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
Natural Numbers together with “zero”
0, 1, 2, 3, 4, 5, . . .
What is zero? Is it a number?
How can the number of nothing be a number? Is zero nothing, or is it
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
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, . . .
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 “–.”
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
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.
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
- 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
- Cannot be expressed as
a ratio of integers.
- As decimals they never
repeat or terminate (rationals always do one or the other)
Irrational (never repeats or terminates)
Irrational (never repeats or terminates)
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.
Imagine a line segment exactly one unit long:
Now draw another line one unit long,
perpendicular to the first one, like this:
Now draw the diagonal connecting the two ends:
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
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
- Rationals + Irrationals
- All points on the
- 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:
a is less than b, (expressed as a < b)
a is equal to b, (expressed as a = b)
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
- 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
- If a
number is positive (or zero), the absolute value function does nothing to
- 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
The correct result is