DECIDING IF A NUMBER IS A WHOLE NUMBER, INTEGER, etc.

Numbers have lots of different names!
Even though [beautiful math coming... please be patient] $\;4\;$ and [beautiful math coming... please be patient] $\;\frac{8}{2}\;$ and [beautiful math coming... please be patient] $\;5-1\;$ and [beautiful math coming... please be patient] $\;4.00\;$ and [beautiful math coming... please be patient] $\;2^2\;$ 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, an arrow is put at the right end of a number line, to show that this is the ‘positive’ direction;
that is, the numbers increase as you move to the right.

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 [beautiful math coming... please be patient] $\;0\;$ and [beautiful math coming... please be patient] $\;1\;$, 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 [beautiful math coming... please be patient] $\;2\;$ 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:

Since there is this ‘pairing’ of real numbers and points on the line, people tend to use the words ‘number’ and ‘point’ interchangeably.
So will this author.

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 [beautiful math coming... please be patient] $\;0\;$ indicates that [beautiful math coming... please be patient] $\;0\;$ 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:   [beautiful math coming... please be patient] $0$, $1$, $2$, $3$, $\dots$

The three lower dots ‘ [beautiful math coming... please be patient] $\;\ldots\;$’ 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 [beautiful math coming... please be patient] $\;3\;$ is a whole number.
The number [beautiful math coming... please be patient] $\;3\;$ has many names, like [beautiful math coming... please be patient] $\;\frac62\;$ and [beautiful math coming... please be patient] $\;2.9+0.1\;$.
So, [beautiful math coming... please be patient] $\;\frac62\;$ is a whole number and [beautiful math coming... please be patient] $\;2.9+0.1\;$ 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, [beautiful math coming... please be patient] $\;2\;$ and $\;3\;$ are consecutive whole numbers;
[beautiful math coming... please be patient] $\;5\;$, $\;6\;$, and $\;7\;$ are consecutive whole numbers;
[beautiful math coming... please be patient] $\;2\;$ 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 [beautiful math coming... please be patient] $\frac{1}{1000}$; one thousandth) or negative (like [beautiful math coming... please be patient] $-\frac{1}{1000}$; negative one thousandth).
Usually, ‘small’ means close to zero, but not equal to zero.

Numbers like [beautiful math coming... please be patient] $\;2\;$ 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 [beautiful math coming... please be patient] $\;-3\;$ 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, [beautiful math coming... please be patient] $\;-x\;$ can be read as either negative ex or the opposite of ex.
Here, the letter [beautiful math coming... please be patient] $\;x\;$ 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 [beautiful math coming... please be patient] $\;-x\;$ as the opposite of [beautiful math coming... please be patient] $\;x\;$.

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: [beautiful math coming... please be patient] $$\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$$

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 [beautiful math coming... please be patient] $\,a\,$ 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 [beautiful math coming... please be patient] $\,a\,$ and $\,b\,$ are just different names for the same number!

Throughout this web site, when the word number is used, it means real number.

EXAMPLES:
QuestionAnswer
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
Master the ideas from this section
by practicing both exercises at the bottom of this page.

When you're done practicing, move on to:
Addition of Signed Numbers

 
 
yes
no
    
(an even number, please)
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 26; there are 26 different problem types.)