Numerals
The ancient Egyptians were using special symbols, known as pictographs, to write down numbers over 3,000 years ago. Later, the Romans developed a system of numerals that used letters from their alphabet rather than special symbols. Today, we use numbers based on the Hindu-Arabic system. We can write down any number using combinations of up to 10 different symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). The ancient Egyptians developed number systems to keep accounts of what was bought and sold.
What is Number?
We use numbers every day and tend to take them for granted. But how did the idea of numbers arise? Did every culture develop the ideas of counting and numbers separately or have these ideas arisen in only a few cultures and then spread, for example, through trade? Is counting intuitive or did it arise to solve particular problems?
Some of the oldest evidence of counting so far discovered comes from ancient artifacts belonging to groups of hunters and gatherers. For example, a wolf bone, dated about 30,000 BC, has been discovered with a series of notches carved in it, which seem to represent a tally of some kind.
Tally Systems and Number Words
Tallying seems likely to be among the earliest methods of keeping a record of quantities and appears in many cultures. But is this really counting? For example, a tally system might be used to keep track of a flock of sheep. A small stone might be put in a pile for each sheep as it is let out to graze in the morning and then a stone could be removed from the pile again for each sheep as it was collected in at night. Any pebbles left over would indicate that some sheep were missing. This is really only a straight comparison between two sets of objects, the stones and the sheep. It does not give any idea of the actual number of sheep in the flock.
Another idea sometimes found at an early developmental stage of a culture is that of different number words being used depending on the context. For example, there might be one word for four people and another for four stones. At some stage an abstract idea of number develops and the concept of, for example, "threeness", where a group of three fish and three stones are perceived to have something in common, is incorporated into the system.
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One, Two, Many Systems
Do humans have an intuitive knowledge of numbers, or do cultures develop numbers and counting to solve particular problems? There is no single answer to this question and it is the subject of much debate. However, there are cultures that have number systems consisting only of words for one, two, and many. For example, certain Australian aboriginal tribes use this system, but, if children from these tribes are taught to use other number systems, there is no significant difference between their mathematical abilities and those of children of other cultures. Perhaps this fact provides some evidence for numbers and counting having arisen in response to particular needs. The words for three in some European languages are very similar to the words for beyond or over. For example, the word for three in Latin is tres, which is similar to the word meaning beyond, trans. Perhaps these systems were also one, two, many systems originally, with words for different types of "many" only being developed later.
The one, two, many system has been extended by some cultures to form the 2-system. This allows the representation of groups of three or four objects by repeating the words for one or two. Thus, three would be two-one, and four would be two-two. We might wonder why, having got this far, counting was not continued indefinitely. However, again, this is not really counting. The words are being used to describe particular states, and do not include the idea of an increasing series of numbers.
Finger Counting
The development of ordinal numbers (first, second, third, etc.) from cardinal numbers (one, two, three, etc.) may have happened as a result of finger counting. Fingers, and sometimes toes, can be used to count on. If this is always done in the same order, then the same finger will always be used to represent, for example, seven. This provides a link between the seven items being counted and the seventh finger in the sequence. This link is not provided if we use stones as a tally. Any pebble in a pile could be the seventh one; they are not counted in any particular order.
Pictorial Numbers
Counting is nearly as old as speech and numerals are as old as writing. In ancient times, and in some instances down to modern times, each language had its series of numerals. Thus the number of systems of notation employed was about the same as the number of written languages, and in some cases a single language had several systems.
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The Pictographic Notation
In pictographic notation the number is given by repetitions of the symbol representing the object in question. For example, "five men" would be represented by the symbol for "man" repeated five times.
Many cultures throughout history have represented numbers by repetitions of a vertical or horizontal stroke, 1 by one stroke, 2 by two strokes, etc. In various forms of speech the number five is expressed by the word "hand" or "the hand finished", and ten by "two hands" or "two hands finished". In some South American languages "all the fingers" means ten; "all the fingers and toes", 20; "fingers and toes of two men", 40; "one of the other hand", six; "foot one", 11; "foot two", 12; and so on.
Hieroglyphic Systems
Hieroglyphic number systems consisted of repetitions of a single unit, with the use of hieroglyphics (pictures or symbols) for higher numbers. These systems sometimes introduced the principle of multiplication when repetitions became too many for practical use.
Egyptian Numbers
The Egyptians had three systems of writing, the hieroglyphic, the hieratic, and the demotic writings. The hieroglyphic system was the sacred writing reserved for formal inscriptions and was usually used for writing on stone. Numbers were seldom used in this system. Hieroglyphics were too complex for everyday use and this led to the development of a simpler system, hieratic or temple writing. This was used by priests and scribes for everyday records and was mostly used for writing on papyrus. Demotic writing developed from hieratic writing and was the system of writing used in everyday life.
Egyptian numbers were written from right to left. They started at one and went up to a million. The numbers one to nine were written as combinations of vertical strokes, ten was represented by a sign that looks like an upturned U, 100 by a coil of rope, 1000 by a sign representing a lotus flower, 10,000 by a vertical finger, 100,000 by a tadpole, and 1 million by a man with upraised arms (see Diagram 1).
Diagram 1: The Egyptian Numbers
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Sumerian Numbers
The Sumerians are thought to have invented writing in the 4th-2nd millennia BC. They used symbols incised in clay with a stylus to record words and numbers. The Sumerian number system was a base 60 or sexagesimal system. We can still see traces of a base 60 system in the current use of hours, minutes, and seconds to measure time and degrees (360 in a circle) to measure angles.
The Sumerians had only two numerals, one and ten. They also had a place-value system. The columns, reading from the right, increased by a factor of 60. Thus the Arabic (base ten) number 46,940 can be represented as in Diagram 2.
Diagram 2
One drawback of the Sumerian system was the lack of a way to represent zero. For example, it would be impossible to tell if the number shown in Diagram 3 represented 421, 4210, 4021, or any other variation, although it might have been possible to tell which meaning was intended from the context.
Diagram 3
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Babylonian Numbers
The Babylonians also used clay for writing. They incised numbers with a stylus that left wedge-shaped marks. This resulted in the writing system being known as cuneiform, from cuneus, meaning a wedge, and forma, meaning a shape. The Babylonian system used a mixture of base ten and base sixty. Base sixty tended to be used for larger numbers.
The numerical notation for small numbers was quite simple; one was represented by a short, straight, vertical stroke, or wedge, two to nine by two to nine short strokes, 10 by an angle, and 100 by a short vertical wedge followed by a short horizontal wedge (see Diagram 4).
Diagram 4: The Babylonian Numbers
Sometimes 10 was represented by a vertical crossed by a horizontal stroke, 20 by a vertical crossed by two horizontals, etc., and the units were represented by horizontal rather than vertical strokes. The tens were sometimes represented by circles, and the units by a kind of crescent. In larger numbers, which were not standardized, the vertical stroke also stood for 60, 3600, and in general for 60n, where n is a positive integer. The angle also stood for 10 × 60, 10 × 3600, etc., and in general for 10 × 60n. The value to be taken depended entirely upon the context. The Babylonians also had a place-value system so it was usually clear what each symbol was being used to represent in a number. For example, 11 would be represented by the symbol for ten followed by the symbol for one (or sixty), whereas 70 would be the represented by the same two symbols in reverse order. Sometimes the symbols for one and sixty were different sizes.
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The Babylonians also developed a way of representing zero, an important advance in the history of numbers since it eliminated any possible confusion over whether a number such as 316 was intended to represent 316, 3160, 3016, 3106, etc. Initially, Babylonian scribes represented zero by leaving a large gap between numerals, but, later, a special symbol was invented. For example, the Arabic number 3608 could be represented by the symbol for 3600 followed by the symbols for 0 and 8, as in Diagram 5.
Diagram 5: The Babylonian Zero
Another development of the Babylonian number system was the occasional use of a minus sign to represent a number that was just less than an exact number of tens. For example 29 could be represented as in Diagram 6.
Diagram 6
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Chinese Numbers
The Chinese had, and still have, various numerical systems. The rod or stick numerals, derived from the wooden sticks used on counting boards, are quite simple. These numerals were frequently written in the monogram form, as in Diagram 7.
Diagram 7: Chinese Rod Numerals
The diagram shows the numbers 0 to 90 and the number 46,431 written in monogram form.
Apart from the rod system, the Chinese have several other systems of numerals, including the "common" numerals (hsiao-hsieh), and the "official" numerals, which are highly decorated versions of the common numerals used on documents to prevent fraud. Since 1955 it has been government policy to introduce Arabic numerals.
The different Chinese systems have a variety of symbols for zero. The symbol for zero in the rod system can be seen in the above diagram.
Mayan and Aztec Numbers
The Mayan and Aztec number systems were vigesimal (base twenty) systems rather than denary (base ten) systems such as those in widespread use today.
In the Mayan system, the numbers 1 to 4 were represented by dots and 5, 10, and 15 by sticks, lines, or bars. The moon may have been used to represent 20. The symbols used for the multiples of 20 (400, 8000, 160,000, etc.) are still uncertain. It is now thought unlikely that the Mayans used a place-value notation.
In the Aztec system, the numbers 1 to 19 were represented by dots or circles, 20 by a religious banner, 400 (20 × 20) by a pine tree, and 8000 (20 × 20 × 20) by an incense-pouch.
The Mayan system also used a symbol to represent zero; this symbol was similar to a shell, having numerous variants. The highest number found in a Mayan inscription is 1,814,639,800 days, corresponding to over 5,100,000 years.
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Alphabetic Numbers
A number of civilizations have used the letters of their alphabets for numerical symbols, taking the first letter for 1, the second for 2, the tenth for 10, the eleventh for 20, and so forth.
Hebrew Numbers
One of these systems is the Hebrew number system. The 22 letters of the Jewish alphabet were used to represent the numbers up to 400. In the Talmud the numbers above 400 are formed by composition, for example, 500 = 400 + 100 and 900 = 400 + 400 + 100; in later times the final forms of the letters k, m, n, p, and ts were used for 500, 600, 700, 800, and 900, respectively. The later Hebrew number system is shown in Diagram 1.
Diagram 1: The Later Hebrew Number System
The thousands were represented by the same letters as the units, sometimes followed by a kind of apostrophe.
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Greek Numbers
The Greeks had two number systems, the earliest of which was used from the 5th to the 1st centuries BC. These numbers were known either as Herodian numbers (after the writer from the 2nd century who described them) or as Attic numbers (after the Attic inscriptions in which they occur). This system uses the initial letters of the number words to represent the numbers. Thus, G (an old form of P, initial of PENTE, pente), was used to represent 5; D (DEKA, deka), 10; H (HEKATON, hekaton), 100; X (XILIO, khilioi), 1000; and M (MYPIOI, murioi), 10,000. These numerals were often combined with the symbol for 5 to create numerals for numbers such as 15, 50, 500, and 5000. The Herodian numbers are shown in Diagram 2.
Diagram 2: The Herodian or Attic Numbers
Tens of thousands were sometimes indicated by dots. Sometimes special symbols were used for fractions, and an accent or a line above the numeral might indicate the fraction.
The later Greek number system used the letters of the Greek alphabet for numerical notation in the same way as the Hebrew system did. Since they had only 24 letters in their classical alphabet, and for a more satisfactory system of numerals they needed 27 letters, they retained the ancient letters digamma (as 6), koppa (as 90), and sampi (as 900). To distinguish the numerals from letters a bar was commonly written over each number, but sometimes the letter was written as if lying on its side. Thousands were often indicated by placing a bar to the left of the number. In modern Greek type, the thousands are indicated as in Diagram 3.
Diagram 3: The Later Greek Number System
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Roman Numbers
Roman numerals are among the most enduring of number systems. Indeed, they are still in use today for certain purposes, such as numbering appendices in books, denoting dates in the credits of films and television programmes, and on watches and clocks. The long usage of Roman numerals could be due to a number of factors, including the widespread influence of the Roman Empire, the force of tradition, and the fact that the system had many advantages over other European systems of the time. For example, Roman numbers had the advantage that the majority of users had to memorize only a few symbols and their values.
The origin of the Roman number system is still obscure. One theory is that it may have been based on the number 5. Thus, the symbol for 5, V, could have its origins in a representation of a hand held flat with the fingers together, and X, for 10, could be a double V. Three of the other symbols could have come from Greek letters not needed in the early Roman alphabet: C (100) from theta (q), M (1000) from phi (f), and L (50) from chi (c, also written as ^). q and f were probably changed gradually, under the influence of the initials of the number words centum (100), and mille (1000), to C and M.
The modern system of Roman numerals uses a subtractive principle. For example, 4 is written as IV, five minus one, rather than IIII, 9 is written as IX, ten minus one, rather than VIIII, and 900 is written as CM, 1000 minus 100, rather than DCCCC. Thus, for example, 1996 is written as MCMXCVI. The modern set of Roman numerals can be seen in Diagram 4.
Diagram 4: Modern Roman Numerals
It is clear that the Romans used some very large numbers. For example, an inscription on the columna rostrata, commemorating the Roman victory off Mylae (Milazzo) in the First Punic war (260 BC), has a symbol for 100,000 repeated 23 times (= 2,300,000).
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Pi
Pi, which is represented by the Greek letter p, is one of the oldest mathematical quantities known to humanity. It denotes the ratio of the circumference of a circle to its diameter. In other words, it is the number of times the diameter of a circle will fit around the edge of the circle. This is the same for all circles and is approximately 3.14. p is the first letter of the Greek word perifereia, meaning circumference.
The Nature of Pi
p is an irrational number; it cannot be expressed as a ratio of two integers, and its expression as a decimal never terminates and never starts recurring. In fact the numbers in the decimal expression of p seem to be almost "random" - there is no perceivable pattern in their order.
Not only is p irrational, it is also a transcendental number. This means that it is not the solution of any polynomial equation with coefficients that are rational numbers.
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The History of Pi
The origins of p are so ancient that they are now untraceable. It is possible that p first became known in Egypt. There is a reference to p in the Egyptian Rhind Papyrus, dated about 1650 BC. The first theoretical calculation is likely to have been carried out by Archimedes around 200 BC. He found that the value of p is between 223/71 and 22/7.
No further advances were made until the 17th century. p was then known as the Ludolphian number, after Ludolph van Ceulen, a German mathematician. In the 17th century the search for mathematical formulas for p began. One of the most well known of these is the infinite series:
This series is sometimes attributed to Gottfried Leibniz but it seems to have been first proposed by James Gregory (1638-1675). It does not actually help us to calculate p at all. To find the first four decimal places of p correctly, we would need to find the first 10,000 terms of the series.
However, the search for mathematical formulas for p continued. In 1706, the English mathematician John Machin found a formula that enabled the calculation of p to any number of decimal places. The calculation was still time-consuming and repetitive, with a single mistake disrupting all further stages of the process. In 1873, Shanks calculated p to 707 decimal places. Unfortunately he made a mistake when calculating the 528th decimal place so that he only got the first 527 correct.
p was proved to be irrational by the Swiss-German mathematician Johann Lambert in 1761. In 1882, the German mathematician Carl Louis Ferdinand von Lindemann proved that p is a transcendental number. The symbol p was first used with its present meaning by the Welsh mathematician William Jones in 1706. Leonhard Euler adopted the notation in 1737 and it quickly became standard.
With the advent of computers, it became possible to find p to many more decimal places. In June 1995, the Japanese mathematician Yasumasa Kanada found p correct to 3,221,220,000 decimal places. In theory, p can be calculated to any desired degree of accuracy because its decimal expansion goes on for ever. In practice, p calculated to 39 decimal places would be enough to allow us to find the circumference of a circle around a sphere with a radius of 15 billion light years (the distance to some of the most remote objects in the universe) to within an accuracy of the radius of a hydrogen atom.
Many mnemonics have been invented to help remember the decimal representation of p. The following one gives the first 15 digits: "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." Each word is replaced by the number of letters in it. Thus, p = 3.14159265358979….
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Arabic Numbers
The Arabic numerals, 0, 1, …, 9, now in general use, are derived from Indian numerals. The name Arabic is used because Western Europeans learned about the system from Arabic writers.
History
The history of Indian-Arabic numerals is only vaguely known because few examples of their early use have survived. Early forms of the numerals were developed in India between the 2nd century BC and the 6th century AD. During that period Indian mathematicians realized that a place-value system of notation (see below), which included a symbol for zero, enabled calculations to be performed by simply writing down numbers instead of using an abacus. The numerals themselves were used in several sites around the Mediterranean but only as simple ways of writing down numbers. Methods of calculation that used the numbers were not understood outside India until the Arabic mathematician Al-Khwarizmi wrote a treatise on them in the early 9th century. Al-Khwarizmi's treatise was not translated into Latin until the early 12th century and the method of calculation by writing down numbers became known as "algorism" (a corruption of al-Khwarizmi). In the late 12th century, Leonardo Fibonacci started to publish books in Pisa that showed the full power of the Arabic number system and it was taken up by the book-keepers and commercial men of northern Italy. For several centuries there was great controversy over whether algorism or the abacus was the fastest method of calculating. With the invention of printing, numerous books describing algorism appeared and, from about 1500 onwards, the Arabic number system became standard in Western Europe. The cumbersome Roman numerals were inadequate for writing out the large and complicated numbers used in astronomy and, increasingly, in other branches of science, and the invention in the early 17th century of logarithms finally ended their use.
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The Place-value System
The most important feature of the Arabic number system is the use of place-value notation. Perhaps the best way to see how this works is to imagine that counting is carried out by threading beads onto wires arranged as in Diagram 1.
Diagram 1: A Representation of the Place-value Notation
As each object is counted it is represented by a single bead threaded on the right-hand end wire. The wire has only enough room for nine beads. To count the tenth object, one bead is put on the next wire and all beads are taken off the end wire. Then the beads are used to fill up the end wire again, and so on. Thus each bead on the second wire represents ten; on the third wire one hundred; on the fourth wire one thousand, and so on. To write this down, all that are needed are nine symbols to represent from one to nine beads (in this case, the Arabic numerals 1 to 9) and one symbol (in this case, 0) to represent an empty wire. The symbols are written out in the same places as the wires: the value of a numeral depends on its place. For example, in Diagram 2, we can see that 4305 is a way of representing the number that consists of 4 thousands, 3 hundreds, 0 tens and 5 units or ones.
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Diagram 2
Ten is called the base, or radix, of this system, which is the one now in everyday use. A notation system with ten as the base is called a denary system. Any other number could be used as a base.
Place-value notation is very difficult without a symbol for zero. If there were no zero symbol in the denary system then 9 could mean nine, ninety, nine hundred, etc. Zero may be a late invention in the history of numbers, although Hindu literature suggests it was in use before the time of Christ.
Who came up with the idea of number systems and for what reasons?
The origins of numbers date back to the Egyptians and Babylonians, who had a complete system for arithmetic on the whole numbers (1,2,3,4,. . . ) and the positive rational numbers.
The Greeks at the time of Pythagoras knew that these number systems (whole numbers and ratios of whole numbers) could not completely describe everything they wanted numbers to describe. They discovered that no rational number could describe the length of the diagonal of a square whose sides were of length 1. They called such lengths "irrational", recognizing that some other kind of number system would be needed in order to describe them, but not knowing what it would be. They did not pursue the matter, for they viewed whole numbers with such awe that anything not expressible in terms of whole numbers was distrusted by them as contrary to nature.
These number systems evolved somewhat during the Middle ages with the notable addition by the Hindus of a convenient notation for zero and negative numbers, concepts which previously had been difficult to deal with due to the lack of notation. The properties of the "real number system" (consisting of both rational and irrational numbers) began to be understood in the 1600's with the development of calculus, and by the end of the 1800's mathematicians such as Dedekind and Cantor were giving rigorous mathematical definitions of this number system, putting it on equal footing with the whole numbers and rational numbers.
It wasn't until the early 1800's, however, that the abstract structure of these number systems was studied. This new area of math, like many other areas of math, arose from a creative new way to answer an old question: how to find the roots of a polynomial (those numbers which, when substituted into it, give zero).
Much was known about polynomials of degree (highest power) less than 5. Italian mathematicians had solved for the roots of the 3rd and 4th degree polynomials in the 1500's. These solutions were always expressible in terms of "radicals" or nth roots of numbers. For a long time no one knew how to solve a general 5th degree polynomial for its root.
Polynomials of lower degrees were still of interest though. In search of a deeper understanding of them, Gauss studied quadratic (2nd degree) polynomials. Through his work, he found that the objects he was considering were related to each other in much the same way that numbers are related under addition or multiplication. In modern terms, he was considering "finite group structures": finite sets which are essentially like a number system, but with only one operation. In many of the groups which he worked with the order in which the operation was performed doesn't matter: a ·b = b ·a. Groups in which the operation commutes in this way are called abelian groups. It is believed that Gauss may have been one of the first to have a rough understanding of the structure of finite abelian groups.
Also related to the study of polynomials is the "theory of substitutions" studied by Lagrange, Vandermonde, and Gauss. A substitution is where the variable of the polynomial is replaced with a different expression (such as a new variable plus a constant). It is possible sometimes to make the "right" substitution and turn a very complicated polynomial into something much easier to handle. This led to the study of the permutations of a set. Also studied by Ruffini and Cauchy, the permutations of a set form a group structure as well, though in this case the order of operation matters and therefore the groups are non-abelian.
Any discussion of the study of polynomials and number systems incomplete without a mention of Galois. He was the first to fully understand the connections between these finite number systems and the behavior of the roots of polynomials. It follows from his work that there is no "nice" formula for the roots of some 5th degree polynomials. While he died in his early 20s in a duel, his work (which was allegedly written in a letter and sent to a friend the day before he died) is still one of the cornerstones of the study of number systems.
>Do you know where the ST, ND, RD and TH come from when we
>write and say ordinal numbers...1ST, 2ND, 3RD etc? I have
>a first grade student who wants to know. Thank You.
This isn't exacatly a math question, but I like languages too!
Like many things in English (or other languages), such as
verb tenses, we have "regular" and "irregular" number words.
The simplest and oldest tend to follow their own odd patterns,
and when you get into the newer and bigger words a pattern
takes over. Clearly "-th" is the standard pattern for an
ordinal number; so where do the other endings come from?
Actually, it's not just the endings but the whole words:
"fir-" and "seco-" don't exactly come from "one" and "two",
though you can see a connection between "third" and "three".
Let's look at each word individually, using the American
Heritage dictionary (which specializes in etymology) to
see where they come from.
FIRST: This comes from Old English "fyrst". This is believed
to come from a Germanic word "furista", which in modern
English survives as "foremost". The "-st" is actually a
superlative ending ("-est").
SECOND: This comes from Latin "secundus", meaning "following"
or "next". It's a participle of the verb "sequor", which means
"to follow" (as in "sequel", "sequence", and "subsequent".
So the ending "-nd" is a Latin gerund ending.
THIRD: This one finally comes from an English number: Old
English "thridda" comes from "thri", three. I don't know enough
Old English to know whether the "-d" is just a variant of the
"-th" in other ordinals ("-t" in Old English), but it appears
that both are merely endings for ordinals.
So we see that "first" and "second" are different because
they weren't originally ordinals at all, but special words
for two special positions in a sequence, the "front-most" and
the "next". "Third" almost falls into the normal pattern, and
then things become regular (and were in Old English as well:
feortha, fifta, ...).
Interesting, isn't it? Thanks for the question.
I would say that the words for cardinals and ordinals indicate not
that one came before the other, but that they originated separately,
with different purposes. And the difference (in English, at least) is
not really a relic of the prehistoric development of Indo-European
languages, but a relatively recent thing (after all, "second" is a
borrowing from Latin). So I don't think we necessarily have any
evidence of the sequence of events in ancient languages, but rather
an evidence that small numbers were for a long time (and maybe still
are?) perceived as something other than numbers in the full abstract
sense. Consider also that in at least some languages "one" is hard to
distinguish from the article "an", which does not carry a strong
sense of number, just mere singularity.