Learning Objectives
Introduction
Even in this day and age of calculators, it is very
important to know
these basic rules of operations on real numbers. Even if you are
using a calculator, you are the one that is putting the information
into
it, so you need to know things like when you are subtracting versus
multiplying
and the order that you need to put it in. Also, if you are using
a calculator you should have a rough idea as to what the answer should
be. You never know, you may hit a wrong key and get a wrong answer (it
happens to the best of us). Also, your batteries in your
calculator
may run out and you may have to do a problem by hand
(scary!!!).
You want to be prepared for those Murphy's Law moments.
We will start with the individual operations and then mix ‘em up using order of operations.
Tutorial
Step 2: Attach
their common sign
to sum .
Example 1: Find the sum -6 + (-8).
The sum of the absolute values would be 14 and their common sign is -. That is how we get the answer of -14.
You can also think of this as money - I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 6 dollars and then having lost another 8 dollars for a total loss of 14 dollars.
The sum of the absolute values would be 14.2 and their common sign is -. That is how we get the answer of -14.2.
You can also think of this as money - I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 5.5 dollars and then having lost another 8.7 dollars for a total loss of 14.2 dollars.
Adding Real Numbers
with Opposite Signs
Step 2: Attach the sign of the number that has the higher absolute value.
The difference between 8 and 6 is 2 and the sign of 8 (the larger absolute value) is -. That is how we get the answer of -2.
Thinking in terms of money: we lost 8 dollars and got back 6 dollars, so we are still in the hole 2 dollars.
*Take the difference of the
numerators
and write over common denominator 6
*Reduce fraction
Thinking in terms of money: we had 2/3 of a dollar and lost 1/6 of a dollar, so we would come out ahead 1/2 of a dollar.
Note that if you need help on fractions, click on this link: Fractions
Now, you do not have to write it out like this if you are already comfortable with it. This just gives you the thought behind it.
Subtracting 5 is the same as adding a -5.
Once it is written as addition, I just follow the rules for addition, as shown above, to complete for an answer of -8.
Subtracting -5 is the same as adding 5.
Once it is written as addition, I just follow the rules for addition, as shown above, to complete for an answer of 2.
Step 1: Multiply or
divide their absolute
values.
Step 2: Put the correct
sign.
If they have opposite signs, the product or quotient is negative.
The product of the absolute values 4 x 3 is 12 and they have opposite signs, so our answer is -12.
Note that if you need help on fractions, click on this link: Fractions
The quotient of the absolute values 10/2 is 5 and they have the same signs, so our answer is 5.
*Mult. num. together
*Mult. den. together
*(+)(-) = -
*Reduce fraction
Note that if you need help on fractions, click on this link: Fractions
a(0) = 0
and
0/a = 0 (when a does not equal 0)
Multiplying any expression by 0 results in an answer of 0.
Dividing 0 by any expression other than 0 results in an answer of 0.
a/0 is undefined
Example 13: Find the quotient 5/0.
Dividing by 0 results in an undefined answer.
x = base, n = exponent
The exponent tells you how many
times a base
appears in a PRODUCT.
Example
14: Evaluate .
Note that we are only interested in the principal root and since 9 is positive and there is not a sign in front of the radical, our answer is positive 3. If there had been a negative in front of the radical our answer would have been -3.
Now we are looking for the third
or cube root
of -1/8, which means we are looking for a number that when we cube it
we
get -1/8.
Since -1/2 cubed is -1/8, our
answer is going
to be -1/2.
Please Parenthesis
or grouping symbols
Excuse Exponents
(and radicals)
My Dear Multiplication/Division
left to right
Aunt Sally Addition/Subtraction
left to right
When you do have more than one mathematical operation,
you need to use
the order of operations as listed above. You may have already heard of
the saying "Please Excuse My Dear Aunt Sally". It is just a way
to
help you remember the order you need to go in when applying the order
of
operations.
Example 19: Simplify .
Using the Order of Operations we get:
*Multiply
*Add
Plugging in values for the variables and evaluating the expression using the Order of Operations we get:
*Exponent in num.
*Multiply in num. and den.
*Add in num. and subtract in
den.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: Find the sum or difference.
1c. -10 - (-2)
(answer/discussion
to 1c)
Practice Problems 2a - 2b: Find the product or quotient.
Practice Problems 3a - 3c: Evaluate.
Practice Problems 4a - 4b: Find the root.
Practice Problems 5a - 5b: Simplify the expression.
Practice Problem 6a: Find the value of the expression when a = 3 and b = -2.
Need Extra Help on these Topics?
http://www.purplemath.com/modules/negative.htm
This webpage goes over addition, subtraction, multiplication, exponents
and roots of real numbers.
http://www.purplemath.com/modules/orderops.htm
This webpage goes over the order of operations.
http://www.mathleague.com/help/posandneg/posandneg.htm#addingpositiveandnegativenumbers
This webpage goes over adding, subtracting, multiplying, and dividing
signed numbers.
http://www.math.com/school/subject1/lessons/S1U1L11DP.html
This webpage helps with adding and subtracting integers.
http://www.mathleague.com/help/integers/integers.htm#addingintegers
This webpage helps with adding integers.
http://www.mathleague.com/help/integers/integers.htm#subtractingintegers
This webpage helps with subtracting integers.
http://www.math.com/school/subject1/lessons/S1U1L12DP.html
This webpage helps with multiplying and dividing integers.
http://www.mathleague.com/help/integers/integers.htm#multiplyingintegers
This webpage helps with multiplying integers.
http://www.mathleague.com/help/integers/integers.htm#dividingintegers
This webpage helps with dividing integers.
http://home.earthlink.net/~djbach/basic.html#anchor938594
This webpage helps with exponents.
http://www.mathleague.com/help/decwholeexp/decwholeexp.htm#exponents
This webpage helps with exponents.
http://www.blarg.net/~math/squares.html
This webpage helps with exponents and square roots.
http://home.earthlink.net/~djbach/basic.html#anchor941779
This webpage helps with order of operations.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on June 10, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.