed patterns.

(27) It is the property of fixidity, which is perceived as patterns.

(28) It means that some pattern or other continues to be exhibited.

Matter:

(29) These patterns provides the idea of matter.

(30) Physical patterns provides the idea of physical matter.

(31) Mental patterns provides the idea of mental matter.

The Universe of Manifestation:

(32) The Universe of manifestation is made up of time, space, energy, and matter.

(33) Research into manifestations is continuing through physical and mental sciences.

Awareness:

(34) There is observation, so there is also the ability to observe.

(35) It is the ability to observe, which is recognized as awareness.

(36) Awareness continues as long as there is something to be aware of.

Perception:

(37) Thus, there is the ability to assess what is there.

(38) It is the ability to assess, which is recognized as perception.

(39) Perception continues as long as there is something to perceive.

Consideration:

(40) Thus, there is the ability to visualize what is assessed to be there.

(41) It is the ability to visualize, which is recognized as consideration.

(42) Consideration continues as long as there is something to consider.

The Universe of Consideration:

(43) The Universe of Consideration is made up of the awareness, perception and consideration of existence, form, motion and pattern.

(44) Research into manifestations and underlying considerations is continuing, primarily, by examining their consistency and inconsistency through mathematics.

Life and Beingness:

(45) It is the presence of abilities, which is perceived as life.

(46) These abilities are manifested  just like other things are manifested.

(47) It is the manifesting of abilities that we may call Beingness.

The universe:

(48) Thus, there are manifestations in terms of existence and beingness.

(49) Thus, there is a universe, which is aware of itself on the whole.

(50) There are considerations of existing and not existing.

(51) There are considerations of static and kinetic.

(52) There are considerations of cause and effect.

(53) What is beyond these considerations may only be speculated upon.

(54) The research into Unknowable thus continues.

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NOTES: Alfred North Whitehead said: “All systematic thought must start from  presuppositions.” The factors above start from what is observable. These may be considered to be presuppositions.

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November 13, 2011 – 11:29 AM Categories: Postulates | Comments (18)