Types of Numbers
Introduction to :
Addition, subtraction, multiplication, division, and exponentiation are all operations on the real numbers, meaning things you do to the real numbers. For complicated arithmetic expressions, it's important to perform operations in the correct order. So the "whatever order I feel like" tactic isn't going to work for you all that well.
This correct order is given by the magical phrase "Please Excuse My Dear Aunt Sally" (PEMDAS). Oh, poor, dear Aunt Sally – she gets a little confused sometimes and needs acronyms to remind her how to go about solving her favorite math equations. The letters stand for Parentheses, Exponentiation, Multiplication and Division, Addition and Subtraction, in the order we want to do them. Notice that "Multiplication and Division" and "Addition and Subtraction" are grouped together. That's because multiplication doesn't necessarily need to be done before division - you just need to have all your multiplication and division wrapped up before you start in on your addition and subtraction. Don't feel bad if you didn't get that at first - Aunt Sally's been trying to grasp that concept for 40 years and it still eludes her.
First we evaluate things in parentheses:
3 x 2 - (4) / 22
Then we evaluate exponents:
Then multiplication and division:
6 - 1
And finally, addition and subtraction:
When evaluating addition and subtraction, we work from left to right. (Check to see which of your shoes has the big "L" written on the sole of it if you're not sure.)
Sample: 4 - 6 - 2 = (4-6) -2 = -2 - 2 = -4
Notice that, if we had performed the subtraction 6-2 first, we would get a different answer:
4 - (6-2) = 4 -4 = 0.
We also work from left to right when evaluating multiplication and division.
Sample: 3 x 4 ÷ 2 ÷ 6 = 12 ÷ 2 ÷ 6 = 6 ÷ 6 = 1
If we worked from right to left, we would get a different answer:
1, 36 - eh, what's the difference?
One way to keep track of your work is to break the problem into pieces, separated by addition or subtraction signs. A rock hammer or mortar and pestle should do the trick.
Now work out each of the pieces:
6 + 2 - 0 - 4
Then combine the answers to the pieces:
6 + 2 - 0 - 4 = 4
3 + 6 x 16 - 6 x 1
3 + 6 × 16 - 6 × 1
= 3 + 96 - 6
Okay, but Please Excuse My Dear Aunt Sally is a really long thing to remember. And we've already spent the last 10 years attempting to block her out. Here at Shmoop, we like to simplify things ("Shmoop" is actually short for "Shmoopalumpagus").
We've seen that subtraction can be replaced by adding a negative, and division can be replaced by multiplying a reciprocal. So all we really need to remember is Please Excuse My Aunt. Do what's inside the Parentheses first, then take all Exponents, then Multiply, then Add. Just remember that division = multiply by reciprocal, and subtract = add a negative. Yeah, either way, you're going to have to remember some stuff. C'est l'algebra.
Let's work out a few samples:
Ok, so let's start by dealing with the stuff inside the parentheses . First, however, we're going to change the division to multiplication by a reciprocal, and the subtraction to adding a negative. We should also probably change the radio station. What is that - Avant-garde Metal?
A pretty roundabout way just to get to the number 1, but we'll take it.
Now that we have finished all operations inside the parentheses, we look for exponents. If they don't immediately present themselves, whistle loudly and shake a bag of treats - they'll come running.
In the first part of the equation we have 12, which of course is just 1. Then we look for multiplication. Since there is none, all that is left is to add -13. The subtraction of such an unlucky number can't be a bad thing.
1+ (-13) = -12. And we're done!
Let's try one more.
Okay - the first thing to notice is that there are a lot of parentheses. This isn't a difficult problem, but you have to clear your mind of all other distractions and concentrate so you don't accidentally do things out of order or omit a step. We'll give you a minute - just let us know when you've entered into a state of Zen.
You there? Excellent. First, let's rewrite this without division and subtraction:
You may notice that there is a different type of bracket being used here than what you're used to seeing. That's just to block off the bit that is being multiplied rather than divided. Notice that the entire expression 22 + (-3)^3 goes in the denominator of our new fraction when we take the reciprocal of the original - not just the 22. If you're starting to feel like parentheses are difficult to understand, just recognize that the first part of the word is "parent." Hm. Makes sense now, eh?
Now let's take care of the junk inside the parentheses. Remember your Aunt Sally when performing your operations!
Joe was asked by a stranger to find the answer to the arithmetic problem:
8 + 5 × 2
Joe wrote the following:
8 + 5 × 2 = 13 × 2 = 26
What did Joe do wrong (other than talking to a stranger, of course)? What should the answer be?
Beth was asked to find 4 - (3-5)2. Beth wrote the following:
4 - (-2)2 = 4 - (-4) = 8
What did Beth do wrong? What should the answer be?
(3 + 2 - 1)3 - (-7 + 3)2 × 8