MA 1128: Lecture 01 --- 1/18/11. Order of Operations And Real Number Operations. (Click Left Mouse button or Enter to Continue). Quit. How these slides work. Each new element of this presentation will appear when you click your left mouse button or hit enter.
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MA 1128: Lecture 01 --- 1/18/11
Order of Operations
And
Real Number Operations
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(23 – 4)3
Inside the ( ), the exponent is the highest priority operation.
= (8 – 4)3
Inside the ( ), subtraction is the highest priority (and only) operation.
= 43
The exponent is the only operation left.
= 64
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30 3 5
Division and multiplication have the same priority, and the division is furthest left, so we do the division first.
= 10 5
Now we do the multiplication.
= 50
30 3 5
If we ignore the left-to-right rule, and do the multiplication first, we’ll get a different answer.
= 30 15
Now the division.
= 2
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As much as is possible, mathematics is supposed to take care of itself.
In fact, most of our notation encourages our order of operations. If you look at
-3x2 + 6x + 5
The highest priority operation is the exponent, and this is emphasized by the 2 being really close to the x.
The next highest priorities are the two multiplications. You can see that the –3 and x2 are close together, as are the 6 and x.
Finally, the additions have the three terms somewhat spread out.
If we substitute x = 2 into the above expression, we don’t really have to think about left-to-right.
-3(2)2 + 6(2) + 5
= -3(4) + 6(2) + 5
= -12 + 12 + 5
= 5
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We can indicate our intended order of operations more naturally without using the division and multiplication symbols.
Consider 30 3 5. If we want the division to go first, then we could write
(
)
___
30
= (10)(5) = 50
(5)
3
If we want the multiplication to go first, then we could write
____
________
30
30
=
= 2
(3) (5)
15
The and are hardly ever used in algebra and higher level math, because it’s easier to indicate what we want using this notation.
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Simplify each of the following expressions as much as you can using the order of operations.
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1) 13; 2) 0; 3) 21; 4) 14; 5) 6.
__20__
(3)
(2)(5)
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The optional text covers real number operations in section 1.2.
I’ll look at the two most important ones here.
Operations with plus and minus signs.
The distributive properties.
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Addition and subtraction are basically the same thing.
You should think of subtracting as the same as adding a negative number.
For example, 4 – 6 is the same as 4 + (6)
In terms of the number line, you should think 4 to the right and 6 to the left.
Since the 6 is bigger, you end up to the left of zero, so the answer is negative, 2.
Hopefully, each of the following makes sense.
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We’ll also see things like
4 – (6)
You’ll want to think of this as 4 + (1)(1)(6).
In either case, negative of a negative is positive, and negative times negative is positive.
Two negatives are positive
Three negatives are negative
Four negatives are positive again, etc.
Whenever we see a double negative, we’ll generally write, or at least think to ourselves
4 – (-6)
= 4 + 6 = 10
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Exponents mean repeated multiplication, and the number of negatives being multiplied determines whether the end result is positive or negative. Consider
(2)4
Our order of operation rules make the “” go with the “2”.
= (2) (2) (2) (2) = 16,
Since four negatives is positive.
Compare this to
24
Here, the exponent is a higher priority than the negative sign (which is like multiplication by 1).
= (2)(2)(2)(2) = 16
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Simplify each of the following expressions as much as you can using the order of operations.
Click for answers.
1) 5; 2) 5; 3) 11; 4) 30; 5) 20; 6) 1; 7) 4; 8) 4.
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The distributive property is central to many of the things we’ll do.
Since we’ll be working with variables and unknowns, we often won’t be able to simplify using the order of operations.
For example, in 3x + 2x, we want to multiply the 3 times the x and the 2 times the x before the addition.
But we can’t, because we don’t know what number x represents.
As you may already know, however, 3x + 2x = 5x.
This is a manifestation of the distributive property.
The distributive property is a rule that allows us to implement the order of operations, without actually following the same steps.
In other words, using the distributive property gives us the same result as we would get using the order of operations.
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Example: Consider the following expression
3(2 + 5 – 3)
= 3(4) = 12.
The Distributive Property states that we will get the same result, if we multiply the 3 times the 2, the 5, and the 3 first.
3(2 + 5 – 3)
= 3(2) + 3(5) – 3(3)
= 6 + 15 – 9 = 12.
If we multiply times a bunch of things added (or subtracted) together,
the distributive property says that that’s equivalent to
multiplying times every one of those things.
And using the distributive property is consistent with the order of operations.
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It is convenient to think of division as a special kind of multiplication.
Division, therefore, should distribute also, and it does.
Consider the following example.
First we’ll simplify using the order of operations.
___________
6 + 9 – 3
____
12
=
= 4
3
3
We get the same result, if we distribute the division by 3.
___
6 9 3
___
___
–
+
= 2 + 3 – 1 = 4
3 3 3
Remember! If you divide into a bunch of things added together,
Then you have to divide into every one of those things.
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(
)
3
___
2
____
____
8
23
Example
=
=
3
33
27
( (3) (2) )4= ( 34 ) ( 24 ) = (81) (16) = 1296
Example
In general, exponents/radicals are higher level operations
than multiplication/division, which are higher level operations
than addition/subtraction.
Each level of operation distributes over the next lower level.
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5 10 + 25
__________
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( )2
__
2
5
3
________
2 + 8 6
2
Rewrite each of the following expressions using the distributive property, but don’t simplify.
Click for answers.
1) (3)(2) + (3)(5); 2) (1)(7) + (1)(7) + (1)(7); 3) (8)(1) (8)(3) + (8)(2);
4) 2/2 + 8/2 – 6/2 (with the bars horizontal) ; 5) 5/5 10/5 + 25/5;
6) (23)(33); 7) 22/32.
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For online classes only:
You should review your practice problems.
When you’re comfortable with those, work through the quiz problems.
You’ll find the link for Quiz 01 right next to the link for this lecture.
All the quizzes are in pdf format. You’ll need Adobe Reader for that.
Write your answers on a piece of paper.
You may ask someone else about the stuff in this lecture, but you shouldn’t discuss the quiz in any specific way.
Check through your work on the quiz, and when you’re ready, go into Desire2Learn and enter your answers.
You may take the quizzes as many times as you’d like, but make sure that you understand what you got wrong on the ones you missed.
End Lecture