Download Whole Numbers and Integers

Whole Numbers and Integers
Whole Numbers
Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)
No Fractions!
Counting Numbers
Counting Numbers are Whole Numbers, but without the zero. Because you
can't "count" zero .
So they are 1, 2, 3, 4, 5, ... (and so on).
Natural Numbers
"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or
"Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.
Integers
Integers are like whole numbers, but they also include negative numbers ...
but still no fractions allowed!
So, integers can be negative {-1, -2,-3, -4, -5, ... }, positive {1, 2, 3, 4, 5, ...
}, or zero {0}
We can put that all together like this:
Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }
Example, these are all integers:
-16, -3, 0, 1, 198
(But numbers like ½, 1.1 and 3.5 are not integers)
These are all integers (click to mark), and they continue left and right
infinitely:
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Some People Have Different Definitions!
Some people (not me) say that whole numbers can also be negative, which
makes them exactly the same as integers.
And some people say that zero is NOT a whole number. So there you go, not
everyone agrees on a simple thing!
My Standard
I usually stick to this:
Name
Numbers
Examples
Whole Numbers
{ 0, 1, 2, 3, 4, 5, ... }
0, 27, 398,
2345
Counting
Numbers
{ 1, 2, 3, 4, 5, ... }
1, 18, 27, 2061
Integers
{ ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,
... }
-15, 0, 27,
1102
And everyone agrees on the definition of an integer, so when in doubt say
"integer".
And when you only want positive integers, say "positive integers". It is not only
accurate, it makes you sound intelligent. Like this (note: zero isn't positive or
negative):

Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }

Negative Integers = { ..., -5, -4, -3, -2, -1 }

Positive Integers = { 1, 2, 3, 4, 5, ... }

Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?)
Integers
Unit 5 > Lesson 1
of 11
Problem: The highest elevation in North America is
Mt. McKinley, which is 20,320 feet above
sea level. The lowest elevation is Death
Valley, which is 282 feet below sea level.
What is the distance from the top of Mt.
McKinley to the bottom of Death Valley?
Solution: The distance from the top of Mt. McKinley
to sea level is 20,320 feet and the
distance from sea level to the bottom of
Death Valley is 282 feet. The total
distance is the sum of 20,320 and 282,
which is 20,602 feet.
The problem above uses the notion of opposites: Above sea level is the opposite of below sea
level. Here are some more examples of opposites:
top, bottom
increase, decrease
forward, backward
positive, negative
We could solve the problem above using integers. Integers are the set of whole
numbers and their opposites. The number line is used to represent integers. This is
shown below.
Definitions






The number line goes on forever in both directions. This is indicated by the arrows.
Whole numbers greater than zero are called positive integers. These numbers are to the right
of zero on the number line.
Whole numbers less than zero are called negative integers. These numbers are to the left of
zero on the number line.
The integer zero is neutral. It is neither positive nor negative.
The sign of an integer is either positive (+) or negative (-), except zero, which has no sign.
Two integers are opposites if they are each the same distance away from zero, but on opposite
sides of the number line. One will have a positive sign, the other a negative sign. In the number
line above, +3 and -3 are labeled as opposites.
Let's revisit the problem from the top of this page using integers to solve it.
Problem: The highest elevation in North America is
Mt. McKinley, which is 20,320 feet above
sea level. The lowest elevation is Death
Valley, which is 282 feet below sea level.
What is the distance from the top of Mt.
McKinley to the bottom of Death Valley?
Solution: We can represent the elevation as an
integers:
Elevation
Integer
20,320 feet above sea level +20,320
sea level
282 feet below sea level
0
-282
The distance from the top of Mt. McKinley to the bottom of Death Valley is the
same as the distance from +20,320 to -282 on the number line. We add the
distance from +20,320 to 0, and the distance from 0 to -282, for a total of
20,602 feet.
Example 1:
Write an integer to represent each situation:
10 degrees above zero
a loss of 16 dollars
a gain of 5 points
8 steps backward
Example 2:
Example 3:
+10
-16
+5
-8
Name the opposite of each integer.
-12
+12
+21
-21
-17
+17
+9
-9
Name 4 real life situations in which integers can be used.
Spending and earning money.
Rising and falling temperatures.
Stock market gains and losses.
Gaining and losing yards in a football game.
Note: A positive integer does not have to have a + sign in it. For example, +3 and 3 are
interchangeable.
Summary: Integers are the set of whole numbers and their opposites. Whole numbers
greater than zero are called positive integers. Whole numbers less than zero
are called negative integers. The integer zero is neither positive nor negative,
and has no sign. Two integers are opposites if they are each the same
distance away from zero, but on opposite sides of the number line. Positive
integers can be written with or without a sign.
Absolute Value
Absolute Value means ...
... only how far a number is from zero:
"6" is 6 away from zero,
and "−6" is also 6 away from zero.
So the absolute value of 6 is 6,
and the absolute value of −6 is also 6
More Examples:

The absolute value of −9 is 9

The absolute value of 3 is 3

The absolute value of 0 is 0

The absolute value of −156 is 156
No Negatives!
So in practice "absolute value" means to remove any negative sign in front of a
number, and to think of all numbers as positive (or zero).
Absolute Value Symbol
To show that we want the absolute value of something, we put "|" marks either
side (they are called "bars" and are found on the right side of a keyboard), like
these examples:
|−5| = 5
|7| = 7
Sometimes absolute value is also written as "abs()", so abs(−1) = 1 is the
same as |−1| = 1
Try It Yourself
© 2015 MathsIsFun.com v0.77
Subtract Either Way Around
And it doesn't matter which way around we do a subtraction, the absolute value
will always be the same:
|8−3| = 5
|3−8| = 5
(8−3 = 5)
(3−8 = −5, and |−5| = 5)
More Examples
Here are some more examples of how to handle absolute values:
|−3×6| = 18
(−3×6 = −18, and |−18| = 18)
−|5−2| = −3
(5−2 = 3 and then the first minus gets you −3)
−|2−5| = −3
(2−5 = −3 , |−3| = 3, and then the first minus gets you −3)
−|−12| = −12
(|−12| = 12 and then the first minus gets you −12)
Number Line
Writing numbers down on a Number Line makes it easy to tell
which numbers are bigger or smaller.
Negative Numbers (-)
Positive Numbers (+)
(The line continues left and right forever.)
Numbers on the left are smaller than numbers on the right.
Examples:

5 is smaller than 8

−1 is smaller than 1

−8 is smaller than −5
Numbers on the right are larger than numbers on the left.
Examples:

8 is larger than 5

1 is larger than −1

−5 is larger than −8
Try this interactive number line (click to mark):
© 2015 MathsIsFun.com v0.77
You can also try the zoomable number line .
An Example
Example: John owes $3, Virginia owes $5 but Alex doesn't owe anything, in fact
he has $3 in his pocket. Place these people on the number line to find who is
poorest and who is richest.
Having money in your pocket is positive.
But owing money is negative.
So John has "−3", Virginia "−5" and Alex "+3"
Now it is easy to see that Virginia is poorer than John (−5 is less than −3) and
John is poorer than Alex (−3 is smaller than 3), and Alex is, of course, the
richest!
Using The Number Line
We can use the number line to help us add. We always move to the right to
add.
We can use the number line to help us subtract. We always move to the left to
subtract.
Read How to Use the Number Line to Add and Subtract .
Footnote: Absolute Value
Absolute Value means to think only about how far a number is from zero.
For example "6" is 6 away from zero, but "−6" is also 6 away from zero.
So the absolute value of 6 is 6, and the absolute value of −6 is also 6