MA 1128: Lecture 01 --- 1/18/11

Order of Operations

And

Real Number Operations

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• And there is nothing wrong with doing the same problem more than once. Practice will make you faster and more accurate.

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Order of Operations

• In algebra, we manipulate groups of mathematical symbols called expressions.

• Like x2 + 5 or 4. • You’re probably familiar with what the individual symbols mean.• For example, + means add.• Adding, subtracting, and raising to an exponent are operations.• What an expression means depends heavily on the order in which the

operations are performed.• So that we’re all talking about the same thing, we must agree on a

standard set of ordering rules.• The rules we’ll use are standard (everyone uses them), but arbitrary

(there is no obvious reason why these were the rules we chose). • While our notation will encourage you to use the correct order of

operations, you still need to know them

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The Order of Operations

1. Grouping Symbols

2. Exponents and Radicals

3. Multiplication and Division

4. Addition and Subtraction

5. If there is a tie

– Go from left to right

• In an expression, the highest priority operation should be done first.

• The next highest priority operation is done second.

• Continue this process until there are no operations left to perform.

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Example

(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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• Consider the expression below. • It has only numbers, so we can simplify it into a single number.• The highest priority operations are inside the parentheses ( ).• So we look inside the ( ) first.

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The left-to-right rule matters when we mix division and multiplication.For example …

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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The left-to-right rule is really bad.

-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.

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-3(2)2 + 6(2) + 5 = -3(4) + 6(2) + 5 = -12 + 12 + 5 = 5

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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

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Example

30___3( ) (5) = (10)(5) = 50

If we want the multiplication to go first, then we could write

30________(3) (5)

30____15

= = 2

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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

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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Practice Order of Operations Problems

Simplify each of the following expressions as much as you can using the order of operations.

1. (3)2 + 6 2 (We’ll talk more about this later, but (3)2 is the same as 32 = (3)(3) = 9.)

2. 8 5 3 (You have to use the left-to-right rule here. We’ll have a better way later.)

3. 5 + (2)(3) + (5)(2)

4. 22 + 32 + 1

5.

Click for answers.

1) 13; 2) 0; 3) 21; 4) 14; 5) 6.

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__20__ (2)(5)

(3)

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Real Number Operations

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 of Signed Numbers

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.

• 4 + 6 = 10

• 4 – 6 = 4 + ( 6) = 2

4 + 6 = ( 4) + 6 = 2

4 – 6 = ( 4) + ( 6) = 10

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Double NegativesWe’ll also see things like4 – (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 positiveThree negatives are negativeFour 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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Also, look out for the following.

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.

1. 3 + (8)

2. 8 + (3)

3. 3 (8)

4. (2)(3)(5)

5. (5)(2)(2)

6. (1)3

7. (2)2

8. 22

Click for answers.

1) 5; 2) 5; 3) 11; 4) 30; 5) 20; 6) 1; 7) 4; 8) 4.

Practice Signed Number Problems

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The Distributive PropertyThe 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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What does the Distributive Property Say?Example: Consider the following expression3(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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Division Distributes Also

6 + 9 – 3___________3

= 12____3

= 4

We get the same result, if we distribute the division by 3.

= 2 + 3 – 1 = 4

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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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 3 3 3

–+______

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____

And Exponents Distribute over Multiplication/Division

Example 2___3( )

3= 23

33 = 8____

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Example ( (3) (2) )4 = ( 34 ) ( 24 ) = (81) (16) = 1296

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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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Rewrite each of the following expressions using the distributive property, but don’t simplify.

1. 3(2 + 5)2. (1 + 1 + 1)(7) 3. 8(1 3 + 2)

4.

5.

6. ( (2)(3) )3

7.

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.

Practice Distributive Property Problems

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________2

2 + 8 6

__________5 10 + 255

( )2

3__2

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Last Slide

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.

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