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Summer 2016 - Session 2
Math 1300
FUNDAMENTALS OF MATH
Section #16535
Monday - Friday, 10am-12pm
Instructor: Dr. Angelynn Alvarez
[email protected]
06/05/2016
Section 1.2 - Integers
Section 1.2 – Integers
Recall: The integers is the set of numbers which consists of
• The natural numbers (the “counting numbers”)
• The number 0
• The negatives of the natural numbers
{…-51,…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…10,...,51,…}
Most of the problems in this class involve integers because they are
one of the “nicest” numbers to work with.
There are 3 major operations on integers that we will cover in this
class:
(1)
Absolute Value
(2)
Addition / Subtraction
(3)
Multiplication / Division
Absolute Value
The absolute value of a number, x, is the distance from 0 on the
number line. In symbols, we write |𝑥|.
With integers on a number line, we can simply count the number of
units from x and 0.
If you do not want to plot a number and count, you can use the
formal definition of absolute value:
which basically says…
• If a number inside the bars is positive or 0, the absolute value
of the number is itself---so we just drop the absolute value
bars.
• If a number inside the bars is negative, we drop the absolute
value bars and the negative sign.
Example: Find the following:
|-14|
|8|
|-91|
|100|
|0|
|2.5|
|1,614|
|-5,140|
Addition of Integers
• When we add two integers, say m and n, we use the “ + "
symbol and write m+n.
• After adding two numbers, the answer is called the sum.
The two rules of addition of integers:
(1)
(2)
If the two integers have the same signs, add their
absolute values and keep the common sign.
If the two integers have different signs, subtract their
absolute values (larger - smaller), and keep the same
sign of the integer with the larger absolute value.
Examples: Add the following numbers:
-6 + 4
-4 + (-18)
10 + 6
10 + (-3)
Addition of Double-digit Integers
The following rules still apply:
(1)
If the two integers have the same signs, add their
absolute values and keep the common sign.
(2)
If the two integers have different signs, subtract their
absolute values (larger - smaller), and keep the same
sign of the integer with the larger absolute value.
Only now, when adding or subtracting the absolute values, you will
need to align the numbers and add or subtract the numbers
column-by-column, from right to left.
Example: Add the following numbers:
−26 + 42
−29 + (−35)
Subtraction of Integers
• When we subtract two integers, say m and n, we use the " − "
symbol and write m-n.
• After subtracting two numbers, the answer is called the
difference.
• Subtraction uses the SAME rules as addition because the
following is equivalent:
𝑚 − 𝑛 = 𝑚 + −𝑛
• After rewriting subtraction in terms of addition, we apply the
same rules as addition.
Example: Subtract -29 – 36.
Addition and Subtraction of More than 2 Integers
Sometimes you will be asked to add or subtract three or more
integers, such as:
20 + 21 − 18
In order to do this, you will need to add or subtract two numbers at a
time, going from left to right.
Example: Evaluate 20 + 21 − 18.
Example: Evaluate −16 − 8 + 12 − 7
Multiplication of Integers
• When multiplying two integers, say m and n, we use the "×"
symbol and write m×n.
• The "×" symbol can sometimes be confused with the variable 𝑥,
so instead we usually write: mn or m(n) or (m)(n).
• After multiplying two numbers, the answer is called the
product.
The two rules of multiplication of integers:
(1)
(2)
If two integers have the same signs, multiply their
absolute values and the answer will be positive.
If two integers have different signs, multiply their
absolute values and the answer will be negative.
Some useful facts…
-Any number multiplied by 0 is always 0---that is, if m is an integer,
then
𝑚 0 = 0,
0 𝑚 =0
-A number multiplied by 1 is always itself---that is, , if m is an
integer, then
𝑚 1 = 𝑚,
1(𝑚) = 𝑚
Example: Multiply the following numbers:
5(6)
-3(2)
(-4)(-3)
Multiplication of Double-digit integers
The following rules still apply:
(1)
(2)
If two integers have the same signs, multiply their
absolute values and the answer will be positive.
If two integers have different signs, multiply their
absolute values and the answer will be negative.
Only now, when multiplying the absolute values, you will need to
align the numbers and multiply the numbers column-by-column.
If both integers are double-digit, then you will need to use a 0 as a
placeholder.
Example: Multiply the following numbers.
16(24)
-12(15)
-15(9)
Multiplication of 3 or more integers
Sometimes you will be asked to multiply 3 or more more integers,
such as:
−5(−4)(3)
In order to do this, you will need to multiply two numbers at a time,
going from left to right.
Example: Evaluate −5(−4)(3)
Example: Evaluate −6(7)(−2)
Tip: (Re-)Memorize the multiplication table!
Division of Integers
• When dividing two integers, say m and n, we use the " ÷ "
symbol and write m÷n.
• Because writing “ ÷ ” can be quite tedious, we usually write
!
!
.
• When dividing two integers, the answer is called the quotient.
The two rules of division of integers:
(1)
If two integers
values and the
If two integers
values and the
(2)
have the same signs, divide their absolute
answer will be positive.
have different signs, divide their absolute
answer will be negative.
Some important facts…
• The number 0 divided by any nonzero number is 0---that is,
0
= 0,
𝑛≠0
𝑛
• If you divide a number by itself, the quotient is always 1---that
is,
𝑛
= 1,
𝑛 ≠0
𝑛
• If you divide a number by 1, it is always itself –that is,
𝑛
=𝑛
1
• Recall the “worst thing you can do” is DIVIDE BY 0.
!
!
DOES NOT EQUAL 0.
!
!
IS UNDEFINED!
You can NEVER have zero on the denominator.