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Presents, a Life with a Plan. My name is Karen Anastasia Placek, I am the author of this Google Blog. This is the story of my journey, a quest to understanding more than myself. The title of my first blog delivered more than a million views!! The title is its work as "The Secret of the Universe is Choice!; know decision" will be the next global slogan. Placed on T-shirts, Jackets, Sweatshirts, it really doesn't matter, 'cause a picture with my slogan is worth more than a thousand words, it's worth??.......Know Conversation!!!

Friday, December 31, 2021

Watch

 


 

The Viracocha in the story of the reverse of the sun would gain in understanding to more than the Gobekli Tepe as the gate.  To the world and the movement of mass it is in the construction of quote unquote the hammer.  As the gate is not in contention as Gobekli Tepe shows the anvil it is the gain to know that the connection is round leaving only mass as subject.  The quote unquote bird men as a mystery should have stayed mysterious yet as the world stood angels appeared and became the precursor to what is man.  The general at-mass should have halted and left once again the mystery to men cannot fly however that is not true.  Due to the zipper and the concept the fly being lord of that saying I can state with implicit gain due to an unfortunate event when I was a child concerning my chin made something happen so fast that the web by design made wings a cage.  The bad boys, girls and hermaphrodites (said with coverage for today's applicable essence) have done something so hideous that I can state that the bone fabric needed to gain wings does happen and had to be pulled-out, the entire action of hideous behavior had to come out backwards to my chin and at such the stripping of those wings can be known.  The hammer in the nail makes this world a game that is on a clock and wings "when the hammer drops".  


 

The evidence of the gate grew on the ancient world leaving I to believe that the chin mechanism had been employed and the evidence built The Annanuki.  To engage that is to understand the wing at the structure of The Pyramid and back to the story of Osiris and the "Four Corners Earth" to greater than and yet should one stop the return to the world chart takes you to the pyramids where the pyramid of the sun has a flattop making four corners earth, a platform, a stage and a location on the other side of the world.  To continue is to understand the reality of how this would continue: The Ark of the Covenant.  Now, knowing that during this fragmented time of the transfer of world powers had thrones one will begin to understand that the ark is represented and the throne would have stopped anyone should the Egyptians been overthrown.  



 

 Gate of the Sun

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The "Gate of the Sun"

The Gate of the Sun, also known as the Gateway of the Sun, is a monolith carved in the form of an arch or gateway at the site of Tiahuanaco by the Tiwanaku culture, an Andean civilization of Bolivia that thrived around Lake Titicaca in the Andes of western South America around 500-950 CE.[1]

Tiwanaku is located near Lake Titicaca at about 12,549.2 ft (3,825.0 m) above sea level near La Paz, Bolivia. The Gate of the Sun is approximately 9.8 ft (3.0 m) tall and 13 ft (4.0 m) wide, and was carved from a single piece of stone. Its weight is estimated to be 10 tons.[2] When rediscovered by European explorers in the mid-19th century, the megalith was lying horizontally and had a large crack through it. It presently stands in the location where it was found, although it is believed that this is not its original site, which remains uncertain.[3]

Some elements of Tiwanaku iconography spread throughout Peru and parts of Bolivia. Although there have been various modern interpretations of the mysterious inscriptions found on the object, the carvings that decorate the gate are believed to possess astronomical and/or astrological significance and may have served a calendrical purpose.[4] In addition, scholars have found that the design below the central figure is meant to represent celestial cycles.[5] Being a later monument to the site in which it stands, the Gateway of the Sun could have also represented a transition from lunar religion to a solar religion based on its positioning to the sun to the West.[6]

Background

The iconography of the Tiwanaku civilization in Bolivia was influential throughout the Andean region. The images found on the Gateway of the Sun can be recognized in other areas and associated with contemporaneous or later civilizations, such as the Wari and Inca.[7] The Andean civilizations did not leave written records other than quipus (fiber recording devices) of their religious belief systems. Despite this, researchers have been able to gather information from the Spanish who documented the Incas, following their conquest of the empire. The Incas themselves had trained memorizers who were responsible for providing history through the oral traditions which the Spanish chroniclers used for their records. One of these chroniclers was a Felipe Guaman Poma de Ayala, who illustrated over 400 depictions of the Inca performing religious rituals and their stories. It is because of these drawings and sources similar to them that the ways of life for the ancient civilizations are not a mystery.[8]

Among the early investigators of the Gate of the Sun were Arthur Posnansky and Edmund Kiss, who each interpreted this monument in the context of erroneous theories of an early contact with Nordic Aryans. The claim by Kiss was asserted to be reinforced by the presence of a Nordic man's head sculpture found during an excavation, though there is some speculation this have been a hoax by Kiss. Their interpretations lacked modern data and methods and are now regarded as pseudoarchaeology.

Figures

The lintel is carved with 48 squares surrounding a central figure. Each square represents a character in the form of winged effigy. There are 32 effigies with human faces and 16 anthropomorphic figures with the heads of condors. All look towards the central motif: the figure of a person whose head is surrounded by 24 linear rays, thought by some to represent rays of the Sun. The styled staffs held by the figure apparently symbolize thunder and lightning. Some historians and archaeologists believe that the central figure represents the “Sun God” and others have linked it with the Inca god Viracocha. The image of the central figure on the gateway is thought to be Thunupa, or Tunupa, a major weather god in Aymara culture and in the Titicaca Basin throughout the Middle Horizon period.[9] The deity is also known as the Gateway God or Staff God and was believed to provide rain, lightning, and thunder to the Titicaca Basin.[10] The mythological emergence of the creator gods from Lake Titicaca is the foundation of Tiwanaku religious belief, making the city of Tiwanaku a significant symbol for the whole Andes region.[11] A central theme of Andean religion, including religion in the Inca Empire was the worship of gods that represented elements related to the Earth, especially for the Tiwanaku and Inca cultures. In Inca culture, the Sun God was known as Inti, depicted as a young boy holding various objects of gold. Scholars have drawn comparisons between the Inca and Tiwanaku icons as proof of Tiwanaku influence had on Inca mythology and iconography.[12]

Historical depictions

References


  • Stone-Miller, Rebecca. (March 1996). Art Of The Andes. Thames & Hudson. ISBN 978-0-500-20286-9. Retrieved 9 October 2011.
    1. D'Altroy, Terence N. (2015). The Incas (2nd ed.). pp. 252, 254.

    External links

    Languages

  • Fernando Cajías de la Vega, La enseñanza de la historia : Bolivia, Convenio Andrés Bello, 1999,p.44.
  • Kolata, Alan L. (December 15, 1993). The Tiwanaku: Portrait of an Andean Civilization. Wiley-Blackwell. ISBN 978-1-55786-183-2.
  • Magli, Giulio. Mysteries and discoveries of archaeoastronomy: From Giza to Easter Island. English trans. NY: Springer Science & Business Media, 2009.
  • Staller, John E.; Stross, Brian (2013). Lightning in the Andes and Mesoamerica, Pre-Columbian, Colonial, and Contemporary Perspectives. p. 86.
  • Quilter, Jeffrey (2014). The Ancient Central Andes. Routledge World Archaeology. p. 205.
  • Cartwright, Mark. "Tiwanaku". World History Encyclopedia. Retrieved 17 March 2020.
  • Littleton, C. Scott (2005). Gods, Goddesses, and Mythology: Inca-Mercury. Marshal Cavendish. p. 730.
  • D'Altroy, Terence N. (2015). The Incas (2nd ed.). p. 254.
  • Staller, John E.; Stross, Brian (2013). Lightning in the Andes and Mesoamerica, Pre-Columbian, Colonial, and Contemporary Perspectives. p. 86.
  • D'Altroy, Terence N. (2015). The Incas (2nd ed.). p. 52.
  • Thursday, December 30, 2021

    Global Warming

     

     

    Global warming:  A main concern of today's economy in the year of 2021 is the essence of speech regarding global warming going into the year of 2022, what is the cause.  The design of such has delivered many closures and more laws delivering to many manufacturers the dictation of exhaust and the time and date of the implication of such, this is to include the dry cleaner, in other words the smog.

     Smog has been another main frame as the blame has been cast per country to announce the hassle of glacier melt and as many countries have been busy pointing the finger the aspect has been left to thought?   This quick brief is to say that global warming may be as simply known as plastic, waste, litter in the oceans worldwide.  Question; has anyone measured the temperature of the plastics in our oceans?

    As decompose happens in the dump what is the collection of a solar panel?  This is just a first look at the admissions of such measure as the tides would increase this vast study and the beaches would dock the ground not showing the heat of such products however just by understanding the hot sand on my beach in San Francisco I know that heat is held on the beach even with our consistent temperature of the water at 58 degrees.  Is the land parched?

    So, is the plastic in our oceans causing glacier melt?


    Sunday, December 12, 2021

    Cantore Arithmetic The Extraction Of Root

     

     "Arithmetic (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art' or 'craft') is a branch of mathematics that consists of the study of numbers, especially concerning the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.[1][2]"

    Cantore arithmetic is at the branch of Cantore mathematics and is not number theory.  This basis is driven by 'Word' and at that the concept is to calculate worldwide physics at Cantore arithmetic.  The on-going will process into the branches of science and physics will enhance to change as geometric shapes must participate as basic geometry met at algebraic equation due to Albert Einstein and his equation of e=mc2.  The plural to Albert Einstein is currently at "four corners earth" written in Isaiah 11:12, the King James Version and said: "And gather the dispersed of Judah from the four corners of the earth" and this will only process at the arithmetic of word leaving the number theory in the bible to counter currently at the popular sciences of physics disregarding mathematics forgetting arithmetic not counting algebra and leaving geometry to Pythagorus from 2500 years ago, leaving mathematics to philosophical.  This is not a problem in Cantore arithmetic as word is currently Cantore mathematics and has proven at word with tone as french is not math.

    To equal age:  The Ages must relate to more that Albert Einstein at e equaled mc squared for the Age of Aquarius must calculate:  "The Age of Aquarius, in astrology, is either the current or forthcoming astrological age, depending on the method of calculation. Astrologers maintain that an astrological age is a product of the earth's slow precessional rotation and lasts for 2,160 years, on average (one Great Year equals 25,920-year period of precession / 12 zodiac signs = 2,160 years).

    There are various methods of calculating the boundaries of an astrological age. In sun-sign astrology, the first sign is Aries, followed by Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pisces, whereupon the cycle returns to Aries and through the zodiacal signs again. Astrological ages proceed in the opposite direction ("retrograde" in astronomy). Therefore, the Age of Aquarius follows the Age of Pisces.[1]".

    The astronomy an average does not participate in Cantore arithmetic and is in use on this page for the proof of Cantore mathematics science department represented as Cantore arithmetic branch.  To calculate the stream a Jet must be represented and four corners earth left to navigation to the added (=) as what must calculate is the giant, ages equal giant?

    To gain this Cantore arithmetic must find the unicorn and not absorb the egg.  To do this at the current Cantore mathematics must add (+) word to discuss the probability of unicorn at the subtraction of body and invite of picture to understand modern methods to dragon, stars, mythology and the basic story.  The story of tell must pixel in order to basic Cantore arithmetic and that will happen at picture, test must be left for Cantore Sciences and thereby Cantore mathematics may Cantore arithmetic today, Sunday, December 12, 2021 based on the most recent show of William Shatner the host of 'The UnXplained' playing on the History Channel Friday nights at 9:00PM West Coast time.

    December 10th, 2021, just last Friday night The UnXplained with host William Shatner played and interesting clip explaining the phenomena of age.  There was a gentleman in India that was well-over 200 years old and that became picture perfect to an older picture that had shown the body of this work with arm length and that gave square to the other story that will provide Cantore arithmetic with explained to the "Ages" and/or The Age of Aquarius as a continuum bringing forward the concept of the unicorn as the time it takes to incubate, articulate, fossil and drain with just the giant at the sequoia itself.

    The bones of Cantore arithmetic:  Devils Tower in Wyoming.  The archaic explanation to Cantore arithmetic is to understand fossil at the mathematics and not the reason.  To engage the ages at the capricious.

    Four Corners Earth:  The pyramid to the Aztec, Mayan, or any other pyramid that has a flat top (Note the show "The UnXplained" for the India gentleman and reserve that gentleman for the show of the reach).

     

     

    "

    Thursday, December 9, 2021

    Cantore Arithmetic: The Olive Branch?

     

     

    The cause of “Theory":  Perhaps the by example of compulsion halts at tendency and picks-up at challenge.  The “word” to the “sign” and not as compulsion (signature piece) rather the “bout”, this bout, the cause to theory will be width to Cantore Mathematics presenting for the first time arithmetic.


    The balance of ‘Proof to Cause’ not “cause and effect” for that would end concept and atmosphere leaving only exit as cause to proof, and that would be mathematics not arithmetic.

     

    To outline this properly I will step into and away from the lineage that Newton’s Law presented as the Law of Physics stands to tell the story as not the law of man.  This character of Nostradamus will prove that Stoker, Bram’s Dracula did place the horseshoe on backwards and gallop forwards to escape capture at his castle in the story told Wikipedia at https://simple.wikipedia.org/wiki/Vlad_III_the_Impaler

     

    The mammoth elephant and the pachyderm as the modern day elephant (pachyderm) will not suffice as the only fact to present the equine to equus ferus as the unicorn does enhance.  This evolution of picture to pixel will present (For the fact of “freak” as I was told that I was a ‘Freak’ repeatedly as a child came to a test and proven by tested by an actual M.D., Louis Vuksinick in San Francisco while working with Dr. Byron Kilgore, M.D. gave rise to actual: Prodigy).  In such there are “Bearded” circuses performers (https://en.wikipedia.org/wiki/Annie_Jones_(bearded_woman)) that have in our history been notoriously shown as “freak” and many have paid to see, laugh, spectacle, and, drive dinner to conversation.  This product management is well –proven and driven many economies to product line.  By this line to the mammoth the “bearded Circus performer” is not an elephant or a pachyderm or a mammoth as Newton’s Law would have had calculation leaving only the outline by difference to same using the outline to out-stretch to Mars.  For that “cause of theory” the saturation must invoke the time to the actual of artificial inheritance to our quick recovery of computation to this day.  This character will only cause to proof as mathematics is left at adding and subtraction and arithmetic to measure such at the current rules.  To that the fact that a movie (Hollywood) has enhanced our lot to enjoy Star Wars Chewbacca must counter the mammoth and not the butterfly to the cotton of cocoon as a mule.  To note: Men, man has never been removed from the “bearded” syndrome as such syndrome until my generation where ‘clean shaven’ was required, this show man has been lost and the return to bearded sex via indent has set.

     

    Notice to ‘Hollywood’ fair and Albert Einstein the return to not from will be by design for Cantore Arithmetic for reason to mathematical.  Thank you for your advance and subtraction to enhance the law to explain that Nostradamus ran to a blank page on role where Cantore mathematics will arithmetic to show that the slow bring from and not to can calculate at Pi (https://en.wikipedia.org/wiki/Pi ) and Fibonacci (https://en.wikipedia.org/wiki/Fibonacci) delivering new mathematics and now new arithmetic.

     

    To invoke “Robocop” as the example to Mars than arithmetic will be met.

     

    Arithmetic

    From Wikipedia, the free encyclopedia
    Jump to navigation Jump to search
    Arithmetic tables for children, Lausanne, 1835

    Arithmetic (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art' or 'craft') is a branch of mathematics that consists of the study of numbers, especially concerning the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.[1][2] Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory.[3]

    History

    The prehistory of arithmetic is limited to a small number of artifacts, which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed.[4]

    The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base, but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board (or the Roman abacus) to obtain the results.

    Early number systems that included positional notation were not decimal, including the sexagesimal (base 60) system for Babylonian numerals, and the vigesimal (base 20) system that defined Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.

    The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic.

    Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from the modern notation. The ancient Greeks lacked a symbol for zero until the Hellenistic period, and they used three separate sets of symbols as digits: one set for the units place, one for the tens place, and one for the hundreds. For the thousands place, they would reuse the symbols for the units place, and so on. Their addition algorithm was identical to the modern method, and their multiplication algorithm was only slightly different. Their long division algorithm was the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known to Archimedes (who may have invented it). He preferred it to Hero's method of successive approximation because, once computed, a digit does not change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60—instead of negative powers of 10 for the fractional part 0.934.[5]

    The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. Since they also lacked a symbol for zero, they had one set of symbols for the units place, and a second set for the tens place. For the hundreds place, they then reused the symbols for the units place, and so on. Their symbols were based on the ancient counting rods. The exact time where the Chinese started calculating with positional representation is unknown, though it is known that the adoption started before 400 BC.[6] The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers. This is explained in the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), which was written by Liu Hui dated back to 2nd century BC.

    The gradual development of the Hindu–Arabic numeral system independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base, and the use of a digit representing 0. This allowed the system to consistently represent both large and small integers—an approach which eventually replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta established the use of 0 as a separate number, and determined the results for multiplication, division, addition and subtraction of zero and all other numbers—except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht (650 AD) said, "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols."[7] The Arabs also learned this new method and called it hesab.

    Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.

    Although the Codex Vigilanus described an early form of Arabic numerals (omitting 0) by 976 AD, Leonardo of Pisa (Fibonacci) was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He wrote, "The method of the Indians (Latin Modus Indorum) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol zero".[8]

    In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

    The flourishing of algebra in the medieval Islamic world, and also in Renaissance Europe, was an outgrowth of the enormous simplification of computation through decimal notation.

    Various types of tools have been invented and widely used to assist in numeric calculations. Before Renaissance, they were various types of abaci. More recent examples include slide rules, nomograms and mechanical calculators, such as Pascal's calculator. At present, they have been supplanted by electronic calculators and computers.

    Arithmetic operations

    The basic arithmetic operations are addition, subtraction, multiplication and division. Although arithmetic also includes more advanced operations, such as manipulations of percentages,[2] square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.[9]

    Addition

    Addition, denoted by the symbol +, is the most basic operation of arithmetic. In its simple form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers (such as 2 + 2 = 4 or 3 + 5 = 8).

    Adding finitely many numbers can be viewed as repeated simple addition; this procedure is known as summation, a term also used to denote the definition for "adding infinitely many numbers" in an infinite series. Repeated addition of the number 1 is the most basic form of counting; the result of adding 1 is usually called the successor of the original number.

    Addition is commutative and associative, so the order in which finitely many terms are added does not matter.

    The number 0 has the property that, when added to any number, it yields that same number; so, it is the identity element of addition, or the additive identity.

    For every number x, there is a number denoted x, called the opposite of x, such that x + (–x) = 0 and (–x) + x = 0. So, the opposite of x is the inverse of x with respect to addition, or the additive inverse of x. For example, the opposite of 7 is −7, since 7 + (−7) = 0.

    Addition can also be interpreted geometrically, as in the following example. If we have two sticks of lengths 2 and 5, then, if the sticks are aligned one after the other, the length of the combined stick becomes 7, since 2 + 5 = 7.

    Subtraction

    Subtraction, denoted by the symbol -, is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: D = MS. Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: D + S = M.[1]

    For positive arguments M and S holds:

    If the minuend is larger than the subtrahend, the difference D is positive.
    If the minuend is smaller than the subtrahend, the difference D is negative.

    In any case, if minuend and subtrahend are equal, the difference D = 0.

    Subtraction is neither commutative nor associative. For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements (as sketched under § Addition), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, ab = a + (−b). The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial) unary operation, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved.

    For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others. For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware (negation). The trade-off is the halving of the number range for a fixed word length.

    A formerly wide spread method to achieve a correct change amount, knowing the due and given amounts, is the counting up method, which does not explicitly generate the value of the difference. Suppose an amount P is given in order to pay the required amount Q, with P greater than Q. Rather than explicitly performing the subtraction PQ = C and counting out that amount C in change, money is counted out starting with the successor of Q, and continuing in the steps of the currency, until P is reached. Although the amount counted out must equal the result of the subtraction PQ, the subtraction was never really done and the value of PQ is not supplied by this method.

    Multiplication

    Multiplication, denoted by the symbols \times or \cdot, is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are simply called factors.

    Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number greater than 1, say x, is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0, in such a way that 1 goes to the multiplicand.

    Another view on multiplication of integer numbers (extendable to rationals but not very accessible for real numbers) is by considering it as repeated addition. For example. 3 × 4 corresponds to either adding 3 times a 4, or 4 times a 3, giving the same result. There are different opinions on the advantageousness of these paradigmata in math education.

    Multiplication is commutative and associative; further, it is distributive over addition and subtraction. The multiplicative identity is 1, since multiplying any number by 1 yields that same number. The multiplicative inverse for any number except 0 is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity 1. 0 is the only number without a multiplicative inverse, and the result of multiplying any number and 0 is again 0. One says that 0 is not contained in the multiplicative group of the numbers.

    The product of a and b is written as a × b or a·b. When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab. In computer programming languages and software packages (in which one can only use characters normally found on a keyboard), it is often written with an asterisk: a * b.

    Algorithms implementing the operation of multiplication for various representations of numbers are by far more costly and laborious than those for addition. Those accessible for manual computation either rely on breaking down the factors to single place values and applying repeated addition, or on employing tables or slide rules, thereby mapping multiplication to addition and vice versa. These methods are outdated and are gradually replaced by mobile devices. Computers utilize diverse sophisticated and highly optimized algorithms, to implement multiplication and division for the various number formats supported in their system.

    Division

    Division, denoted by the symbols \div or /, is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than or equal to 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

    Division is neither commutative nor associative. So as explained in § Subtraction, the construction of the division in modern algebra is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced in § Multiplication. Hence division is the multiplication of the dividend with the reciprocal of the divisor as factors, that is, a ÷ b = a × 1/b.

    Within the natural numbers, there is also a different but related notion called Euclidean division, which outputs two numbers after "dividing" a natural N (numerator) by a natural D (denominator): first a natural Q (quotient), and second a natural R (remainder) such that N = D×Q + R and 0 ≤ R < Q.

    In some contexts, including computer programming and advanced arithmetic, division is extended with another output for the remainder. This is often treated as a separate operation, the Modulo operation, denoted by the symbol {\displaystyle \%} or the word mod, though sometimes a second output for one "divmod" operation.[10] In either case, Modular arithmetic has a variety of use cases. Different implementations of division (floored, truncated, Euclidean, etc.) correspond with different implementations of modulus.

    Fundamental theorem of arithmetic

    The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number as the product of prime factors), excluding the order of the factors. For example, 252 only has one prime factorization:

    252 = 22 × 32 × 71

    Euclid's Elements first introduced this theorem, and gave a partial proof (which is called Euclid's lemma). The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss.

    The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number. Other reasons include the sieve of Eratosthenes, and the definition of a prime number itself (a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.).

    Decimal arithmetic

    Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal") positional notation; however, any numeral system based on powers of 10, e.g., Greek, Cyrillic, Roman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".

    Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indorum" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2).

    The concept of 0 as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci[11] using the Hindu–Arabic numeral system.

    Algorism comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value 9, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one (if not zero). This adjustment is termed a carry of the value 1.

    The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds 9, the carry adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from 1 to 8 (9 × 9 = 81). Additional steps define the final result.

    Similar techniques exist for subtraction and division.

    The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the exponent for the radix (base) of 10 increases by 1 (to the left) or decreases by 1 (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10n with integer n. The list of values corresponding to all possible positions for a single digit is written as {..., 102, 10, 1, 10−1, 10−2, ...}.

    Repeated multiplication of any value in this list by 10 produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of number theory.

    Compound unit arithmetic

    Compound[12] unit arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches; gallons and pints; pounds, shillings and pence; and so on. Before decimal-based systems of money and units of measure, compound unit arithmetic was widely used in commerce and industry.

    Basic arithmetic operations

    The techniques used in compound unit arithmetic were developed over many centuries and are well documented in many textbooks in many different languages.[13][14][15][16] In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions:

    • Reduction, in which a compound quantity is reduced to a single quantity—for example, conversion of a distance expressed in yards, feet and inches to one expressed in inches.[17]
    • Expansion, the inverse function to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to 1 lb 8 oz.
    • Normalization is the conversion of a set of compound units to a standard form—for example, rewriting "1 ft 13 in" as "2 ft 1 in".

    Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic.

    Principles of compound unit arithmetic

    There are two basic approaches to compound unit arithmetic:

    • Reduction–expansion method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach is suited for automated calculations. A typical example is the handling of time by Microsoft Excel where all time intervals are processed internally as days and decimal fractions of a day.
    • On-going normalization method in which each unit is treated separately and the problem is continuously normalized as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations. An example of the ongoing normalization method as applied to addition is shown below.
    MixedUnitAddition.svg

    The addition operation is carried out from right to left; in this case, pence are processed first, then shillings followed by pounds. The numbers below the "answer line" are intermediate results.

    The total in the pence column is 25. Since there are 12 pennies in a shilling, 25 is divided by 12 to give 2 with a remainder of 1. The value "1" is then written to the answer row and the value "2" carried forward to the shillings column. This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column. The intermediate total is divided by 20 as there are 20 shillings in a pound. The pound column is then processed, but as pounds are the largest unit that is being considered, no values are carried forward from the pounds column.

    For the sake of simplicity, the example chosen did not have farthings.

    Operations in practice

    A scale calibrated in imperial units with an associated cost display.

    During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings, and ready reckoners, which are books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet[18] that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".

    The cumbersome nature of compound unit arithmetic has been recognized for many years—in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth")[19] in which he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time. In the modern era, many conversion programs, such as that included in the Microsoft Windows 7 operating system calculator, display compound units in a reduced decimal format rather than using an expanded format (e.g. "2.5 ft" is displayed rather than "2 ft 6 in").

    Number theory

    Until the 19th century, number theory was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned primality, divisibility, and the solution of equations in integers, such as Fermat's Last Theorem. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophisticated methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic.

    Arithmetic in education

    Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (using the decimal place-value system). This study is sometimes known as algorism.

    The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.[20]

    Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth. This was done in a book entitled The Best of Arithmetic by Abd-al-Fattah-al-Dumyati.[21]

    The book begins with the foundations of mathematics and proceeds to its application in the later chapters.

    See also

    Related topics

    Notes


  • "Arithmetic". Encyclopedia Britannica. Retrieved 2020-08-25.
    1. al-Dumyati, Abd-al-Fattah Bin Abd-al-Rahman al-Banna (1887). "The Best of Arithmetic". World Digital Library (in Arabic). Retrieved 30 June 2013.

    References

    External links

    Languages


  • "Definition of Arithmetic". www.mathsisfun.com. Retrieved 2020-08-25.

  • Davenport, Harold, The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed.), Cambridge University Press, Cambridge, 1999, ISBN 0-521-63446-6.

  • Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4.

  • The Works of Archimedes, Chapter IV, Arithmetic in Archimedes, edited by T.L. Heath, Dover Publications Inc, New York, 2002.

  • Joseph Needham, Science and Civilization in China, Vol. 3, p. 9, Cambridge University Press, 1959.

  • Reference: Revue de l'Orient Chretien by François Nau pp. 327–338. (1929)

  • Reference: Sigler, L., "Fibonacci's Liber Abaci", Springer, 2003.

  • Tapson, Frank (1996). The Oxford Mathematics Study Dictionary. Oxford University Press. ISBN 0-19-914551-2.

  • "Python divmod() Function". W3Schools. Refsnes Data. Retrieved 2021-03-13.

  • Leonardo Pisano – p. 3: "Contributions to number theory" Archived 2008-06-17 at the Wayback Machine. Encyclopædia Britannica Online, 2006. Retrieved 18 September 2006.

  • Walkingame, Francis (1860). "The Tutor's Companion; or, Complete Practical Arithmetic" (PDF). Webb, Millington & Co. pp. 24–39. Archived from the original (PDF) on 2015-05-04.

  • Palaiseau, JFG (October 1816). Métrologie universelle, ancienne et moderne: ou rapport des poids et mesures des empires, royaumes, duchés et principautés des quatre parties du monde [Universal, ancient and modern metrology: or report of weights and measurements of empires, kingdoms, duchies and principalities of all parts of the world] (in French). Bordeaux. Retrieved October 30, 2011.

  • Jacob de Gelder (1824). Allereerste Gronden der Cijferkunst [Introduction to Numeracy] (in Dutch). 's-Gravenhage and Amsterdam: de Gebroeders van Cleef. pp. 163–176. Archived from the original on October 5, 2015. Retrieved March 2, 2011.

  • Malaisé, Ferdinand (1842). Theoretisch-Praktischer Unterricht im Rechnen für die niederen Classen der Regimentsschulen der Königl. Bayer. Infantrie und Cavalerie [Theoretical and practical instruction in arithmetic for the lower classes of the Royal Bavarian Infantry and Cavalry School] (in German). Munich. Archived from the original on 25 September 2012. Retrieved 20 March 2012.

  • Encyclopædia Britannica, I, Edinburgh, 1772, Arithmetick

  • Walkingame, Francis (1860). "The Tutor's Companion; or, Complete Practical Arithmetic" (PDF). Webb, Millington & Co. pp. 43–50. Archived from the original (PDF) on 2015-05-04.

  • Thomson, J (1824). The Ready Reckoner in miniature containing accurate table from one to the thousand at the various prices from one farthing to one pound. Montreal. ISBN 9780665947063. Archived from the original on 28 July 2013. Retrieved 25 March 2012.

  • O'Connor, John J.; Robertson, Edmund F. (January 2004), "Arithmetic", MacTutor History of Mathematics archive, University of St Andrews

  • Mathematically Correct: Glossary of Terms

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