Numbers have lots of different names!
Even though
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$\;4\;$ and
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$\;\frac{8}{2}\;$ and
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$\;5-1\;$ and
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$\;4.00\;$ and
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$\;2^2\;$ all look different,
they are all representations for the same number.
Although numbers come in lots of different sizes, and have lots of different names,
here's the good news:
all these numbers live on the ‘line’ shown below.
This is called a real number line,
and is the subject of this section.
A real number line is determined by three pieces of information:
Sometimes, arrows are put at both ends, to suggest that the line extends forever in both directions.
Sometimes, there are no arrows at all: this is the simplest representation, and is the one that will be used most often on this website.
With choices made for
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$\;0\;$ and
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$\;1\;$, the locations
of all other numbers are uniquely
determined.
For example, the locations of these three numbers are shown below:
Even though two different numbers are required to determine where all the other numbers live, people occasionally get lazy.
If there's only a single number that is currently of interest,
then a ‘number line’
may be drawn showing only that particular number.
For example, all numbers to the right
of
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$\;2\;$ might be illustrated like this:
(The hollow dot at [beautiful math coming... please be patient] $\;2\;$ indicates that [beautiful math coming... please be patient] $\;2\;$ is not to be included.)
A number line provides us with a picture of a collection of numbers
referred to as the real numbers.
It is a conceptually perfect picture, in the following sense:
The symbol [beautiful math coming... please be patient] $\;\mathbb{R}\;$ (‘blackboard bold R’) is used to denote the set of real numbers.
The numbers to the right of zero are called the positive real numbers;
the numbers to the left of zero are the negative real numbers.
The number zero is not positive (since it doesn't lie to the right
of zero), and not negative (since it doesn't lie to the left of
zero).
Zero is the only real number with this ‘neutral’ status;
every other real number is either positive or negative.
Which real numbers are not negative?
Zero isn't negative.
Also, the positive numbers are not negative.
These numbers—zero, together with all real numbers to the
right of zero—are called the nonnegative real numbers,
and are shaded below.
The solid (filled-in)
dot at
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$\;0\;$ indicates that
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$\;0\;$ is being included; the arrow to the
right indicates that the shading is to continue for all numbers to
the right of zero.
A nonzero real number is one that is not zero; the nonzero real numbers are shaded below.
There are some important subcollections of the real numbers that are given special names.
The whole numbers are the subcollection containing:
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$0$, $1$, $2$, $3$, $\dots$
The three lower dots ‘
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$\;\ldots\;$’ indicate that the established pattern is
to be repeated ad infinitum
(pronounced odd in-fi-NIGHT-um or
add in-fi-NIGHT-um; means forever and ever).
Thus, [beautiful math coming... please be patient] $\;127\;$ is a whole number, but [beautiful math coming... please be patient] $\;\frac12\;$ isn't.
Be careful! Numbers have lots of different names.
Either a number is a whole number, or it isn't.
The particular name being used doesn't matter.
For example, the number
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$\;3\;$ is a whole number.
The number
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$\;3\;$ has many names, like
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$\;\frac62\;$ and
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$\;2.9+0.1\;$.
So,
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$\;\frac62\;$ is a whole number and
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$\;2.9+0.1\;$ is a whole number.
Don't let the name being used lead you astray!
Consecutive whole numbers are whole numbers that follow one
after the other, without gaps.
The phrase can refer to just two numbers, or more than two.
Thus,
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$\;2\;$ and $\;3\;$ are consecutive whole numbers;
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$\;5\;$, $\;6\;$, and $\;7\;$ are consecutive whole numbers;
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$\;2\;$ and $\;5\;$ are not consecutive whole numbers.
Notice how sparse the whole numbers are, as they sit in the collection of real numbers!
Between any two consecutive whole numbers are an infinite (IN-fi-nit) number of real numbers,
that are NOT whole numbers.
There are two different concepts frequently used to compare numbers:
The size of real numbers is discussed next.
Order will be discussed in the future section
I Live Two Blocks West Of You.
Roughly, a number is big or large if it is far from zero:
Note that big/large numbers can be positive (like [beautiful math coming... please be patient] $1{,}000{,}000$; one million) or negative (like [beautiful math coming... please be patient] $-1{,}000{,}000$; negative one million).
Roughly, a number is small if it is close to zero:
Note that small numbers can be positive (like
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$\frac{1}{1000}$; one thousandth)
or negative (like
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$-\frac{1}{1000}$; negative one thousandth).
Usually, ‘small’ means close to zero, but not equal to zero.
Numbers like
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$\;2\;$ and $\;-2\;$ are called opposites:
they have the same distance from zero, but are on opposite sides of zero.
The opposite of a positive number is a negative number.
The opposite of a negative number is a positive number.
The opposite of zero is zero: zero is the only real number that is its own opposite.
Whenever you add a number to its opposite, you get zero as a result:
[beautiful math coming... please be patient] $$3 + (-3) = 0$$ [beautiful math coming... please be patient] $$(-2) + 2 = 0$$ [beautiful math coming... please be patient] $$5.1 + (-5.1) = 0$$ [beautiful math coming... please be patient] $$0 + 0 = 0$$
The number
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$\;-3\;$ can be read as either negative three
or the opposite of three.
Many people favor negative three, because it's faster.
However, both ways are correct.
Try not to read it as minus three,
because the word minus is reserved for the operation of subtraction.
Similarly,
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$\;-x\;$ can be read as either negative ex
or the opposite of ex.
Here, the letter
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$\;x\;$ is being used to represent a number:
such use of letters to represent numbers is
discussed in the section Holding This, Holding That.
We'll see in this future section that it is preferable for beginning
students of mathematics to read
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$\;-x\;$
as the opposite of
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$\;x\;$.
When we take the whole numbers, and throw in their opposites,
then we get the important subcollection of the real numbers that is called the integers (IN-teh-jers).
Thus, the integers are the subcollection:
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$$\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$$
The symbol [beautiful math coming... please be patient] $\;\mathbb{Z}\;$ is used to denote the set of integers.
One important property of the real numbers is that they are dense;
that is, between every two different real numbers (no matter how close they are),
there is another real number.
This is often called the density property of the real numbers.
Indeed, between every two different real numbers,
there are an infinite number of real numbers!
If two numbers live at the same place on a real number line,
then we say that they are equal.
And, if two numbers are equal,
this means that they live at the same place on a real number line.
The mathematical sentence
‘$\;a = b\;$’ is read as ‘$\,a\,$ equals $\,b\,$’
or ‘$\,a\,$ is equal to $\,b\,$’.
This sentence is true if
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$\,a\,$ and $\,b\,$ live at the same place on a real number
line; otherwise, it's false.
Note that if the sentence ‘$\;a = b\;$’ is true,
then you're being told that
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$\,a\,$ and $\,b\,$ are just different
names for the same number!
Throughout this web site, when the word number is used, it means real number.
Question | Answer |
Is [beautiful math coming... please be patient] $\,-3\,$ a real number? | yes |
Is [beautiful math coming... please be patient] $\,-3\,$ an integer? | yes |
Is [beautiful math coming... please be patient] $\,-3\,$ a whole number? | no |
Is [beautiful math coming... please be patient] $\,-3\,$ a positive number? | no |
Is [beautiful math coming... please be patient] $\,-3\,$ a negative number? | yes |
Is [beautiful math coming... please be patient] $\,-3\,$ a nonzero number? | yes |
Is [beautiful math coming... please be patient] $\,-3\,$ a nonnegative number? | no |
Is [beautiful math coming... please be patient] $\,-3\,$ a nonpositive number? | yes |