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The real number system evolved
over time by expanding the
notion of what we mean by
the word “number.” At first,
“number” meant something you
could count, like how many
sheep a farmer owns. These
are called the natural
numbers, or sometimes
the counting numbers.
or
“Counting Numbers”
1,
2, 3, 4, 5, . . .
- The
use of three dots at the
end of the list is a common
mathematical notation to
indicate that the list keeps
going forever.
At some point, the idea of
“zero” came to be considered
as a number. If the farmer
does not have any sheep, then
the number of sheep that the
farmer owns is zero. We call
the set of natural numbers
plus the number zero the whole
numbers.
Natural Numbers together
with “zero”
0,
1, 2, 3, 4, 5, . . .
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What
is zero? Is it a number?
How can the number of
nothing be a number?
Is zero nothing, or
is it something?
Well,
before this starts to
sound like a Zen koan,
let’s look at how we
use the numeral “0.”
Arab and Indian scholars
were the first to use
zero to develop the
place-value number system
that we use today. When
we write a number, we
use only the ten numerals
0, 1, 2, 3, 4, 5, 6,
7, 8, and 9. These numerals
can stand for ones,
tens, hundreds, or whatever
depending on their position
in the number. In order
for this to work, we
have to have a way to
mark an empty place
in a number, or the
place values won’t come
out right. This is what
the numeral “0” does.
Think of it as an empty
container, signifying
that that place is empty.
For example, the number
302 has 3 hundreds,
no tens, and 2 ones.
So
is zero a number? Well,
that is a matter of
definition, but in mathematics
we tend to call it a
duck if it acts like
a duck, or at least
if it’s behavior is
mostly duck-like. The
number zero obeys most
of the same rules of
arithmetic that ordinary
numbers do, so we call
it a number. It is a
rather special number,
though, because it doesn’t
quite obey all the same
laws as other numbers—you
can’t divide by zero,
for example.
Note
for math purists: In
the strict axiomatic
field development of
the real numbers, both
0 and 1 are singled
out for special treatment.
Zero is the additive
identity, because
adding zero to a number
does not change the
number. Similarly, 1
is the multiplicative
identity because
multiplying a number
by 1 does not change
it.
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Even more abstract than zero
is the idea of negative numbers.
If, in addition to not having
any sheep, the farmer owes
someone 3 sheep, you could
say that the number of sheep
that the farmer owns is negative
3. It took longer for the
idea of negative numbers to
be accepted, but eventually
they came to be seen as something
we could call “numbers.” The
expanded set of numbers that
we get by including negative
versions of the counting numbers
is called the integers.
Whole numbers plus negatives
.
. . –4, –3, –2, –1, 0, 1,
2, 3, 4, . . .
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How
can you have less than
zero? Well, do you have
a checking account?
Having less than zero
means that you have
to add some to it just
to get it up to zero.
And if you take more
out of it, it will be
even further less than
zero, meaning that you
will have to add even
more just to get it
up to zero.
The
strict mathematical
definition goes something
like this:
For
every real number n,
there exists its opposite,
denoted – n,
such that the sum of
n and – n
is zero, or
n + (–
n) = 0
Note
that the negative sign
in front of a number
is part of the symbol
for that number: The
symbol “–3” is one object—it
stands for “negative
three,” the name of
the number that is three
units less than zero.
The
number zero is its own
opposite, and zero is
considered to be neither
negative nor positive.
Read
the discussion of subtraction
for more about the meanings
of the symbol “–.”
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The next generalization that
we can make is to include
the idea of fractions. While
it is unlikely that a farmer
owns a fractional number of
sheep, many other things in
real life are measured in
fractions, like a half-cup
of sugar. If we add fractions
to the set of integers, we
get the set of rational
numbers.
All numbers of the form  ,
where a and b
are integers (but b
cannot be zero)
Rational numbers include
what we usually call fractions
- Notice
that the word “rational”
contains the word “ratio,”
which should remind you
of fractions.
The
bottom of the fraction is
called the denominator.
Think of it as the denomination—it
tells you what size fraction
we are talking about: fourths,
fifths, etc.
The
top of the fraction is called
the numerator.
It tells you how many
fourths, fifths, or whatever.
- RESTRICTION:
The denominator cannot be
zero! (But the numerator
can)
If
the numerator is zero, then
the whole fraction is just
equal to zero. If I have zero
thirds or zero fourths, than
I don’t have anything. However,
it makes no sense at all to
talk about a fraction measured
in “zeroths.”
- Fractions
can be numbers smaller than
1, like 1/2 or 3/4 (called
proper fractions), or they can be numbers bigger than 1 (called
improper fractions),
like two-and-a-half, which
we could also write as 5/2
All integers can also be thought
of as rational numbers, with
a denominator of 1:
This means that all the previous
sets of numbers (natural numbers,
whole numbers, and integers)
are subsets of the rational
numbers.
Now it might seem as though
the set of rational numbers
would cover every possible
case, but that is not so.
There are numbers that cannot
be expressed as a fraction,
and these numbers are called
irrational because
they are not rational.
- Cannot
be expressed as a ratio
of integers.
- As
decimals they never repeat
or terminate (rationals
always do one or the other)
Examples:
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Rational (terminates) |
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Rational (repeats) |
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Rational (repeats) |
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Rational (repeats) |
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Irrational (never repeats or terminates) |
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Irrational (never repeats or terminates) |
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It
might seem that the
rational numbers would
cover any possible number.
After all, if I measure
a length with a ruler,
it is going to come
out to some fraction—maybe
2 and 3/4 inches. Suppose
I then measure it with
more precision. I will
get something like 2
and 5/8 inches, or maybe
2 and 23/32 inches.
It seems that however
close I look it is going
to be some fraction.
However, this is not
always the case.
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Imagine
a line segment
exactly one unit
long:
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Now
draw another line
one unit long,
perpendicular
to the first one,
like this:
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Now
draw the diagonal
connecting the
two ends: |
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Congratulations!
You have just drawn
a length that cannot
be measured by any rational
number. According to
the Pythagorean Theorem,
the length of this diagonal
is the square root of
2; that is, the number
which when multiplied
by itself gives 2.
According
to my calculator,
But
my calculator only stops
at eleven decimal places
because it can hold
no more. This number
actually goes on forever
past the decimal point,
without the pattern
ever terminating or
repeating.
This
is because if the pattern
ever stopped or repeated,
you could write the
number as a fraction—and
it can be proven that
the square root of 2
can never be written
as
for
any choice of
integers for a
and b. The proof
of this was considered
quite shocking when
it was first demonstrated
by the followers of
Pythagoras 26 centuries
ago. |
- Rationals + Irrationals
- All
points on the number line
- Or
all possible distances on
the number line
When we put the irrational
numbers together with the
rational numbers, we finally
have the complete set of real
numbers. Any number that represents
an amount of something, such
as a weight, a volume, or
the distance between two points,
will always be a real number.
The following diagram illustrates
the relationships of the sets
that make up the real numbers.
The real numbers have the
property that they are ordered,
which means that given any
two different numbers we can
always say that one is greater
or less than the other. A
more formal way of saying
this is:
For any two real numbers
a and b, one
and only one of the following
three statements is true:
1.
a is less than b, (expressed as a < b)
2.
a is equal to b, (expressed as a = b)
3.
a is greater than b, (expressed as a > b)
The ordered nature of the
real numbers lets us arrange
them along a line (imagine
that the line is made up of
an infinite number of points
all packed so closely together
that they form a solid line).
The points are ordered so
that points to the right are
greater than points to the
left:
- Every
real number corresponds
to a distance on the number
line, starting at the center
(zero).
- Negative
numbers represent distances
to the left of zero, and
positive numbers are distances
to the right.
- The
arrows on the end indicate
that it keeps going forever
in both directions.
When we want to talk about
how “large” a number is without
regard as to whether it is
positive or negative, we use
the absolute value
function. The absolute value
of a number is the distance
from that number to the origin
(zero) on the number line.
That distance is always given
as a non-negative number.
In short:
- If
a number is positive (or
zero), the absolute value
function does nothing to
it:
- If
a number is negative, the
absolute value function
makes it positive:
WARNING:
If there is arithmetic to
do inside the absolute value
sign, you must do it before
taking the absolute value—the
absolute value function acts
on the result of whatever
is inside it. For example,
a common error is
(WRONG)
The correct result is
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