A History of Numbers
Our modern understanding of numbers as quantities that can be positive or negative, whole numbers, fractions or irrationals, infinitely big or infinitely small, and so on, is so familiar that it seems almost self-evident. However, it actually evolved slowly over thousands of years and required huge intellectual leaps.
Our innate understanding of numbers is surprisingly primitive but our intelligence has allowed us to expand our conception of numbers far beyond our natural understanding of it.
Number Sense:
Humans have ‘number sense’ – an innate ability to understand that a collection contains a specific amount of things. For example, a dog sees only a group of birds but a human sees that there are exactly four. Some other animals also have number sense. For example, most birds will continue to tend to a nest if the number of eggs it contains decreases from four to three, but they will often desert it if the number of eggs drops from three to two.
However, human number sense is very basic and doesn’t extend naturally beyond collections containing only a few things. This is reflected in the languages of primitive tribes which often only have words for ‘one’, ‘two’ and ‘many’. Many European languages have fossil evidence of this as well – for example, the words ‘thrice’ in English and ‘ter’ in Latin mean both ‘three times’ and ‘many’, and the Latin and French words for three are related to their words for ‘beyond’ and ‘very’.
Our innate understanding of number is also specific and practical rather than general and abstract, and this is also reflected in our languages. For example, there are lots of old-English words for collections of specific things such as a ‘flock’ of birds, a ‘herd’ of cattle, a ‘pack’ of wolves, etc., but general terms like ‘collection’ and ‘group’ both come from foreign languages and don’t have any old-English equivalents. There are also unnecessarily many words for collections of two things such as ‘pair’, ‘brace’, ‘couple’, etc., which suggests that the people who created those words didn’t understand that there was an abstract quality that they all have in common.
We are so familiar with the abstract concept of number that it’s almost impossible to imagine not realising that all collections of the same size have something fundamental in common. However, it’s obviously much less intuitive than it seems to us now.
Cardinal Numbers and Ordinal Numbers:
Numbers can be conceived of in two different ways – ‘cardinally’ and ‘ordinally’. A cardinal number is the number of items in a collection, and an ordinal number is the position of an item in that collection. For example, the cardinality of the alphabet is twenty-six and the ordinality of the letter ‘b’ is two.
The size of a collection can be measured cardinally by placing it in one-to-one correspondence with something else. For example, a shepherd may record the size of a flock entering a field cardinally by adding a pebble to a pile for every sheep that passes through a gate and then record the size of the flock leaving the same field by removing a pebble for every sheep that passes back through the gate in the other direction. The amount of pebbles and the amount of sheep are in one-to-one correspondence, so the shepherd knows if he has all his sheep if there are no pebbles left over when they leave the field.
The size of a collection can also be measured ordinally by assigning a position to each of the items it contains. For example, the shepherd could number each sheep passing through the gate – the first sheep passing through the gate would be one, the next sheep would be two, and so on – and the number assigned to the last sheep is the size the flock.
Today, we generally quantify collections ordinally but the impractical cardinal system of tallying was used until surprisingly recently. As late as the nineteenth century the UK Treasury recorded the public finances on thousands of tally sticks. When the tallying system was finally replaced with written ledgers, the sticks were burned. However, there were so many that the fire grew out of control and destroyed the Houses of Parliament. A few years later, Charles Dickens recalled the tragicomic episode in a public address on administrative reform:
‘Ages ago, a savage mode of keeping accounts on notched sticks was introduced into the Court of Exchequer and the accounts were kept much as Robinson Crusoe kept his calendar on the desert island. A multitude of accountants, bookkeepers, and actuaries were born and died…Still official routine inclined to those notched sticks as if they were pillars of the Constitution, and still the Exchequer accounts continued to be kept on certain splints of elm-wood called tallies. In the reign of George III, an inquiry was made by some revolutionary spirit whether, pens, ink and paper, slates and pencils being in existence, this obstinate adherence to an obsolete custom ought to be continued, and whether a change ought not be effected. All the red tape in the country grew redder at the bare mention of this bold and original conception, and it took until 1826 to get these sticks abolished. In 1834 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur to any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighbourhood. However, they never had been useful, and official routine required that they never should be, and so the order went out that they were to be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over-gorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost thereof.’
Number Systems:
Both cardinal and the ordinal numbers were used to create early number systems.
Some primitive societies generalised their cardinal measuring systems by using standard collections of a specific size to compare a given collection with – e.g., the two wings of a bird, the three leaves if a clover, the four legs of a dog, etc. In these societies, number words were derived from the words for these standard collections – i.e., bird, clover, dog, etc.
In other societies, number words were derived ordinally from the word for each of the fingers. In other words, the word for the first finger they used when counting was used for one, the word for the second finger was used for two, the word for the third finger was used for three, and so on.
When written number systems were created, some were based on cardinal numbers and some were based on ordinal numbers.
The ancient Egyptian and Babylonian number systems were cardinal. Each number from one to nine was represented by a symbol that was made up of the corresponding number of some repeated unit. For example, these are the first ten numerals in the Babylonian number system:
However, other ancient civilisations like the Greeks and the Phoenicians used an ordinal number system which used letters of the alphabet as numerals – i.e., the first letter of the alphabet was also used as the symbol for one, the second letter was used as the symbol for two, and so on.
Both of these types of written number system made arithmetic very difficult, so mechanical systems like the abacus were used instead. An abacus has beads arranged in parallel rows, where each row represents a different power of ten – i.e., units, tens, hundreds, etc. This system is more practical because it’s positional. In other words, the value of a bead in an abacus depends on which row it’s on – e.g., two beads represent two on the top row, twenty on the second row, two hundred on the third row, and so on. This allows addition, subtract, multiplication and division to be performed using a few simple rules about how to move the beads.
Surprisingly, it took a very long time for these mechanical positional systems to inspire written positional systems. This is because there was no symbol to represent an empty column which makes positional number systems ambiguous. For example, without a symbol for an empty column, ‘23’ could mean 23, 203, 230, 2030, 2300, etc. This problem was solved when the ancient Indians invented a symbol for zero to represent an empty column and used it to create a positional number system. These numerals were later adopted by the Arab world from where they spread to Europe, which is why they are known to us as Arabic numerals.
Number Theory:
Today, number theory is the most rigorous branch of maths but ironically numbers were first studied by people who believed that they had mystical properties – a superstition known as ‘numerology’.
A common form of numerology is the belief that certain numbers are omens. For example, ancient Jews believed that the number forty was significant because it was the number of days and nights of the Biblical flood and Moses’s meeting with God on Mount Sinai, as well as the number of years the Israelites wandered in the wildness. Similarly, the number seven recurs in Christian theology as the number of virtues, deadly sins, and spirits of God. Today, this idea persists as the belief that the number thirteen is unlucky.
In civilisations which used their alphabet as numerals, another common form of numerology was ‘gematria’ – the association of words with numbers. In gematria, a word’s number is the sum of the numbers represented by the letters it contains. Gematria was used by Talmudic scholars to interpret Jewish theology, and the ancient Greeks to interpret myths. For example, in the Iliad the names of Patroclus, Hector and Achilles add up to 87, 1225 and 1276 respectively, which was taken as evidence of their relative strengths.
In ancient Greece, Pythagoras led a religious cult called the Pythagoreans who worshiped numbers and studied their properties. They believed that even numbers were feminine and earthly, whilst odd numbers were masculine and celestial. They also believed that specific numbers represented specific things. For example, one represented reason because it was unchangeable, two represented opinion, four represented justice because it was first perfect square, five represented marriage because it was the sum of the first masculine number and the first feminine number (they didn’t consider one to be a number, rather the source of all numbers).
Through their worship of numbers, the Pythagoreans discovered many of their properties. For example, they discovered that some numbers are the sum of their factors. For example, the factors of 6 are 1, 2, and 3, which add up to 6 and the factors of 28 are 1, 2, 4, 7, and 14, which add up to 28. They named these numbers ‘perfect numbers’ and they were particularly significant to the Jews and Christians who believed that God had chosen them specifically for the number of days of creation and the lunar cycle.
The Pythagoreans also discovered pairs of numbers whose factors add up to each other. For example, the factors of 284 are 1, 2, 4, 71, and 142, which add up to 220, and the factors of 220 are 1, 2, 4, 5 10, 11, 20, 22, 44, 55, 110, which add up to 284. They named these pairs ‘amicable numbers’, and many people believed they were a good omen. There was a medieval story of a prince, whose name in gematria was 284, who wanted a bride whose name in gematria was 220 because he believed this would guarantee a happy marriage.
Irrational Numbers:
The Pythagorean concept of number only included the natural numbers – i.e., positive whole numbers. They discovered many right-angle triangles whose lengths were a ratio of whole numbers such as 3:4:5, 5:12:13 or 8:15:17, and they believed that the sides of all right-angle triangles could be written as a ratio of whole numbers. There were some triangles whose sides didn’t seem to fit this pattern but the Pythagoreans had faith that the sides of these triangles were just ratios of very big numbers which they hadn’t discovered yet.
They later discovered what we today call Pythagoras’s theorem, that the square of the longest side of a right-angle triangle is equal to the sum of the squares of the other two sides – i.e., a2+b2=c2. They were not the first people to discover this theorem and they probably didn’t have a rigorous proof of it, but they attached enormous significance to its simplicity and elegance. However, it led to the discovery that the right-angle triangle formed by drawing a diagonal line through the middle of square doesn’t have sides which are a ratio of whole numbers.
If sides b and c of this triangle are the same length – say, 1 unit long – then:
For c to be written as the ratio of two whole numbers, x and y, there must be two integers, x and y, such that x2/y2=2, where the fraction x2/y2 is in the lowest terms. For x2/y2 to be in the lowest terms, at least one of x2 or y2 must be odd.
x2/y2=2 can be rearranged as x2=2y2, which means x2, and therefore x, must be even. If x is even, then x=2n, where n is another whole number. Substituting 2n for x in x2=2y2, gives 4n2=2y2 or 2n2=y2 which means y2 is also even. However, at least one of x2 or y2 must be odd for x2/y2 to be in its lowest terms. Therefore, there are no two integers, x and y, which satisfy the equation x2/y2=2, so the square root of 2 can’t be written as a fraction.
Today, these numbers are called ‘irrational’, but the Pythagoreans were so horrified by them that they called them ‘alogon’ – the unspeakable. They believed they revealed an imperfection in the work of God, and they were forbidden from telling outsiders about them.
Euclid:
The Pythagoreans bridged the gap between numerology and number theory and Greek mathematicians continued to study numbers but steadily abandoned the Pythagoreans’ mystical beliefs. The most significant of these mathematicians was Euclid who proved lots of general properties of numbers. For example, he proved that there is an infinite number of prime numbers.
His proof involves ‘factorials’, which are the products of the first n consecutive numbers, notated as n! – e.g., 7!=1x2x3x4x5x6x7. Euclid used factorials to show that there is an infinite number of primes because there is no highest prime. The proof has two parts:
First, two consecutive numbers can’t have any common factors because if a number, x, can be divided into y equal parts, then x+1 can’t also be divided into y equal parts because there will be 1 left over. Therefore, the numbers n! and n!+1 can’t have any common factors because they’re consecutive.
Second, if n is prime, then n!+1 must be either a prime or the product of different prime factors to n!. These prime factors must be greater than n because n! is the product of all the numbers smaller than n. Therefore, n!+1 is a either a prime or the product of primes greater than n and, in either case, n isn’t the largest prime.
Algebra:
In ancient Greece, mathematical statements were expressed in words. For example, the Greeks described Pythagoras’s theorem as ‘the square of the longest side of a right-angle triangle is equal to the sum of the squares of the other two sides’, rather than ‘a2+b2=c2’. The development of the ‘symbolic algebra’ we use today from the the ‘rhetorical algebra’ of the ancient Greeks has happened many times in many different cultures and has tended to go through an intermediate stage called ‘syncopated algebra’. Syncopated algebra involves shortening the words used for unknown quantities and mathematical operations, and it transitions into symbolic algebra when those shorthands become symbols which have no resemblance to the original words. For example, subtraction was originally written as the word ‘minus’, then it was shortened to the letter ‘m’ with a line above it, then just the line.
The creation of the Western symbolic algebra began with a sixteenth century French mathematician called François Viète who proposed using vowels for unknown quantities and consonants for known quantities. A few decades later, another French mathematician, René Descartes, replaced this with a system which used the first letters of the alphabet for known quantities and the last letters of the alphabet for unknown quantities, which is why we use x, y and z as variables today. This, along with the creation of symbols for the operations addition, subtraction, multiplication, division, and so on, produced a mathematical notation which could express every conceivable mathematical statement as a series of symbols.
Symbolic algebra wasn’t just much simpler than rhetorical algebra, its generality led to an extension of the conception of number. Prior to the invention of symbolic algebra, medieval mathematicians didn’t believe fractions, irrationals and negatives were real numbers. For them, these equations were meaningful:
But these equations weren’t:
However, a purely symbolic algebra can generalise those equations like this:
Mathematicians found that using these equations didn’t lead to any contradictions, so they were increasingly accepted as valid in general, which implied the legitimacy of fractions, irrationals and negatives.
Imaginary Numbers:
In 1545, the Italian mathematician Gerolamo Cardano created a formula to solve this seemingly impossible pair of simultaneous equations:
However, his formula gave him this apparently meaningless solution:
These values of x and y satisfy the simultaneous equations but the inclusion of the square root of -15 seems to invalidate the solution because all square numbers are positive, so there is no number, x, such that x2=–15. Cardano published the result but acknowledged that the solution was impossible.
However, a few decades later another Italian mathematician, Rafael Bombelli, applied Cardano’s formula to this equation:
which has the solutions:
Bombelli found that Cardano’s formula gave the ‘solution’:
Bombelli then showed that this simplifies to:
and these values of x sum to 4, which is one of the equation’s ‘real solutions’. Thus, he found that whilst these negative square numbers seemed impossible, they were nonetheless useful.
Today, we call the square root of a negative number ‘imaginary’ and numbers which have a ‘real’ component and an ‘imaginary’ component in the form a+bi, where i is the square root of -1, are known as ‘complex numbers’. However, in the centuries following Cardano and Bombelli’s discoveries, more of their uses became known but they still weren’t accepted as legitimate numbers.
However, in 1831 the German mathematician Carl Friedrich Gauss developed a rigorous model to interpret complex numbers. His model mapped the complex numbers onto a graph like this:
According to this system, the real component of a complex number corresponds to a position on the x axis and the imaginary component corresponds to a position on the y axis. Thus, imaginary numbers exist along a different dimension to real numbers and complex numbers are essentially two-dimensional numbers. Quantities which can’t be expressed by a single number are called vectors and the line from centre of the graph, O, to the (x,y) coordinates of the complex number, A, corresponds to the vector OA. Adding, subtracting, multiplying and dividing complex numbers are essentially different geometric transformations of the complex numbers’ vectors.
Adding Complex Numbers:
Take the two complex numbers, 1+i and 3+i, represented by the vectors OA and OB.
When complex numbers are added together the real components and the imaginary components are added separately. So, (1 +i) + (3+i)=4+2i, which can be represented by the vector OC.
As the graph below shows, this is equivalent to adding the vectors OA and OB together:
Multiplying a complex number by a real number:
Take the complex number 2+i represented by the vector OA.
When the complex number 2+i is multiplied by the real number 2, both the real and the imaginary components are multiplied by 2. So, 2(2+i)=4+2i, which can be represented by the vector OB. This is equivalent to stretching the vector OA by a factor of 2.
Multiplying a real number by a complex number:
Take the real number 1 represented by the vector OA.
Multiplying it by i is equivalent to rotating the vector OA anti-clockwise through 90° like this:
Conclusion:
Gauss’s system established complex numbers as vectors and mathematical operations such as addition, subtraction, multiplication, division, etc., as geometric transformations of them.
Transfinite Numbers:
In 1636, Galileo published the first description of the properties of collections of an infinite number of things – known as ‘infinite sets’. Galileo argued that infinite sets lead to paradoxes by comparing the number of natural numbers (positive whole numbers) and the number of square numbers. In the first n natural numbers, there are only root n square numbers. For example, there are 10 square numbers up to 100, 100 square numbers up to 10,000, and so on. However, there must also be as many square numbers as there are natural numbers because every natural number is the square root of some square number. Therefore, there must simultaneously be more natural numbers than square numbers and as many square numbers as there are natural numbers.
However, this apparent paradox is actually a defining property of infinite sets – namely, that the whole of the set can be put into one-to-one correspondence with a part of itself. From his ‘paradox’, Galileo concluded that the concepts ‘greater than’, ‘less than’ and ‘equal to’ can’t be applied to infinite sets. However, even though infinite sets can’t be quantified ordinally by ordering them in a sequence and taking the position of the final member, they can be compared cardinally by placing them in one-to-one correspondence with each other. For example, this shows the one-to-one correspondence between the natural numbers and the square numbers:
This process can continue indefinitely with every square number paired with a unique natural number, and every natural number paired with a unique square number, without ever exhausting either set. Therefore, the two sets are the same size.
This is the smallest type of infinity and it’s represented by the transfinite number N01 – called ‘aleph-null’. The aleph numbers go from N0 to N∞ and each aleph number describes a different size of infinity.
Any set which can be placed into one-to-one correspondence with the natural numbers is also equal to N0. Placing a set into one-to-one correspondence with the natural numbers is equivalent to creating a system which can assign a position to each element in the set. As a result, infinite sets equal to N0 are described as ‘countably infinite’. For example, the set of all integers (all whole numbers) is countably infinite because the integers can be placed in a sequence like this:
The set of all natural numbers and the set of all integers are ‘incremental’ because their elements increase by a finite amount – i.e., 1. However, the set of all rational numbers is ‘everywhere dense’ which means that between any two rational numbers there is an infinite number of rational numbers. For example, between the numbers and 1/10 and 2/10 there are the numbers 11/100, 12/100, 13/100, 14/100, 15/100, 16/100, 17/100, 18/100, 19/100 and then 111/1000, 112/1000, 113/1000 and so on.
However, despite being everywhere dense, the set of all rational numbers is also countably infinite. This was proved by the German mathematician Georg Cantor who designed a system for assigning a position to each rational numbers. To do this, he mapped every rational number to a position on a graph. On this graph, the x and y axes each go from 0 to ±∞ and the coordinates (x,y) represent the fraction x/y. Then he designed a path that will go through every value of (x,y), where x and y are whole numbers, like this:
This path will go through every rational number and the order in which it passes through them can be used to assign each one a position. This allows the rational numbers to be placed in one-to-one correspondence with the natural numbers. This process will record duplicates of the same rational number, such as 1/1 and 2/2, but the sequence can be simplified by only including the first occurrence of each rational number. The one-to-one correspondence between the set of natural numbers and the set of rational numbers produced by this system looks like this:
Therefore, despite being everywhere dense, the set of all rational numbers is a countably infinite set of size N0.
The set of all real numbers (all rational and irrational numbers) is also everywhere dense. However, it is not countably infinite and Cantor proved this by showing that any system for ordering the real numbers will necessarily be incomplete.
Take an arbitrary system for ordering the real numbers and place the numbers in a table like this:
Then generate a number, rn, whose first numeral is one less than the first numeral of r1, whose second numeral is one less than the second numeral of r2, whose third numeral one less than the third numeral of r3, and so on to r∞, like this:
This method produces a real number which is different from every other real number in the table. This is because two identical numbers must have the same numeral in every position, but the number produced by this method will always have at least one numeral different to every other number in the table. Therefore, any system for ordering the real numbers will necessarily be incomplete, which means the set of real numbers is uncountably infinite.
The size of the infinite set of real numbers is denoted by the symbol C. C is larger than N0, but there is an on-going controversy about whether C is equal to N1.
Cantor proved that there are an infinite number of transfinite numbers from N0 to N∞ by showing that an infinite set and the set of the subsets of that infinite set aren’t the same size.
The set of the subsets of a set is called its ‘power set’. If a finite set, called ‘Set(A)’, contains the elements {a,b,c}, then its power set, called ‘PowerSet(A)’, contains the elements {{a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}, {}}, where {} is the ‘empty set’.
The elements of a set and the elements of its power set are the same size if they can be placed in one-to-one correspondence with each other. As the table below shows, the elements of Set(A) and the elements of PowerSet(A) can’t be placed into one-to-one correspondence.
Now consider the infinite set, Set(X), and its power set, PowerSet(X). When placing the elements of Set(X) in one-to-one correspondence with the elements of PowerSet(X), each element of Set(X) will either be contained in the subset of PowerSet(X) it’s been paired with, or it won’t be. For example, if {a} is paired with {a,b} then {a} is contained in the subset it’s been paired with, but if {a} is paired with {b,c} then {a} isn’t contained in the subset it’s been paired with.
In PowerSet(X), there is a Subset(Y) which is the set of all elements which aren’t contained in the subset they’re paired with. For example, if {a} is paired with {a,b} then {a} isn’t contained in Subset(Y) because {a} is contained in {a,b}, but if {a} is paired with {b,c} then {a} is contained in Subset(Y) because {a} isn’t contained in {b,c}.
However, there is no system for establishing one-to-one correspondence between the elements of Set(X) and the elements of PowerSet(X) which can pair any element of Set(X) with Subset(Y).
Imagine {a} is an arbitrary element of Set(X) and Subset(a) is the subset of PowerSet(X) that {a} has been paired with.
If {a} is an element of Subset(a) then {a} isn’t an element of Subset(Y), and if {a} isn’t element of Subset(a) then {a} is an element of Subset(Y). In other words, if {a} is a member of one it can’t be a member of the other. That means that Subset(a) and Subset(Y) can’t be the same set, but Subset(a) is the set that {a} has been paired with, so that’s just another way of saying that {a} can’t be paired with Subset(Y). The same logic applies to every element of Set(X), so no element of Set(X) can be paired with Subset(Y).
Another way of thinking about this is to imagine pairing each element of Set(X) with a subset of PowerSet(X) and placing each element of Set(X) which isn’t contained in the subset of PowerSet(X) it has been paired with in Subset(Y). Now imagine that {a} is paired with Subset(Y). Before being paired with Subset(Y), {a} wasn’t a member of Subset(Y) so it should now be placed in Subset(Y). However, as soon as {a} becomes a member of Subset(Y) then Subset(Y) is no longer the set of all elements which aren’t contained in the subset they’re paired with because {a} will now be both paired with it and contained in it. Therefore, once any element of Set(X) has been paired with Subset(Y) then Subset(Y) ceases to have the defining property of Subset(Y).
This shows that there is no system for pairing the elements of Set(X) and PowerSet(X) which can pair any element of Set(X) with Subset(Y). Therefore, the elements of Set(X) and PowerSet(X) can’t be placed in one-to-one correspondence with each other which means that PowerSet(X) is a bigger infinity than Set(X). This proves that there are an infinite number of transfinite numbers because this same process can be repeated infinitely by taking the power set of PowerSet(X), and then taking the power set of the power set of PowerSet(X), and so on.
Conclusion:
For the majority of human history, we didn’t have any abstract understanding of number. This seems unimaginable to us now but it’s obviously much less intuitive than it seems.
Over the last few thousand years our understanding of number has become increasingly abstract and also increasingly diverse. In other words, not only can the numbers π, i and N0 only exist as abstractions but they also seem to have nothing in common with each other.
The majority of this history is taken from Number by Tobias Dantzig, which is one of the most interesting books I’ve read. Cantor’s proof of the infinite number of transfinite numbers was explained to me by my friend Sam.
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