See also the
Dr. Math FAQ:
negative x negative
3D and higher
Browse Middle School Negative Numbers
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Inequalities and negative numbers.
Positive/negative integer rules.
- Decreasing by Degrees [01/07/2016]
A teacher wonders about her choice of context for a problem designed
to elicit subtraction of a negative number. Doctor Peterson provides
troubleshooting linguistic, mathematical, and instructional.
- Distributive Property with Negatives [10/13/2004]
When using the distributive property with negative numbers, I don't
understand when, why, and how the operator changes (from addition to
subtraction, or the other way). Sometimes it changes, sometimes it
doesn't. Can you explain when to change it, and why?
- Dividing Negative Numbers in Real Life [03/03/2004]
Why and when do we divide by a negative number in real life? I can't
find a convincing example to make my child happy.
- Division by a Negative and a Definition of Division [09/18/2006]
If 6 is divided by 3 it means that 6 is being divided into 3 equal
parts. So what does 6 divided by -3 mean? How do you divide something
into -3 equal parts?
- Drawing Equations on the Number Line [05/09/2002]
How can I draw an equation like 2 - (-5) = 7 on a number line?
- Equations and Negatives [05/23/1999]
How would I solve the following problem: 6(-N) = 3(N)+72 ?
- Explaining Subtraction of Integers as Adding the Opposite [02/16/2007]
I'm trying to help my 7th grade class understand why subtracting an
integer is the same as adding its opposite, particularly in cases like
(-5) - (-3). Can you give real life examples or ideas on how to
explain this to them?
- Exponents Containing Negative Integers [12/15/1998]
Order of operations and raising a negative number to a power.
- Factorials of Negative Integers [06/27/2002]
Is it possible to compute factorials for negative integers?
- Inequalities and Negative Numbers [11/7/1994]
When multiplying both sides of an inequality by a negative number, why
does the inequality sign change?
- Integers in Daily Life [05/31/2002]
How are integers (especially negative integers) used in daily life?
- Introduction to Division with Negative Numbers [05/05/2004]
I don't know how to do division with negative numbers, like +18/-9
or -24/-8 or +21/-3.
- Mod Function and Negative Numbers [04/28/2000]
What is the correct value for the function MOD(-340,60)? Microsoft Excel
returns the value 20, while Lotus 1-2-3 returns the value -40. Can you
explain the difference?
- Multiplying and Dividing Negatives [07/20/1998]
Can you explain multiplication and division with signed numbers? For
example what is the sign when you multiply negative numbers?
- Multiplying Decimals by Negatives [03/18/2002]
I can't find decimals on my calculator.
- Multiplying Negative Integers [11/07/2001]
Multiplying negative integers by breaking each negative number into a
positive number multiplied by -1.
- Multiplying or Dividing by a Negative Number [04/08/1999]
When x(x-4) is less than 0...
- Multiplying Positive and Negative Numbers [10/27/2003]
In a problem like -3 x 7 x 4 = ?, what do I do with the negative numbers?
- Multiplying Two Negative Numbers [01/16/1997]
Can you give me an example why when you multiply two negative numbers you
get a positive number?
- Negative 5 and Minus 5 [10/05/2002]
Can you tell me the difference between negative five and minus five?
- Negative Exponents: Who Needs Them? [09/19/2011]
A student wonders about the utility of negative exponents. Doctor Peterson elaborates
on several instances when they come in convenient.
- Negative Fractional Powers [12/08/1998]
How do you do problems like 3^(-2/3), where your power is a negative
- Negative/Negative [8/4/1996]
Why is a negative divided by a negative a positive?
- Negative Number History [11/25/2001]
What happened in the 16th century that caused people to need to use
negative numbers, and what was their image of a negative number before
- Negative Number or Subtraction? [12/16/2003]
When I see a '-' in front of a number, how can I tell whether it's
supposed to indicate a negative number, or a subtraction?
- Negative Numbers Combined with Exponentials [03/09/2001]
Why in the order of operations is negation a multiplication done after
exponentiation, rather than as a part of the base? What about
- Negative Numbers in Base 2, 16, etc. [01/24/2003]
Are there negative numbers in alternate bases? Our calculator says
that 1-2 in base16 is FFFFFFFFFF. This can't be right!
- Negative Numbers in Real Life [06/25/2008]
I'm finding it very difficult to imagine negative numbers outside of
math. In the real world, I can't have -$100 in my bank account. I
can't give away 6 apples when I only have 5 and be left with -1 apple.
How can the math be accurate but impossible?
- Negatives and Inequalities [09/10/1998]
Why when we multiply or divide an inequality by a negative number do we
reverse the inequality sign?
- Negatives in Absolute Value Expressions and Equations [10/15/2005]
How do I solve an equation like |x - 3| - 4 = 0? How do I evaluate
absolute value expressions with negatives, like |-7 - 3|?
- Negative Squared, or Squared Negative? [10/18/2002]
In -3^2, isn't -3 actually a number, and not -1*3? If so, then
shouldn't the value be -3*-3=9?
- A Negative Times a Negative [06/02/1998]
Can you show me why is a negative number times a negative number a
- Negative Two Kids [09/27/2002]
Let's say 3 x 2 = 6. Three is how many kids there are in a 'group.'
Two is how many 'groups' there are, and six is the answer. Now let's
say 3 x (-2) = -6. How can you have negative two 'groups' of three
- Order of Operations and Adding Negative Numbers [08/08/1997]
What does (-9 + -10) mean?
- Placement of the Negative Sign in a Negative Mixed Number [10/25/2004]
When you have a negative improper fraction, can the negative sign be
put anywhere in the fraction? For instance, is -1 3/5 the same as 1
-3/5 and 1 3/-5?
- Positive and Negative Multiplication [7/30/1996]
I would like to know why: + times + is +, + times - is -, and -
times - is +.
- Practical Applications of Negative Numbers [10/13/2002]
Why were negative numbers invented?
- Product of Negative Numbers [02/04/1997]
Can you give a proof of why the product of two negative numbers is
- Proving That (-1) * (-1) = 1 [01/30/2006]
At the high school level it is difficult to prove that -1 x -1 = 1. Is
there a published proof somewhere that is appropriate for kids that age?
- Put the Variable in Parentheses [10/15/2001]