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Infinity 
The History of Infinity
Infinite Sets Decimal NumbersDecimal FractionsTypes of DecimalsRecurring DecimalsRepresenting a Recurring Decimal
Cyclic NumbersBinary NumbersNumber Bases
Binary and Hexadecimal
Calculations in Base Systems
FractionsMagic SquaresNumerology
Adding Magic SquaresThe word infinity comes from the Latin infinitus, a combination of in, meaning no or without, and finis, meaning end or limit. We use the term finite to describe something that can be counted. In other words, a set of objects is finite if, when we count them, we come to an end having counted them all. If the counting can proceed indefinitely with no imaginable end, then the set is infinite. Some sets of objects might appear infinite because they are so large, such as all the hairs on a head or all the molecules of oxygen in the world, but, in theory, these could be counted and so are not infinite. The counting numbers (natural numbers) themselves are an infinite set. However large a number we have, it is always possible to add 1 to the number: we never reach a limit.
Throughout history, studies of infinity have combined philosophy and
mathematics. One of the first people to consider the notion of the infinite was
the Greek philosopher Anaximander, who used the term "boundless" (apeiron).
Infinity has historically been divided into two types: that conceived as actual,
such as space, time, or God, and that which exists only in the mind, such as
mathematical and logical concepts. This second type of infinity can also be
called quantitative infinity.
The study of quantitative infinity in ancient Greece was dominated by Aristotle.
He showed that a finite line with fixed endpoints could be divided an infinite
number of times by dividing the line at its midpoint and then dividing one of
the segments at its midpoint, and so on. In theory, this process could be
continued endlessly, but, in practice, it eventually becomes impossible to
devise any method of carrying the division further (what could be used to mark
two points on an atom?). However, mathematics does not need to be tied to the
physical world. It is generally convenient to assume that an interval can be
divided indefinitely.
Many famous philosophers have considered the nature of infinity. In the period
following Aristotle's observations, infinity increasingly came to be identified
with divine perfection. Thomas Aquinas agreed with this concept in the 13th
century. In the 17th century, Ren้ Descartes said "To attempt to know the
infinite would be to thereby make of it only a quasi-infinity", and stated
that the difference between the infinite and the finite is the same as between
God and creature.
The "love-knot" symbol, ฅ, that we use to represent infinity today
was not introduced until 1650. English mathematician John Wallis, who first used
this symbol to mean infinity, did not add anything to the understanding of
infinity, but, by providing a more convenient notation, he made it easier for
other mathematicians to use it in their work.
Augustin-Louis Cauchy gave a definition of infinity in terms of limits. For
example, as x tends to 0, 1/x tends to ฅ (gets closer to
ฅ). This is the same
as saying the limit of 1/x as x tends to 0 is ฅ. Limits form the basis of
modern calculus.
The idea of an infinite set leads immediately to two problems. Firstly, it is
impossible to think of a practical realization of such a set. We can imagine
numbers going on for ever but they cannot all be written out for us to look at.
Secondly, despite the idea of "unendingness", mathematicians talk of
an "infinite set" as though an unending quantity of objects had been
neatly parceled up for examination. An example of such a neat parcel is the set
of all fractions between 0 and 1. It is quite easy to show that there is an
unending number of such fractions. Intuitively this is obvious because, given
any two fractions a/b and c/d, one can always say that there is a third and
different fraction between them, namely (a/b + c/d)/2.
Infinite sets also have other strange properties. For example, suppose we take
two sets, one containing all the natural numbers (1, 2, 3, 4, etc.) and the
other containing the squares of all the natural numbers (1, 4, 9, 16, etc.). For
each number in the first set there must be a number in the second set (because
the second set just contains the squares of the first). Therefore, there must be
the same number of numbers in both sets. However, all the numbers in the second
set are in the first set, but the first set also has the extra numbers in
between. So there must be more numbers in the first set than in the second. This
paradox was pointed out by Galileo Galilei in 1638, but it was not until the
19th century that George Cantor used it to provide a new type of arithmetic for
dealing with infinite sets.
Decimal (or denary) numbers are widely used for counting and in monetary systems. The decimal system is a place-value system in base 10. It uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) whose values depend upon their position in the number. Moving left from the decimal point, the columns have values 1 (=100), 10 (=101), 100 (=102), and so on. The symbol 0 is used to indicate an empty column. The number 586 (when it is written as a decimal number) is made up of five lots of 100, eight lots of 10, and six lots of 1.
Manipulating vulgar fractions such as 23/450 and 123/500 can be difficult. As
a result of this it was found to be necessary to devise some simpler system of
notation for fractions. This was done by extending the decimal (or denary)
number system.
In the decimal system, the number 125 represents 5 units, or ones, (5 ื 100), 2
tens (2 ื 101), and 1 hundred (1 ื 102). In other words, going from right to
left, each column increases by a factor of ten. To extend this system to
represent fractions, a dot, the decimal point, is placed after the units and the
process is continued in the other direction. Thus, moving right from the decimal
point, each column decreases by a factor of ten. The first column after the
decimal point represents tenths (1/10 = 10-1), the second column, hundredths
(1/100 = 10-2), the third column, thousandths (1/1000 = 10-3), and so on. Thus,
0.1 represents 1/10.
The Decimal System
In the number in the above diagram, the 1 represents 1/10, the 5 represents 5/100, and the 6 represents 6/1000. 1/10 is the same as 100/1000, and 5/100 is the same as 50/1000, so 0.156 is the same as 100/1000 + 50/1000 + 6/1000 = 156/1000. Hence, the number in the diagram is
A finite or terminating decimal is a decimal with a limited number of digits after the decimal point. Short finite decimals, that is, decimals with only a few digits after the decimal point, can easily be converted to a vulgar fraction, as we saw in the example above. Some further examples are given by 0.46 = 46/100 and 0.2573 = 2573/10,000. These fractions may be able to be simplified by canceling the numerator and the denominator by a common factor. For example, 46/100 can be simplified to 23/50, but 2573/10,000 cannot be simplified.
Not all fractions can be represented as a finite decimal. For example, 1/3 = 0.3333 , where the dots indicate that the 3's continue for ever. This type of decimal, which has an unlimited number of digits after the decimal point (not necessarily all the same), is called an infinite decimal. An infinite decimal can either be recurring, that is, consist of a series of digits repeated endlessly, or nonrecurring , that is, with no repeating pattern to the sequence of digits. Recurring decimals are always rational numbers. For example, 2/7=0.285714 where the dots over the 2 and 4 indicate that 285714 will be repeated endlessly. Nonrecurring infinite decimals are always irrational numbers. For example, p = 3.14159265 is an irrational number. The decimal representation of p continues for ever but with no repeated pattern to the series of digits.
A recurring or periodic decimal has the same series of digits repeated over
and over again. For example, 0.326326326
, where the dots indicate that 326
keeps on repeating forever, is a recurring decimal. All recurring decimals are
infinite decimals. In other words, they have an unlimited number of digits after
the decimal point.
The period of a recurring decimal is given by the number of digits in the block
that is repeated. For example, 0.326326326
has a period of 3 because the
repeating block, 326, consists of 3 digits.
Recurring decimals are always rational numbers. For example, 1/13=0.076923076923.
We use dots, either over the first and last digits in the repeating block, or over every digit in the repeating block, to indicate that the decimal recurs. For example, we write 0.3 for the decimal representation of 1/3 = 0.3333 . Similarly 0.4285 would indicate the recurring decimal 0.428542854285 .
When we convert 1/7 to a decimal we get the recurring decimal 0.142857142857
.
The repeating block of this decimal, 142,857, is called a cyclic number because,
when it is multiplied by 1, 2, 3, 4, 5, or 6, the product contains the same six
digits:
142,857 ื 1 = 142,857
142,857 ื 2 = 285,714
142,857 ื 3 = 428,571
142,857 ื 4 = 571,428
142,857 ื 5 = 714,285
142,857 ื 6 = 857,142
Not only do the products contain the same six digits, but we can see that the
digits retain their order; if we imagine the six digits written around a ring,
as in the following diagram, then cutting the ring between each pair of digits
and then reading off the numbers clockwise, starting from the cut, will give the
six products.
Finding the Products of Cyclic Numbers
Cutting the ring here would give the product 142,857 ื 3 = 428,571.
When 142,857 is multiplied by 7 (that is, when it is multiplied by the prime
number used to generate it), the product is a string of nines: 142,857 ื 7 = 999,999
Similar strings of 9s are generated when any cyclic number is multiplied by the
prime that generates it.
When 142,857 is multiplied by a number greater than 7, the product has more than
six digits, and so cannot be a simple permutation of the original digits.
However, whatever the whole number that 142,857 is multiplied by, and however
many digits the product has, the product can be reduced to one of the first
seven multiples of 142,857. Starting from the units column, the product is split
into groups of six digits, with a group of five or fewer digits left over at the
end. These groups of digits are added together as numbers. If the sum has more
than 6 digits, then the splitting and summing process is repeated, until a
six-digit number is obtained. This number will be one of the first seven
multiples of 142,857. For example:
142,857 ื 987,654,321 = 141,093,333,335,097
Splitting the product into groups of six digits and adding, we obtain:
335,097 + 093,333 + 141 = 428571
All cyclic numbers are the repeating blocks of recurring decimals, formed when
vulgar fractions with certain prime numbers as their denominators are converted
to decimals. Cyclic numbers always contain one less digit than the prime number
used to generate them. The cyclic number generated by 7 has 6 digits, that
generated by 17 has 16 digits, that generated by 19 has 18 digits, and so on.
1/17 = 0.0588235294117647
, and has the 16-digit repeating block
0588235294117647, which is a cyclic number. When multiplied by any number from 1
to 16, the products contain the same digits in cyclic permutations. Nine prime
numbers less than 100 generate cyclic numbers: 7, 17, 19, 23, 29, 47, 59, 61,
and 97. There are likely to be an infinite number of cyclic numbers, although
this has never been proved.
The binary number system is a place-value system in base 2 that employs the
two symbols 0 and 1. Reading from right to left, the columns have values 1 (=20), 2
(=21), 4 (=22), 8 (=23), and so on. For example, 1011 in binary is
(1ื1)+(1ื2)+(0ื4)+(1ื8)=11 in decimal. Digital computers encode data and
instructions in binary, as a stream of electrical pulses where 1 =
"on" and 0 = "off".
Binary numbers are used in binary code - a computer language.
A base system is a place-value system where the values of the places are not
arbitrary, but determined by a very simple rule. A place-value system has a
finite number of symbols or digits that change their meaning according to their
position in a sequence of other digits. The most widely used place-value system
today is the decimal system. This uses the familiar Arabic numerals: 0, 1, 2, 3,
4, 5, 6, 7, 8, and 9. Any number is represented by a string of these digits, for
example, 3401. The value of the number is calculated by adding up the values of
each digit in the string. The value of each digit in the string is calculated by
multiplying the digit itself by the value of its position or place in the
string. Each place has a different value: in 3401, the 3 is in the thousands (103) position, the 4 is in the hundreds
(102) position, the zero is in the tens (101) position, and the 1 is in the units (or ones,
100) position. Thus 3401
represents three lots of one thousand, plus four lots of one hundred, plus one
lot of one. In general, each time we move one place to the left the value of the
position increases by a factor of ten; hence this is a base ten system.
In base B, where B is any number, the right-hand place is worth 1 (B0), the
first place to the left of this is worth B (B1), the second place to the left is
worth B squared (B2), the third place to the left is worth B cubed (B3), the
fourth is worth B to the power of four (B4), and so on. So in base ten, the
string of digits 600,349 means 6 lots of ten to the power of five, plus 3 lots
of ten squared, plus 4 lots of ten, plus 9 units (ones). In base three, the
string of digits 2021 stands for 2 lots of three cubed, plus 2 lots of three,
plus 1. In base ten this would be 54 (2 ื 33), plus 6, plus 1, or 61. We ensure
that every number has a unique representation in a given base B by insisting
that no digits with a value greater than or equal to B are used. Thus in base
three, for instance, the only digits are 0, 1, and 2. This results in the number
nine being written as 100 in base three, and never as 30 or as 9.
Base systems of counting have a long history: the Babylonians used base 60 for
larger numbers, while the Mayans and Aztecs used base 20.
Base ten, or decimal, is the most widely used system today. Other commonly used systems are bases two and sixteen, which are called the binary and hexadecimal systems, respectively. These systems are used in computer programming for designating different parts of a computer's memory. In hexadecimal we need extra digits: A stands for ten, B for eleven, and so on, up to F, which stands for fifteen. In binary we only have two digits, 0 and 1. This can make some "small" numbers rather long. For example, 42 in decimal is written as 101001 in binary. Hexadecimal is used to make such numbers more concise and easier to read in a computer program. The decimal number 42 would be written as 2A (2 ื 161 + 10 ื 1) in the hexadecimal system. This is much more concise than the binary representation, 101001. The other advantage of the hexadecimal system is that translation to the binary system (and from binary to hexadecimal) is fairly simple: 10 (1 ื 161) in hexadecimal is 1000 (1 ื 24) in binary because sixteen is two to the power of four.
Calculations in base systems are generally easier compared to those in non-base systems. It is difficult to do even fairly simple additions in a non-base number system such as Roman numerals. We can use the same simple methods of arithmetic that we use for base ten for any other base. Fundamental to this is the idea of carrying. For example, to add 58 to 74 in base ten, we first add the units: 4 plus 8 equals 12. We then write the units part of the result, 2, in the units column of the sum. We also carry the tens part, 1, across to the tens column of the sum, and add it to the 5 and 7 already there (see Diagram 1).
Diagram 1
Then 5 plus 7 plus the carried 1 equals 13, so we write down the 3 and carry the 1, and so on (see Diagram 2).
Diagram 2
The key here is that the "and so on" really means "and we can do
the same thing no matter what column of the sum we are adding the digits
in". This is because no matter what column we are working in, we can always
write the sum of the digits in the base we are using, keep the units, and carry
the rest in the familiar way.
We use the same idea in other bases. Suppose we are working in base three, and
want to add 221 to 122. The units column of the sum is 2 plus 1, equalling 10 in
base three, so we write 0 in the units column of the result and carry the 1 (see
Diagram 3).
Diagram 3
The next column is then 2 plus 2 plus the carried 1, which is five, written 12 in base three, so we write the 2 in the threes (31) column of the result and carry the 1 (see Diagram 4).
Diagram 4
Finally we add the carried 1 to the 2 and 1 in the nines column of the sum, to give 11 in base three, so we write that down in the sum (see Diagram 5).
Diagram 5
So the answer, in base three, is 1120. We can see that this is correct by translating 1120, 221 and 122 into base ten:
221 = 2 ื 32 + 2 ื 31 + 1 ื 30
= 2 ื 9 + 2 ื 3 + 1 ื 1
= 18 + 6 + 1 = 25
122 = 1 ื 32 + 2 ื 31 + 2 ื 30
= 1 ื 9 + 2 ื 3 + 2 ื 1
= 9 + 6 + 2 = 17
1120 = 1 ื 23 + 1 ื 32 + 2 ื 31 + 0 ื 30
= 1 ื 27 + 1 ื 9 + 2 ื 3
= 27 + 9 + 6
= 42
= 25 + 17
Other basic calculations, such as subtraction, multiplication, and division, can
also be carried out in other bases in the same way as they can in base ten.
The similarity of calculations in different bases is further illustrated by
considering the abacus. Addition on an abacus is mechanical, and carrying is
easy: when we run out of beads we move to another wire. An abacus is a good
example of using a place-value representation for numbers. If the number of
beads is the same on each wire, then the place-value system is also a base
system. Knowing how to use a base ten abacus (nine beads on each wire), we could
learn to do sums on a base seven abacus (six beads on each wire). Mental
calculation in base seven would be more difficult, though this is largely a
matter of practice - our whole way of learning and thinking about numbers and
counting is so deeply founded on base ten that it is quite difficult to work in
other bases without carefully following mechanical procedures.
We can also work in bases other than ten with very small numbers, in particular for fractions. A decimal fraction is a fraction in base ten. The notation is an extension of the place-value system. To the right of the decimal point, the first place is worth a tenth (1/10 = 1/101), the second, a hundredth (1/100 = 1/102), the third, a thousandth (1/1000 = 1/103), and so on. In base B, the nth place of a fraction is worth 1/Bn. This is equal to B-n. For instance, the binary fraction 0.110101 is the same as writing
Of course we can combine fractions in any base with whole numbers: in base three, the number 22.121 is the same as writing
Magic squares are sets of numbers arranged in the form of a square in such a manner that the sum of the numbers in each row, column, and diagonal is the same. Diagram 1 shows two examples of magic squares. The first has row, column, and diagonal sums of 15, and the second has row, column, and diagonal sums of 65.
Diagram 1: Examples of Magic Squares
Magic squares have interested mathematicians for over 2,000 years. They have
been intimately associated with the numerologies and mythologies of various
cultures for most of that time. For example, magic squares used to be thought to
protect against plague.
According to Chinese legend, in the 29th century BC the Emperor Fu-hsi found
some footprints in the sand. These were later identified as those of the
"heavenly dragon horse". In the 21st century BC the Emperor Yu met a
"divine tortoise" with strange markings on its back. Both the
footprints and the markings on the tortoise's back were interpreted to be divine
messages to the rulers. Centuries later they were converted to magic squares
that had great spiritual significance. They were used in rituals and for finding
horoscopes for hundreds of years.
If we add two magic squares together by adding the numbers in corresponding positions we will always get another magic square. The new magic square will have row, column, and diagonal totals equal to the sum of the row, column, and diagonal totals of the two magic squares that were added. We add magic squares in the same way that we add matrices. For example, suppose we want to add together the two magic squares in Diagram 2.
Diagram 2
We start by adding the numbers in the top left corners of the magic squares together and placing the sum in the top left corner of the new square. Thus, 9 + 17 = 26. We then repeat this process for the second number in the top row (2 + 6 = 8), then for the third number (25 + 5 = 30), and so on. This gives us the magic square in Diagram 3.
Diagram 3
The row, column, and diagonal totals of both of the two magic squares we added are 65. The row, column, and diagonal totals of the magic square that is the sum are 130 (65 + 65).