Integers: The set of integers is the set that contain the numbers
{· · · − 5, −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . }
In other words, integers include the positive and negative whole numbers, and
the number 0.
Notice that every natural number is an integer, but not every integer is a natural
number.
We can graph the integers on a number line:
Example:
Graph the integers −3, −1, 0, 5 on a number line:
−7 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
The absolute value of a number is the distance the number is from zero.
Example: To find | − 3|, since −3 is three (3) units from zero, | − 3| = 3.
Example: Since 5 is five (5) units from zero, |5| = 5.
Example: |0| = 0.
In general, if a number is positive, then its absolute value is just the number
itself. If a number is negative, then its absolute value is its opposite (which will
be a positive number).
Given a number a, its additive inverse is the number −a.
Example: The additive inverse of 4 is −4.
Example: The additive inverse of −3 is 3.
The negative (opposite) of a positive number is negative. The negative (opposite)
of a negative number is positive.
Example: −(−3) = 3
To add two negative numbers, add the absolute value of the two numbers, and
turn the result into negative.
E.g. −4 + −3 = −(4 + 3) = −7
To add a positive number with a negative number. First determine which number
has a larger absolute value. If the negative number has a larger absolute value, the
result is negative, otherwise the result is positive. Subtract the smaller absolute
value from the larger absolute value and assign a sign as given above.
E.g. −5 + 3 = −(5 − 3) = −2
To subtract a negative number is to add the positive:
E.g. 3 − −4 = 3 + 4 = 7
To add a negative is to subtract the positive:
4 + −6 = 4 − 6 = −2
If we multiply two numbers with the same sign (both negative or both positive),
the result is a positive number.
If we multiply two numbers with opposite sign (one positive and one negative),
the result is a negative number.
The same result holds true if we divide two numbers.
Example:
−5(−3) = 13
4(−2) = −8
−1(4) = −4
−12
= −4
3
21
= −3
−7
−16
=8
−2
9
= −9
−1
The order of operation involving integers is the same as that involving natural
numbers. The absolute value and radical has the same priority as that of a
grouping symbol (parenthesis).
Keep in mind that exponents have a higher priority than multiplication, pay
special attention when raise a negative number to an exponent:
Example: −42 = −16
Example: (−4)2 = 16
Example: −32 = −9
Example: (−3)2 = 9
Example: − 32 = −9
Then radical of a negative number is undefined:
√
Example: 25 = 5
√
Example: −25 = undefined.
√
Example: − 25 = −5
Example:
14
+ 3(5 − 8) − 22
−3 −
7
14
= −3 −
+ 3(−3) − 22
7
14
= −3 −
+ 3(−3) − 4
7
= −3 − 2 + 3(−3) − 4
= −3 − 2 − 9 − 4
= −5 − 9 − 4
= −14 − 4
= −18
Example:
4 − | − 2 − 3| − 7(−3) − 42
= 4 − | − 5| − 7(−3) − 42
= 4 − 5 − 7(−3) − 42
= 4 − 5 − 7(−3) − 16
= 4 − 5 − (−21) − 16
= 4 − 5 + 21 − 16
= −1 + 21 − 16
= 20 − 16
=4
Example:
√
−15
5 + 16 − 7 + |6 − 9| −
5
√
−15
= 5 + 9 + |6 − 9| −
5
−15
= 5 + 3 + |6 − 9| −
5
−15
= 5 + 3 + | − 3| −
5
−15
=5+3+3−
5
= 5 + 3 + 3 − (−3)
=5+3+3+3
=8+3+3
= 11 + 3
= 14