The counting numbers are 1, 2, 3, 4, and so on. Natural numbers are the
same as counting numbers. These are just two different names for the same
numbers. Note that zero is not a natural or a counting number.
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The integer numbers are the whole numbers and their additive opposites,
the negative version of the numbers. We can list the integers as
..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
where the elispsis (...) indicates that
the pattern continues. Note that -5 is the additive opposite (or additive inverse) of 5
because -5 + 5 is 0.
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There are two, equivalent, ways of describing rational numbers. We can say that
rational numbers are all numbers that can be expressed as the quotient of two integers, p and q,
where q is not zero. Using this definition, 2/3, 19/7, -8/5, and -323/45 are all rational numbers.
So is 5 (which can be expressed as 5/1) and so is 0 (which can be expressed as 0/7). In fact,
for any rational number there will be an inifinte number of ways of expressing it. We have
4/5, 8/10, 32/40, and -48/-60 as four different names for the same rational number.
Another definition of rational number is to say that it is a terminating or a repeating
decimal. Thus, 1.02 and -8.345921 as "terminating decimals",
and 0.3333..., and -4.27272727... as "repeating decimals" are all rational numbers.
And, on this definition, so is 4 (which can be expressed as 4.0).
The definitions are equivalent because any fraction of integers can
be "divided out" to produce either a terminating or a repeating decimal, and
any terminating or repeating decimal can be converted into a fraction of integers.
Using the examples above
fraction
decimal
2/3
0.66666...
19/7
2.7142857142857...
-8/5
-1.6
-323/45
-7.17777...
102/100 or 51/50
1.02
-8345921/1000000
-8.345921
1/3
0.3333...
-423/99 or -47/11
-4.27272727...
Rational numbers have a feature that is not found in whole numbers, counting
numbers and integers. That feature is that between any two rational numbers
there is always another rational number.
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Irrational numbers are decimal numbers that are neither terminating nor repeating.
For example, we know that the number that we call "pi" is an irrational number.
We can "approximate it" by such values as 22/7 or 3.14 or 3.1415926, but these are
approximations. The real value of "pi" is a decimal value that does not form a
repeating pattern. Therefore, "pi" is an irrational number. Another irrational number
is the square root of 2, that is, the number that when multiplied by itself gives the
product 2. We can approximate the square root of two as 1.414 or as 1.4142135, but,
again, the true answer is a decimal value that does not form a repeating pattern.
Another example of an irrational number is the following
3.010010001000010000010000001 ...
Here there is a pattern, but it is not a pattern of a repeating sequence of digits.
It may seem like there are only a few irrational numbers. After all, it is easy to think of
the multitude of rational numbers represented by the fraction p/q where p and q are integers (q not zero).
There is no similarly easy way to think of the irrational numbers. However,
as mathematicians count things, there are even more irrational numbers than there are
rational numbers. As a hint of this, consider that "pi" is an irrational number, but so is "pi"
plus any non-zero rational number, and so is "pi" times any rational number other than zero, and so is
"pi" times any rational number (other than zero or 1) plus any non-zero rational number. All of
those new irrational numbers are different, and all of them came from using just the one
irrational number "pi" in combination with rathional numbers and the operations of addition and
multiplication.
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The real numbers are all decimal numbers.
The real numbers are a combination (a union) of the rational and the irrational numbers.
When we talk about a number line, we are talking about a correspondence between the
real numbers and the points on the line. For every point there is a real number and for
every real number there is a point. In theory, on a real number line we could identify all
of the points associated with rational numbers. This would leave holes at all of the
points associated with irrational numbers. Similarly, we could identify all of the points
associated with irrational numbrs, and that would leave holes at all of the
points associated with rational numbers. But if we identify all of the points
associated with real numbers then there are no holes.
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