1.8+Order+of+Operations

Section 1.8 - Order of Operations

 * Some Things Must Be Done in a Particular Order**

When you're making bread, some ingredients must be combined in a particular order, while other do not need to. Putting the salt in before the flour is probably just fine, but adding yeast at the end instead of early on, would not produce very good bread.

In the earliest sections you learned a couple of rules, or properties, that said it doesn't matter which order you add numbers or multiple numbers in - as long as they are all being added or multiplied!

However, for many math problems, which combine addition, subtraction, multiplication, division, and maybe other things like exponents, the order absolutely matters. There are some very specific rules and those rules always apply. They apply for whole numbers, for fractions, for decimals, for all problems!

We call the rules the Order of Operations.


 * PEMDAS**

You may have learned the Order of Operations as PEMDAS - Please Excuse My Dear Aunt Sally - with the initials representing Parentheses, Exponents, Multiply, Divide, Add, Subtract.

Whether you learned the order of operations that way or not, please carefully review and learn these important rules. There are some tricky parts, and they must be followed to ensure correct answers to problems.


 * Step 1: Parentheses and Other Grouping Symbols**

The rule is to do the math within parentheses or other grouping symbols first - starting with the innermost parentheses and moving outward.

There are several kinds of grouping symbols that would work:, [ ], { } are all forms of parentheses. Sometimes these grouping symbols are "nested" one inside the other. An example would be: (5 + (3 + 4)) In this case you would do the 3 + 4 first because they are in the innermost parentheses.

Within parentheses, you still follow the other rules in the Order of Operations. (3 + 4 ( 6 - 5)) Do the 6 - 5 first because they are inside the innermost parentheses. However you will be tempted to add 3 and 4 next, but that would be wrong because multiply comes before add. After subtracting 6 - 5, multiply the answer by 4, then add 3.

Absolute Value symbols ( | | ) are also considered grouping symbols. That means ( 3 + | 2 - 8 |) would start with 2 - 8, which is -6, then take the absolute value of that, which is 6, then add that result to 3 to get a final answer of 9.

There are two other situations that we might describe as "implied parentheses" - two times when you do them first, even though there aren't parentheses shown.
 * 1) When you have a fairly complex division problem - math on the top of a division bar and math below. In this case, you treat the problem as though the top had parentheses around it and the bottom does as well. Do all of the math up top, then do all of the math on the bottom, and as a final step, divide the two results.
 * 2) When there is math to be done underneath a radical (square root sign). You would do all of the math beneath the radical, whether it is addition or division, etc., and then finally take the square root of that answer before moving on.


 * Step 2: Exponents and Roots**

Next, calculate all exponents and square roots.


 * Step 3: Multiply and Divide**

Then, go through the problem from left to right, looking for (and calculating) any multiplication or division problems. Don't do all multiplication before division - do them in the order that they come, left to right.


 * Step 4: Add and Subtract**

Finally, go back through the problem again from left to right and do all addition and subtraction problems in the order you find them. Don't do all the addition and then all of the subtraction - do either in the order you find them, left to right.

Intro video first, more complex one follows: media type="youtube" key="ClYdw4d4OmA?fs=1" height="385" width="640" media type="youtube" key="piIcRV2dx7E?fs=1" height="385" width="640"

**Where to from here?**
1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Chapter One Review**
 * Chapter One Homework**