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Numeracy: Numbers, Numerals & Counting

I believe the phrase "told you so" is the most appropriate one here, Min.

Gordon
 
I maintain that we have never discussed this. Off to the memory thread for one of us! ;)
 
min_bannister said:
I maintain that we have never discussed this. Off to the memory thread for one of us! ;)
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Ooh we so did!

We tlaked about it a billion times! I was suprised you had not encountered the American useage becoming the accepted one. Senility in one so young, it's such a shame.

Gordon
 
How Humans Invented Numbers—And How Numbers Reshaped Our World

Once you learn numbers, it’s hard to unwrap your brain from their embrace. They seem natural, innate, something all humans are born with. But when University of Miami associate professor Caleb Everett and other anthropologists worked with the indigenous Amazonian people known as the Pirahã, they realized the members of the tribe had no word used consistently to identify any quantity, not even one.

Intrigued, the researchers developed further tests for the Pirahã adults, who were all mentally and biologically healthy. The anthropologists lined up a row of batteries on a table and asked the Pirahã participants to place the same number in a parallel row on the other side. When one, two or three batteries were presented, the task was accomplished without any difficulty. But as soon as the initial line included four or more batteries, the Pirahã began to make mistakes. As the number of batteries in the line increased, so did their errors.

The researchers realized something extraordinary: the Pirahã’s lack of numbers meant they couldn’t distinguish exactly between quantities above three. As Everett writes in his new book, Numbers and the Making of Us, “Mathematical concepts are not wired into the human condition. They are learned, acquired through cultural and linguistic transmission. And if they are learned rather than inherited genetically, then it follows that they are not a component of the human mental hardware but are very much a part of our mental software—the feature of an app we ourselves have developed.” ...
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FULL STORY: http://www.smithsonianmag.com/innov...sand-how-numbers-reshaped-our-world-180962485
 
The concept of 'zero' revolutionized counting and number systems. Its proliferation was affected as much by cultural influences as by practical or technical ones, and at some times in some places (most particularly anal-obsessive medieval Europe) it was forbidden math. As it turns out, the history of the concept touches on the origin of two words we use regularly: algorithm and cipher. That's a lot of impact for 'nothing'!

Who Invented Zero?
Though people have always understood the concept of nothing or having nothing, the concept of zero is relatively new; it fully developed in India around the fifth century A.D. Before then, mathematicians struggled to perform the simplest arithmetic calculations. Today, zero — both as a symbol (or numeral) and a concept meaning the absence of any quantity — allows us to perform calculus, do complicated equations, and to have invented computers. ...
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FULL STORY: https://www.livescience.com/27853-who-invented-zero.html
 
More ado about nothing (i.e., the history of zero) ...

Carbon dating pushes earliest timeframe for documentation of zero back at least as far as 3rd / 4th Century CE.

Carbon dating reveals earliest origins of zero symbol

Carbon dating shows an ancient Indian manuscript has the earliest recorded origin of the zero symbol.
The Bakhshali manuscript is now believed to date from the 3rd or 4th Century, making it hundreds of years older than previously thought.
It means the document, held in Oxford, has an earlier zero symbol than a temple in Gwailor, India.
The finding is of "vital importance" to the history of mathematics, Richard Ovenden from Bodleian Libraries said. ...
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SOURCE: http://www.bbc.com/news/uk-england-oxfordshire-41265057
 
A follow-up to the story about the origins of zero cited above ...

The Oxford analysis claimed the manuscript in question dated back to the 3rd / 4th century CE and mentioned 'zero' in a non-calculating / non-mathematical manner.

Both these conclusions have now been challenged ...

The Bakhshali manuscript: The world's oldest zero?
... An international group of historians of Indian mathematics has now challenged Oxford's findings.

The team, which includes scholars from universities in the USA, France, Japan, New Zealand and the University of Alberta in Canada, has published a peer-reviewed article that refutes several of the Library's key assertions.

The scholars argue that the work written on the leaves of the Bakhshali manuscript is a unified treatise on arithmetic that must have been written at the time of the latest of the manuscript's leaves, not the earliest. The treatise shows no signs of being a jumble of fragments from different periods. Both the handwriting and the topic being discussed are continuous across the boundary of the first two dated leaves. It looks very much as if the scribe, who may have lived at the end of the eighth century, wrote out his treatise on a group of leaves that had been manufactured at very different times.

But of greater significance for the history of mathematics is the authors' evidence showing that the Bakhshali treatise does indeed know the "true" zero, and contains calculations like long multiplication that would have necessitated using zero as an arithmetical number. Furthermore, the treatise even contains a statement saying, "having added one to zero...," thus proving that the early Sanskrit author was thinking about zero in a numerical way.

The zero in the Bakhshali treatise is younger, but more important than Oxford claimed. ...
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FULL STORY: https://www.eurekalert.org/pub_releases/2017-10/uoa-tbm102517.php

LINK TO TEAM ARTICLE ABSTRACT: https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22
 
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I'm a linguist, not a mathematician. But still, every so often, I catch a glimpse of the beauty and jaw-dropping wonder that mathematicians claim is inherent in their subject. For example, I recently came across Graham's number - see this engaging introduction to the idea. This is a number so large that the entire universe would be too small to contain its standard decimal representation, assuming that each digit was written so small that it only occupied one Planck volume. Not only that, you couldn't even use a similar hypothetical technique just to write out the total number of digits in Graham's number. And that holds true for several iterations - I confess I don't have sufficient grasp of the maths to tell you how many. You can keep your minuscule googolplexes: Graham's number is, truly, mind-bogglingly large. What an incredible species we are, to be capable of abstract thought on that level, in addition to the trivia and idiocies that seem to preoccupy us for so much of the time.
 
in the united states the national debt is now over 20 trillion dollars

http://usdebtclock.org/

on the top of that site there's a switch that also shows other industrial nations' ever-increasing debts (world debt clocks)
 
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I was watching a very dated programme a few minutes ago and wondered when it waa made, at the end of the programme the Roman Numerals came up, i then thought, is TV production and horology the only place they are still used? Then i thought about the date of the programme MIMV, 1995, i then thought about the number 8 in RN, VIII and why it wasnt IIX which is more economical with the amount of digits, i couldnt come up with an answer, then i thought, maybe i should stop drinking, then f*ck it and poured another vodka and Dr. Pepper. England were fecking hopeless against Scotland.
 
Souleater said:
But why is 8, VIII not IIX?
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The standard form / format for Roman numerals uses no more than 1 numeral inserted before a larger numeral in "subtractive notation." For example, "IX" means "one subtracted from ten."

There were, however, non-standard forms used for subtractive notation in certain places or by certain parties. See:

https://en.wikipedia.org/wiki/Roman_numerals#Standard_form
https://en.wikipedia.org/wiki/Roman_numerals#Irregular_subtractive_notation
 
Souleater said:
I was watching a very dated programme a few minutes ago and wondered when it waa made, at the end of the programme the Roman Numerals came up, i then thought, is TV production and horology the only place they are still used? Then i thought about the date of the programme MIMV, 1995, i then thought about the number 8 in RN, VIII and why it wasnt IIX which is more economical with the amount of digits, i couldnt come up with an answer, then i thought, maybe i should stop drinking, then f*ck it and poured another vodka and Dr. Pepper. England were fecking hopeless against Scotland.
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I dunno. YMMV.
 
Insurance and financial enterprises are looking for ways to promote understanding of basic quantitative concepts, because survey data indicates circa 40% of the population simply isn't confident in using numbers in their everyday lives
Four in 10 people 'not confident using numbers in their everyday lives'

FOUR in 10 (40 per cent) people do not feel confident about using numbers in their everyday lives, a survey has found.

And one in five (20 per cent) would avoid jobs that involve using numbers often, according to the research published by the Association of British Insurers (ABI).

More than a third (37 per cent) said that having easier access to online numeracy tools would help them, and a quarter (25 per cent) feel that clearer explanations would increase their confidence in dealing with numbers. ...
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FULL STORY: https://www.irishnews.com/business/...ing-numbers-in-their-everyday-lives--2594773/
 
This 2007 summary of the zero's history attributes the concept's origin to Sumeria circa 5,000 years ago.

The first evidence we have of zero is from the Sumerian culture in Mesopotamia, some 5,000 years ago. There, a slanted double wedge was inserted between cuneiform symbols for numbers, written positionally, to indicate the absence of a number in a place (as we would write 102, the '0' indicating no digit in the tens column). ...

The first recorded zero appeared in Mesopotamia around 3 B.C. The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. Zero reached western Europe in the 12th century. ...

The symbol changed over time as positional notation (for which zero was crucial), made its way to the Babylonian empire and from there to India, via the Greeks (in whose own culture zero made a late and only occasional appearance; the Romans had no trace of it at all). Arab merchants brought the zero they found in India to the West. After many adventures and much opposition, the symbol we use was accepted and the concept flourished ...
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What is the origin of zero? How did we indicate nothingness before zero?
https://www.scientificamerican.com/article/what-is-the-origin-of-zer/
 
Part of my understanding of "0" was that it was developed as a base holder for our base 10 standard. Sometime in high school math, this wonderous idea (that there are different counting bases) was presented to us. Before then I didn't even consider that counting could be done in any other way. Of course, using any other base counter method greater than 10 does require more figures to represent those numbers.

My dad always told me that human's cannot estimate amounts accurately (I think he meant by sight) greater than 3, ie all other combinations of amount more than 3 explicitly require the brain to group items into groups of 3.
 
Zero became necessary when positional notation began to be used. This is notation where the position of numerals in a string exceeded the number system's base stands for different orders of magnitude. For example, "111" means "one hundred and one ten and one single / one". If you removed the middle "1" (subtract the ten; meaning "one hundred one") and wrote it as "11" you'd mislead a reader into thinking you meant eleven ("one ten and one one").

Zero became critical as a placeholder for positions (and associated orders of magnitude) that were empty within the notational protocol.
 
The hunt for the origin of zero continues, and it's searching the Far East now. The latest developments center on evidence from Cambodia and a set of inscriptions on stones in Sumatra. This Scientific American opinion piece provides an overview of the Cambodian and Sumatran developments.
The Elusive Origin of Zero

Sūnya, nulla, ṣifr, zevero, zip and zilch are among the many names of the mathematical concept of nothingness. Historians, journalists and others have variously identified the symbol’s birthplace as the Andes mountains of South America, the flood plains of the Tigris and Euphrates Rivers, the surface of a calculating board in the Tang dynasty of China, a cast iron column and temple inscriptions in India, and most recently, a stone epigraphic inscription found in Cambodia.

The tracing of zero’s heritage has been elusive. For a country to be able to claim the number’s origin would provide a sense of ownership and determine a source of great nationalistic pride.

Throughout the 20th century, this ownership rested in India. That’s where an inscription was discovered, holding the number “0” in reference to land measurement inside a temple in the central Indian city of Gwalior. In 1883 the renowned German Indologist and philologist, Eugen Julius Theodor Hultzsch copied and translated the inscription into English, dating the text to the year C.E. 876. And this has been accepted as the oldest known date for the appearance of zero. However, a series of stones in what is now Sumatra, casts India’s ownership of nothingness in doubt, and several investigators agree that the first reference of zero was likely on a set of stones found on the island. ...
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FULL STORY: https://www.scientificamerican.com/article/the-elusive-origin-of-zero/
 
mejane said:
Apologies for the cryptic title - couldn't think of anything better.


I've been taught that we use base 10 as the standard counting system because that is the number of fingers humans have on both hands. But is that really true?

I've been thinking about this (yes, I have a headache now!) and it seems to me that base 5 would be the more logical choice - count on one hand, use a abacus/tool/club/whatever in the other :confused:

Another factoid to back up this theory comes from experimental evidence that most people can only count up to 5 before resorting to memory tricks.

eg:

( & 6

(3 characters, easy!


:blah: ;) :confused:

(still easy)


* & % % $ & :cross eye:



Answers on a postcard, please...

Jane.
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The Mayans used base 5 and base 20. The babelonians used base 12, but I am not sure if it was because they mostly had 6 fingers (something that happens sometimes nowdays) or for some other reason. It is only about the number of symbols used to count or depict numbers. Computer science uses base 2, base 8 & base 16.

As for most people being only able to count up to 5 before resorting to memory tricks, what kind of brain damaged people are you talking about? Counting is language and 2 year olds can count to 20, without knowing what that means. They usually know how many 3 is but they can count to 20.
 
Anome said:
He also covers the many number systems that people have used over the last six thousand years, or however long it's been (since writing started, not since the creation). Base 10 and base 5 are both common, as is base 20 (fingers and toes), but also base 60 (which is preserved in the divisions of time). I'm not entirely clear on why base 60 was useful (it's a while since I read that part of the book), but it clearly has more problems over base 10 or even 20.
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I always thought the "base 60" for divisions of time was actually base 12 from the babylonians times 5. But I never researched it.

https://soldatwatch.com/blogs/journal/why-does-an-hour-have-60-minutes

"What is documented is that the ancient Babylonians began to divide time as we know it today. They divided the year according to the moon cycles into twelve months, day and night into twelve parts of equal length = 24 hours." base 12, the day divided by 2 12 hour segments....
 
JahaRa said:
The Mayans used base 5 and base 20. The babelonians used base 12, but I am not sure if it was because they mostly had 6 fingers
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12 was used because it is divisible by more numbers.
 
This is odd ... Interesting, but odd ... There seem to be demonstrable correlations between the way a set of numbers is spatially arranged and how efficiently humans can process the numbers in a given task.

People most often arrange (or engage an arrangement of ... ) numbers listed horizontally. Prior research has demonstrated most people can process horizontal sets of sorted numbers more efficiently when smaller numbers are to the left and larger ones to the right. Newly-published research results indicate people can process number sets even more efficiently and effectively when the numbers are arranged vertically, with larger numbers above and smaller numbers below.
Counting from left to right feels ‘natural’ – but new research shows our brains count faster from bottom to top

When asked to write the numbers from one to ten in a sequence, how do you order them? Horizontally? Vertically? Left to right? Top to bottom? Would you place them randomly?

It has been often been assumed, and taught in schools in Western countries, that the “correct” ordering of numbers is from left to right (1, 2, 3, 4…) rather than right to left (10, 9, 8, 7…). The ordering of numbers along a horizontal dimension is known as a “mental number line” and describes an important way we represent number and quantity in space.

Studies show humans prefer to position larger numbers to the right and smaller numbers to the left. People are usually faster and more accurate at comparing numbers when larger ones are to the right and smaller ones are to the left, and people with brain damage that disrupts their spatial processing also show similar disruptions in number processing.

But so far, there has been little research testing whether the horizontal dimension is the most important one we associate with numbers. In new research published in PLOS ONE, we found that humans actually process numbers faster when they are displayed vertically – with smaller numbers at the bottom and larger numbers at the top. ...
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FULL STORY: https://theconversation.com/countin...brains-count-faster-from-bottom-to-top-189339
 
Here are the bibliographic details and abstract from the published research report. The full report is accessible at the link below.

Vertical versus horizontal Spatial-Numerical Associations (SNA): A processing advantage for the vertical dimension
Luke Greenacre, Jair E. Garcia, Eugene Chan, Scarlett R. Howard, Adrian G. Dyer
PLOS ONE. Published: August 25, 2022
https://doi.org/10.1371/journal.pone.0262559

Abstract
Humans have associations between numbers and physical space on both horizontal and vertical dimensions, called Spatial-Numerical Associations (SNAs). Several studies have considered the hypothesis of there being a dominant orientation by examining on which dimension people are more accurate and efficient at responding during various directional SNA tasks. However, these studies have difficulty differentiating between a person’s efficiency at accessing mental representations of numbers in space, and the efficiency at which they exercise motor control functions, particularly bilateral ones, when manifesting a response during an explicit directional SNA task. In this study we use a conflict test employing combined explicit magnitude and spatial directional processing in which pairs of numbers are placed along the diagonal axes and response accuracy/efficiency are considered across the horizontal and vertical dimensions simultaneously. Participants indicated which number in each pair was largest using a joystick that only required unilateral input. The experiment was run in English using Arabic numerals. Results showed that directional SNAs have a vertical rather than horizontal dominance. A moderating factor was also found during post-hoc analysis, where response efficiency, but not accuracy, is conditional on a person’s native language being oriented the same as the language of the experiment, left to right. The dominance of the vertical orientation suggests adopting more vertical display formats for numbers may provide situational advantages, particularly for explicit magnitude comparisons, with some domains like flight controls and the stock market already using these in some cases.

SOURCE / FULL REPORT: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0262559
 
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