# Order of Operations And Real Number Operations

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Order of Operations And Real Number Operations
MA 1128: Lecture /30/14 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. The green buttons will take you to the previous slide, the next slide, the last slide, or out of PowerPoint. The first few lectures should be review, and we’ll use them to help you get used to these lectures. You should read through the lectures, do the practice problems as you come across them, and when you’re comfortable with these, you should be ready to take the quiz. If you can do the quiz problems easily, you should have no problems with the tests. And there is nothing wrong with doing the same problem more than once. Practice will make you faster and more accurate. Next Slide

Quit Order of Operations In algebra, we manipulate groups of mathematical symbols called expressions. Like x 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 Next Slide

The Order of Operations
Quit The Order of Operations Grouping Symbols Exponents and Radicals Multiplication and Division Addition and Subtraction 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. Next Slide

Example Consider the expression below.
Quit Example 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. (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 Next Slide

Now we do the multiplication. = 50
Quit 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 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 Next Slide

The left-to-right rule is really bad.
Quit The left-to-right rule is really bad. 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 = = 5 Next slide

Quit Example 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 (5) = (10)(5) = 50 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. Next Slide

Practice Order of Operations Problems
Quit Practice Order of Operations Problems Simplify each of the following expressions as much as you can using the order of operations. (3)2 + 6  2 (We’ll talk more about this later, but (3)2 is the same as 32 = (3)(3) = 9.) 8  5  3 (You have to use the left-to-right rule here. We’ll have a better way later.) 5 + (2)(3) + (5)(2) Click for answers. 1) 13; 2) 0; 3) 21; 4) 14; 5) 6. __20__ (3) (2)(5) Next Slide

Real Number Operations
Quit 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. Next Slide

Addition and Subtraction of Signed Numbers
Quit 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 (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 = 2  4 – 6 = ( 4) + ( 6) =  10 Next Slide

Double Negatives We’ll also see things like 4 – (6)
Quit Double Negatives 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) = = 10 Next Slide

Also, look out for the following.
Quit 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 Next Slide

Practice Signed Number Problems
Quit Practice Signed Number Problems Simplify each of the following expressions as much as you can using the order of operations. 3 + (8) 8 + (3) 3  (8) (2)(3)(5) (5)(2)(2) (1)3 (2)2 22 Click for answers. 1) 5; 2) 5; 3) 11; 4) 30; 5) 20; 6) 1; 7) 4; 8) 4. Next Slide

The Distributive Property
Quit The Distributive Property 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. Next Slide

What does the Distributive Property Say?
Quit What does the Distributive Property Say? 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) + 3(5) – 3(3) = – 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. Next Slide

Division Distributes Also
Quit Division Distributes Also 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. ___ ___ ___ + = – 1 = 4 Remember! If you divide into a bunch of things added together, Then you have to divide into every one of those things. Next Slide

And Exponents Distribute over Multiplication/Division
Quit And Exponents Distribute over Multiplication/Division ( ) 3 ___ 2 ____ 23 ____ 8 Example = = 3 33 27 Example ( (3) (2) )4 = ( 34 ) ( 24 ) = (81) (16) = 1296 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. Next Slide

Practice Distributive Property Problems
Quit Practice Distributive Property Problems Simplify each of the following expressions as much as you can using the order of operations. 3(2 + 5) ( )(7) 8(1  3 + 2) ( (2)(3) )3 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. ________ 2 2 + 8  6 __________ 5  5 ( )2 3 __ 2 Next Slide

Last Slide You should review your practice problems.