As I mentioned in the article about working with positive and negative numbers, students in both high school and college make more sign mistakes than any other kind of mistake.
After completing an equation solving problem, simply taking a few extra seconds to follow the signs in the problem would catch most of those mistakes. When solving equations, you are not finished until you have checked it.
Remember that always!
I this article, we extend our look at "signed numbers" to include negative of a negative, multiplication of signed numbers, and distributing a negative.
Properties of Negation: (Note: Remember that an equation is true in both directions.
If a property doesn't make sense to you in the given format, try turning it around.
That just might help.
)
1.
(-1) x = - x This says that multiplying a number by -1 will change the sign. And, by the definition of multiplication, taking (-1), x times would be -x.
Example: (- 1) (4) = - 4 or taking (-1) four times would be negative four.
Turn this property the other way around and it says: -x = (-1)(x).
This way, it means that any negative number can be considered as negative one times the positive.
Example: -5 = (-1)(5).
This will be the most helpful way to think about this.
2.
- (- x) = x In words, this really says that the negative of a negative is positive.
This is easier to understand if you think of the first negative sign as "opposite of.
" So, - (- x) = x could be read: The opposite of negative x is x Example: - ( - 4) = 4. The opposite of (- 4) is +4.
(Remember double negatives from English class.
One negative undoes the other.
)
3.
( - x)(y) = (x)(- y) = -(xy) This makes more sense if you see each negative sign as a -1. This then becomes: (-1)(x)(y) = (x)(-1)(y) = (-1)(x)(y) Since multiplication is commutative, the order can be changed in each case so that the -1 is always in the front making them all identical.
Example: (-3)(4) = (3)(- 4) = - (3)(4) and each is -12.
4.
(-x)(-y) = xy Again, let's write in the -1's. So, (-x)(-y) becomes (-1)(x)(-1)(y) and using the commutative property, it becomes (-1)(-1)(x)(y). Since the opposite of -1 is +1, we end up with xy.
Example: (-3)(- 4) = 12.
Note: #3 and #4 can be summarized as: When multiplying two signed numbers, if the signs are alike, either + + or - -, then the answer (product) is positive.
If the signs are different, either + - or - +, the product will be negative.
These Properties of Negation are for multiplication only.
(Not true for addition)
Summarizing:
When multiplying two numbers:
Like Signs = Positive
Different Signs = Negative
#5 should look vaguely familiar.
5.
- (x + y) = -x + (-y) = -x - y This is essentially the distributive.
You can either follow the normal Order of Operations or you may distribute the negative sign if you wish.
-(3 + 4) = - 7 or distributing the negative sign: - (3 + 4) = -3 + (-4) = -7 or -(3 + 4) = -3 - 4 = -7
A similar situation which is often missed is - (3 - 4)
By Order of Operations, - (3 - 4) = - (-1) = +1 (You don't have to write the + sign.
I am just stressing that it is positive.
) By "distributing the negative" - (3 - 4) becomes -3 + 4 = 1
Remember the phrase "distribute the negative.
" Many Algebra students remember to apply the negative to the first number but forget about the second, so they end up with a sign mistake.
A few words about calculator use and negative signs.
Until we start graphing equations, calculators should only be used for very difficult computations-not for sign issues. The calculator has one button for subtraction and a different button to mean "make negative" and students often get these confused.
When just dealing with simple problems involving positive and negative numbers, the calculator causes more problems than it solves. Learn the rules and use what you know.
Think before you pick up a calculator.
A calculator is a GIGO device.
Garbage In Garbage Out.
Trust your brain!
By the way: I promise! The next article is "What Does It Mean To Solve An Equation?" Do you think you know? Write your answer on a piece of paper, hang it on the fridge, and check yourself when the article comes out.
Just so you know, I have NEVER had a student answer this correctly on their first try.
I doubt if you will find the answer online. This is another one of those "assume you know" topics.
Good luck! (To help you along, the answer is NOT "to find the answer.
")
No comments:
Post a Comment