Virtually everyone is familiar with numbers as those 'counting things',
which answer the question "How many..?". The simple counting numbers
1, 2, 3, 4, 5, ..., are consecutively associated with things in a
collection to be counted, such as some apples.
1
2
3
4
5
Most people are also familiar with those numbers representing 'divisions
of...' such as halves or thirds - i.e. fractions. So if five apples
had to be divided evenly between two people, we know that they would
get two and a half apples each. Further, some people recognise that our
number system is decimal (based on our ten counting fingers), and that
fractional quantites can also be expressed decimally, e.g. two and a
half is 2.5 (two point five, 'point five' meaning 'five tenths').
In this article, we'll briefly review the simple (basic) number concepts
and then overview the other kinds of number.
A number system is
any of various sets of symbols and the rules for using them to express
quantities as the basis for counting, comparing amounts,
performing calculations, determining order, making measurements,
representing value, setting limits, abstracting quantities,
coding information, and transmitting data.
The most elementary representation of numbers is the tally or unitary
system of notation.
It originally consisted of single strokes that were put into one-to-one
correspondence with the items being counted;
later these were combined into groups of five or more.
Mathematicians now recognise many different kinds of numbers.
Natural Numbers
The simplest number system is the natural numbers
(0, 1, 2, 3,... ; some mathematicians exclude 0)
which is used for counting.
The natural numbers are also called the whole numbers,
positive integers, or positive rational integers.
The positive integers are generated by adding 1 to each number in an
unending series, so that each number in the sequence is one more than
its immediate predecessor. Different civilizations throughout history
have developed different kinds of number systems. One of the most
common is the one used in all modern cultures, in which objects are
counted in groups of ten.
In the base 10 system,
integers are represented by digits expressing various powers of 10.
For example, take the number 2497.
Every digit in this number has its own place value,
and the place values increase by another power of 10 as they move to the left.
The first place value is a unit value (here, 7);
the second place value is 10 (here, 9 x 10, or 90);
the third place value is 10 x 10, or 100 (here, 4 x 100, or 400);
and the fourth place value is 10 x 10 x 10, or 1000 (here, 2 x 1000, or 2000).
The incorporation of negative numbers (-1, -2, -3,... )
leads to the integers.
The set of natural numbers is closed under addition and
multiplication:
the sum and product of two natural numbers are always natural numbers.
This means that when you add (or multiply) two natural
numbers, you always get another natural number.
However, it is not closed under division or subtraction.
For example,
if you divide 3 by 4, you get 0.75, which is not a natural number.
Rational Numbers
The quotient (the result of dividing) of two natural numbers,
is not always a natural number,
so it is convenient to introduce the positive fractions to represent
the quotient of any two natural numbers.
The system that includes all fractions p/q in which p and q are
integers and q is not zero comprises the rational numbers.
That is,
any number which can be represented as a fraction
(in other words, as a ratio of integers) is called a rational number.
Rational numbers can also be written as decimals.
These decimals are either finite, or infinite and repetitive.
The integers are really forms of fractions (1/1, 2/1, 3/1, etc).
The natural number n is identified with the fraction n/1.
Rational numbers can be positive or negative.
Because the difference of two positive fractions is not always a
positive fraction,
it is convenient to introduce the negative fractions
(including the negative integers) and the number zero (0).
Since every fraction (ratio of integers) is a rational number and
every rational number can be written as a fraction,
the terms "fraction" and "rational number" are often used synonymously.
Rational numbers are closed under addition, subtraction, multiplication,
and division.
That is, the sum, difference, product, or quotient
of two rational numbers is always a rational number.
Every rational number can be represented as a repeating or periodic
decimal—
that is, as a number in the decimal notation,
which after a certain point consists of the infinite repetition of a
finite block of digits.
Conversely, every repeating decimal represents a rational number.
Thus, 617/50 = 12.34000 ..., and 2317/990 = 2.34040 ....
The first expression is usually written as 12.34,
omitting the infinite repetition of the block consisting of the single
digit 0.
The first type of expression,
where the infinitely repeating block consists of the digit 0,
is called a finite or terminating decimal,
and the second type of expression is called a repeating decimal.
Irrational Numbers
The ancient Greeks discovered that some naturally occurring geometric
lengths, such as the diagonal of a unit square,
cannot be expressed as rational numbers.
This discovery led eventually to the notion of irrational numbers
(numbers that cannot be expressed as a ratio of integers).
Any number that cannot be expressed as a fraction a/b is called an
irrational number.
The development of geometry revealed the need for more types of
numbers;
the length of the diagonal of a square with sides one unit long cannot
be expressed as a rational number.
Similarly,
the ratio of the circumference to the diameter of a circle is not a
rational number.
These and other needs led to the introduction of the irrational numbers.
Some examples of irrational numbers are:
square roots of whole numbers that aren�t perfect squares;
for example,
decimal numbers that don't repeat or terminate.
e = 2.718281828459... and π = 3.1415926535...
are irrational numbers,
and their decimal expansions are necessarily nonterminating and
nonperiodic.
Some irrational numbers can be expressed as square roots;
they can be also written as decimals but the decimal is always
infinite and never repeats.
The word "irrational" suggests that these numbers were "wrong" in some
way. In fact, many early mathematicians like Pythagoras were
unwilling to accept that such numbers could exist.
Nowadays, irrational numbers are accepted as perfectly "proper."
Real Numbers
The totality of the rational and irrational numbers makes up the real
number system; a real number is
any rational or irrational number or
any number which can be written as a decimal.
Real numbers can most
simply be defined as numbers whose decimal expansion may not terminate,
for example, the number π,
whose decimal expansion 3.14159... goes on forever.
The set of real numbers includes everything mentioned so far,
including rational numbers, integers, and whole numbers.
The set of real numbers is closed under addition, subtraction,
multiplication, and division.
There is sometimes more than one way to write a decimal number.
For example, 1.0 and 0.99999� are the same number.
Transcendental Numbers
There are irrational numbers which have decimals that are infinite and
non repeating but cannot be written as roots.
The best example of this type of number is π
(pronounced pi and used extensively in trigonometry)
which has a value beginning with 3.14159... but cannot
be written either as a fraction or a root.
Complex Numbers
What is the square root of 4? In other words, what
number when multiplied by itself gives 4. The answer is, of course, 2
because 2 � 2 = 4. But if you remember your multiplication,
-2 � -2 is also 4 (because two negatives make a positive).
So we can say that 4 has two square roots, +2 and -2.
Every positive number has two square roots.
Every time you square a number you end up with a positive number.
So, with that fact in mind:
What is the square root of -1?
No real number when multiplied by itself gives -1!
The equation x2 = -1 has no solutions in the real number system.
If such a solution is desired, new numbers must be invented.
The number that has been invented to be the square root of -1 is
called i (for imaginary).
In fact it is no more imaginary than any
other number but the name has stuck.
Let us have a look at some of
the properties of this strange number.
i2 = i � i = -1
i3 = i � i � i = i � (-1) = -i
i4 = i2 � i2 = -1� -1 = 1
i5 = i � i4 = i � 1 = i
It is possible to have a combination of real and imaginary numbers.
Here are some examples.
2 + i
-3 + 2i
1 - 5i
All numbers of the form a + bi,
in which a and b are real numbers,
belong to the complex number system.
If b is not 0,
the complex number is called an imaginary number;
if b is not 0 but a is 0,
the complex number is called a pure imaginary number;
if b_ is 0, the complex number is a real number.
Imaginary numbers are extremely useful in the theory of alternating
currents and many other branches of physics and natural science.
Whereas real numbers represent points on a line,
complex numbers can be placed in correspondence with the points on a plane.
To represent the complex number a + bi geometrically,
the x-axis is used as the axis of the real number a,
and the y-axis serves as the axis of the pure imaginary bi;
the complex number, therefore,
corresponds to the point P with the rectangular coordinates a and b.
The line
joining the origin with the point P is the diagonal of a rectangle
with the sides a and bi.
If the complex number a + bi is multiplied by -1,
the vector OP is rotated through 180�,
and the point P falls in the third quadrant;
a rotation of 90� is achieved by multiplying the complex number by
i.
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