1,2,3,4,5,6,7,...
With them we can count: books, friends, money, etc. If we adjoin their
negatives and zero, we obtain the integers:
...,-3,-2,-1,0,1,2,3,...
When we try to measure length, weight, or voltage, the integers are
inadequate. They are spaced too far apart to give sufficient precision. We
are led to consider quotients (ratios) of integers, numbers such as
1/2, -7/8, 21/5, 19/-2, 16/2, and -15/1
Note that 16/2 and -15/1 was included though they would normally be
written as 8 and -15 respectively, since they are equal to the latter by
the ordinary meaning of division. The quotient 5/0 or -9/0 was not
included because it is impossible to make sense out of these symbols.
Imagine trying to divide something into nothing, and you'll see why.
Arithmetic Operations: Given two real numbers x and y, we may add or
multiply them to obtain two new real numbers x+y and x*y (also written
simply as xy). Addition and multiplication have the following familiar
properties, called field properties:
1. Commutative laws. x+y = y+x and xy = yx
2. Associative laws. x+(y+z) = (x+y)+z and x(yz) = (xy)z
3. Distributive law. x(y+z) = xy + xz
4. Identity elements. There are two distinct numbers 0 and 1 satisfying
x+0 = x and x*1 = x
5. Inverses. Each number x has an additive inverse (also called a
negative), -x, satisfying x+(-x) = 0. Also, each number x except 0 has a
multiplicative inverse (also called a reciprocal), x^-1, satisfying x*x^-1
= 1. (Note: the ^ character will be used to denote powers. So x^3 =
x*x*x)
x-y = x+(-y)
and
x/y = x * y^-1
Most of algebra rests on these five field properties and the definition of
subtraction and division.
** Calculus, Dale Verberg, Edwin J. Purcell, 7'th edition
*** Webster's Encyclopedic Unabridged Dictionary, Gramercy Books
in.fin.i.ty (in fin`i te), n., pl. -ties. 1. the quality or
state of being infinite. 2. something that is infinite. 3.
infinite space, time, or quantity. 4. an infinite extent, amount,
or number. 5. an indefinitely great amount or number. 6.
a. the assumed limit of a sequence, series, etc., that increases without
bound. b. infinite distance or an infinitely distant part of space.***
in.fi.nite (in`fe nit), adj. 1. immeasurably great: an
infinite capacity for forgiveness. 2. indefinitely or
exceedingly great: infinite sums of money. 3. unlimited or
unmeasurable in extent of space, duration of time, etc. 4.
unbounded or unlimited; boundless; endless: God's infinite mercy.
5. can be put into one-to-one correspondence with a subset that is
not the given set. --n. 6. something that is infinite. 7.
Math. an infinite quantity or magnitude. 8. the boundless regions
of space.***
Infinity is boundless. An immeasurably large sequence or series. A
function whose domain consists of just the positive integers (or some
other subset of the integers) is called a sequence. A sequence of
natural numbers, for example, is the simplest sequence of all:
Numbers that can be written in the form m/n, where m and n are integers
with n greater or less than 0, are called rational numbers. Do the
rational numbers serve to measure all lengths? No. This surprising fact
was discovered by the ancient Greeks. They showed that while the square
root of 2 (sqrt(2)) measures the hypotenuse of a right triangle with legs
of length 1, it cannot be written as a quotient of two integers. Thus,
sqrt(2) is an irrational (not rational) number. So are sqrt(3), sqrt(5),
sqrt(7), pi, and a host of other numbers.**
Real numberse are the set of all numbers (rational and irrational) that
can measure lengths, together with their negatives and zero. The real
numbers may be viewed as labels for points along a horizontal line. There
they measure the distance to the right or left (the directed distance)
from a fixed point called the origin, labeled 0:
<---6---5---4---3---2---1---0---1---2---3---4---5---6--->
Although all possible labels cannot be shown, each point does have a
unique real number label. This number is called the coordinate of the
point. And the resulting coordinate line is referred to as the real line.
You may also already know that the number system can be enlarged still
more--to the complex numbers. These are numbers of the form a+(b
sqrt(-1)), where a and b are real numbers.**
Subtraction and division are defined by:
Sources:
maddox@xmission.com