Study Guide

# Types of Numbers - Things to Do with Real Numbers

## Things to Do with Real Numbers

### Compare Them

When comparing real numbers, it can help to think of the number line. Ah, how fondly we remember it. Our favorite line of all the lines.

The numbers on a number line get bigger as we go further right on the number line, and smaller as we go further left. All positive numbers are bigger than zero, and all negative numbers are less than zero. Doesn't mean they can't be friends.

### Picturing Addition on the Number Line

The addition of real numbers can be visualized using the number line. Just one more reason this number line thingie is nice to have around.

Adding two positive real numbers is like starting at 0 and then going for two walks to the right. In the first walk, take as many steps as the first number says, and in the second walk take as many steps as the second number says. In between the two walks, make sure to hydrate.

For example, we already know that 3 + 5 = 8. That's some pretty basic addition. But picture it on the number line: go for a walk of 3 steps to the right of 0, then a second walk of 5 steps after that.

Adding two negative numbers is like starting at 0 and going for two walks to the left. Maybe because you're having a fight with the guy who just started off on a walk to the right. In the first walk, take as many steps to the left as the first number says, and in the second walk take as many steps to the left as the second number says. Watch your step and don't accidentally step on any negative signs. Those things can go right through your shoes and give you splinters.

For example, (-4) + (-3) = -7. We're starting at 0, then walking 4 steps to the left to get to -4 on the number line. Then we're walking another 3 steps left after that, which brings us to our final answer of -7.

Adding one positive and one negative number is, again, like going for two walks, only this time one walk will be to the right and one will be to the left. We apologize in advance, because we know how much you hate backtracking.

4 + (-5) = -1

See? We walked 4 steps to the right, then turned on our heel and walked backwards another 5 steps.

Notice that switching the order of the walks doesn't change the answer:

(-5) + 4 = -1

### Sample Problem

5 + __ = -17

This question is asking how many steps it takes to get from 5 to -17 on the number line, and in which direction those steps need to be taken. Picture the number line: you can see that if you walked off in a huff and got to the 5, but then your friend who you're having a spat with yells at you from the -17 and says he's sorry, it would take you 22 steps to the left to walk back over to him and hug it out.

5 + (-22) = -17

No, addition isn't some incredibly wealthy mathematical operation that owns multiple houses in multiple states. Although if it were, it would probably be building additions onto each of them.

If we add two real numbers, we get the same answer no matter which number comes first. This means that addition of real numbers is commutative: the numbers can commute (travel past each other with a cup of coffee while listening to their favorite morning radio station) without changing the answer. Wait, no, that's not right. "Commute" just means "switch places" when we're talking about math.

To write "addition of real numbers is commutative" in symbols—because goodness knows we love us some symbols—we say that for real numbers a and b:

a + b = b + a

Both terms can swap spots without changing the total sum.

Addition of real numbers is also associative: we don't care how the numbers associate with each other. As long as there isn't bloodshed. When adding more than two numbers, you'll get the same answer no matter which end you start adding from. If you have three friends, one of whom tells you 2 jokes, one of whom tells you 4 jokes, and one of whom tells you 7 jokes, you'll end up having heard 13 jokes. Regardless of the order you hear them in, you'll be the one who has the last laugh.

### Sample Problem

1 + 2 + 3 = ?

If we first add up 1 and 2, we get:

(1 + 2) + 3 = 3 + 3 = 6

If we first add up 2 and 3, we get:

1 + (2 + 3) = 1 + 5 = 6

Same deal either way. To write "addition of real numbers is associative" in symbols, we say that for real numbers a, b, and c:

(a + b) + c = a + (b + c)

Addition has an additive identity: the number 0. If you add 0 to any number, that number keeps its identity. For example, 4 + 0 = 4.

In other words, if you take 4 steps, and then 0 steps, you've still taken only 4 steps overall. By the way, you look pretty silly trying to take 0 steps.

If you walk a certain distance on the number line and then walk the same distance in the opposite direction, you'll end up back where you started. This means that if you add a real number n and its negative -n, you end up back at 0 again. Who knows why you went all the way back to 0? Maybe you forgot your keys.

n + (-n) = 0

(-n) + n = 0

2 + (-2) = 0

(-8.7) + 8.7 = 0

If n is a real number (positive or negative or zero), we call -n the additive inverse of n. "Additive inverse of n" is just a fancy way of saying "the number we add to n to get back to zero." Remember that -n is only negative if n is positive. For example if n = -3, then n is negative and -n is positive. It's a tad confusing, but if you get confused, just ask Tad.

• ### Subtraction

Subtraction is really another way of showing that you're adding the additive inverse. In Subtractionland, it's always opposite day. Or is it?

For example, 4 – 3 is just another way of writing 4 + (-3).

Subtraction exists so that instead of writing a plus sign, a negative sign, and parentheses, we can just write a minus sign. Because mathematicians are generally lazy, and they don't want to have to go writing all those extra symbols if they can help it. In fact, the symbol "₳" means "please bring me a donut so I don't have to get up from the couch."

On the number line, subtraction means walking to the first number, then walking in the opposite direction specified by the second number. Told you it was opposite day. Or did we?

What does it mean to subtract a negative number?

### Sample Problem

4 – (-18) = ?

Well, adding (-18) would mean we walk 18 to the left, so subtracting (-18) means we walk 18 to the right. So 4 – (-18) really means 4 + 18 = 22. Think of it this way: if you take the minus sign and the negative sign and cross them, you get a plus sign. If you found the preceding sentence helpful, please check this box: □.

Another way to approach this problem is to remember what subtraction is abbreviating, then rewrite the whole thing as an addition problem:

4 – (-18) = 4 + (-(-18))

Since the negative of a negative is positive, -(-18) = 18.

4 – (-18) = 4 + 18 = 22

Be Careful: Honestly, the notation here can get really confusing. So take off your confused cap and put on your unbefuddled derby. A minus sign and a negative sign look similar, but mean different things. The trick is to see whether the little horizontal line has to do with one number or two.

A negative sign is a horizontal line in front of just one number, and it tells us to reflect the number across zero on a number line:

-5

-(-3)

A minus sign is a horizontal sign in between two numbers, and it tells us to walk to the first number, then walk in the opposite direction of the second number:

4 – 5 = -1

To help avoid uncertainty, you can put parentheses around negative numbers. For example, write (-4) instead of -4. Hopefully you don't use a lot of emoticons when doing your math homework, because something like this can get awfully confusing:

(-:3 - 7;-0):-)

You can also make your negative signs smaller and higher up than minus signs. Don't put them so high up that you can't get them back down when you need them, though.

Be Careful: Subtraction does not commute! (It works from home.) This is because subtraction changes the direction we're walking on the number line for the second number only.

8 – 10 = -2
10 – 8 = 2

So 8 – 10 ≠ 10 – 8.

Subtraction is also not associative. The order in which we perform multiple subtractions changes the final answer.

(3 – 4) – 2 = -3
3 – (4 – 2) = 1

You're always going to want to perform subtraction from left to right: 3 – 4 – 2 = -3. This should be easy to remember, because that's also the direction in which you read, as well as the direction in which you open the chocolates on your Advent calendar. Or, if you're Jewish, the direction in which you light the candles of your menorah. Or, if you're Buddhist, the direction in which you align your chakras.

Another way to think about this is that subtraction is really just the addition of a negative. This way, you can rewrite the problem as:

3 + (-4) + (-2)

Now it's an addition problem, so it doesn't matter what order you add them in!

If we're subtracting a bigger number from a smaller number (for example, 13 – 25), one way to find the answer is to pretend the problem is structured in reverse: evaluate 25 – 13 to get 12. Since we know 13 – 25 will be to the left of zero, stick a negative sign onto the front of it to get -12. Make sure you use Gorilla Glue so that thing really stays on there.

The reason this method works is that whether you go 13 to the right and 25 to the left, or 25 to the right and 13 to the left, you'll end up the same distance from zero. To give the correct final answer, use common sense to figure out which side of zero the answer is on. If you're lacking in the common sense department, then maybe don't use this method. Also, don't rest your palm on one of the eyes of a stove when it's glowing orange.

• ### Multiplication

So far, we've been working a lot with the number line. Now it's time for some pictures in a different dimension. Imagine, if you will, a bunch of boxes. If p and q are positive, the multiplication p × q means "take p groups of q things and see how many things you end up with." This can be pictured nicely with a rectangle divided into smaller boxes.

### Sample Problem

3 × 4

Here's one group of 4 things: ...and here's 3 groups of 4 things: Count up all the things (each of the small boxes) and we get 12. Notice that another way to think of this picture is that we took a rectangle with one side of length 4 and one side of length 3, and then found the area of the rectangle. But that way's not as much fun, because then we don't get to overuse the word "things."

Even if the numbers aren't all whole numbers, a similar diagram will still work.

### Sample Problem

3.5 × 6.2 Simply draw a rectangle with side lengths 3.5 and 6.2, and 3.5 × 6.2 is the area of that rectangle. If we're thinking about it in a real-life scenario, imagine that we have 3.5 of something and 6.2 of something else. Let's just hope they're not living things, or this could get messy.

If we switch the order of the numbers being multiplied, our rectangular box gets turned onto one end (apparently the multiplication symbol didn't notice the "FRAGILE" warning written on the side of it), but the final answer is still the same. 4 × 3 = 12

This means that multiplication is commutative: the order in which we write the numbers doesn't matter. They can "commute" past each other without changing the final answer.

a × b = b × a

Man, a and b sure are lucky they're at the beginning of the alphabet. They get to be in everything.

Multiplication is also associative: we don't care how the numbers associate with each other. In symbols, for real numbers a, b, and c:

(a × b) × c = a × (b × c)

At least c got a little love that time. We still can't help but feel bad for w though.

### Sample Problem

5 × 2 × 3

With multiplication, we can start from either end. Either way we slice it, we'll still get the same answer.

(5 × 2) × 3 = 10 × 3 = 30
5 × (2 × 3) = 5 × 6 = 30

If one or more of the numbers we're multiplying together has a negative sign, we first multiply the absolute values of the numbers together. Then, for each negative sign, we reflect our answer to the other side of the number line. If we come across enough negative signs, we might find ourselves doing more reflecting than Lindsay Lohan during her next jail sentence.

### Sample Problem

2 × (-3)

First multiply 2 by 3 to get 6. Since we have just one negative sign, we reflect 6 to the other side of the number line once to get our final answer: -6. Kind of hard to draw the rectangle for that one, so we recommend that you don't even try.

### Sample Problem

(-6) × (-4)

First multiply 6 by 4 to get 24. Since we have two negative signs, we reflect 24 across 0 and back again. Our final answer is positive 24.

There are several different symbols used for multiplication. We can write "a times b" as:
a × b
ab
(a)(b)

Be Careful: 4 × - 3 doesn't mean anything. It might have some sentimental value to you, but it certainly doesn't mean anything mathematically: "4 times subtract 3?" What in the world is that? Let's put parentheses around the second term to keep track of the negative sign:

4 × (-3)

Also keep in mind that it's really easy to multiply by the number 1. We say 1 is the multiplicative identity, because multiplying by 1 allows numbers to keep their identities. Even if some of them are a little embarrassed by their identities and would prefer to trade them in. Like 37, for example. Terrible self-image.

1 × 8 = 8
9 × 1 = 9
1 × 1 = 1
-π × 1 = -π

In symbols, we say that if n is any real number, then:

1 × n = n × 1 = n

Any real number n, except for 0 (sorry, 0, you'll get 'em next time), has a multiplicative inverse or reciprocal, written 1/n. This is the number that, if we multiply it by n, we get back to 1.

### Sample Problem

What's the multiplicative inverse of 2?

The multiplicative inverse of 2 is , because . Got two bagel halves? We bet they'll make a full bagel if you press them together.

So why doesn't 0 have a multiplicative inverse? Can't a number get a break around here? The number 0 is special in multiplication, because 0 "kills" everything. Wow. Real nice, 0. That's the last time we ever stick up for you.

For any real number n:

0 × n = n × 0 = 0

This means there can't be any real number n where 0 × n = 1. Zero kills everything because if we take n groups of zero things, or zero groups of n things, we don't have any things at all. Or at least that's the defense zero's lawyer is using at trial.

This can be very useful. If you're multiplying a lot of numbers together and one of the numbers happens to be 0, you don't have to do any work. Since anything multiplied by 0 is 0, all you have to do is write down 0 as your answer! Or, if you're taking the SAT, fill in that 0-shaped bubble next to the 0.

### Sample Problem

789,234 × 67,623,746,374 × 23,432,432 × 0 × 234,872,384,723 = 0

That lone 0 in the mix turns the entire product into 0, no matter how gnarly the other terms are.

• ### Division

In symbols, "p divided by q" can be written as , or p ÷ q.

Remember the names for the different parts of a division problem:

• The dividend, p, is the thing being divided up. And sprinkled onto your salad, if you so choose.

• The divisor, q, is the thing that performs the dividing. It may also perform the conquering.

• The quotient is the answer.

Here's a little insider info: "p divided by q" means the same thing as "q divided into p." When the numbers p and q are both whole numbers, division can be thought of as dividing up a bunch of things into smaller groups in either of two ways:

• If we split p into groups of size q will be how many groups (and perhaps fractions of a group) we get. By the way, "faction" is a synonym of "group," so you technically may be looking for the fraction of a faction. We hope you're not mad at us for throwing that in there just to torture you. We wouldn't want there to be any friction.

• If we split p into q groups, will be the size of each group.

For instance, "20 divided by 4" could mean "20 split into groups with 4 things in each group" or "20 split into 4 groups."

The idea is the same when p and q are real numbers instead of whole numbers, except that now the groups can have partial objects. Like if you're grouping chocolate chip cookies and some of them appear to have nibbles taken out of them. There's also the matter of signs, which we didn't have to worry about with whole numbers.

When dividing one positive real number into another, we just do the division like normal: 4.2 ÷ 2.1 = 2.

When one of the real numbers (either the divisor or the dividend) has a negative sign, perform the division while ignoring the signs, and then afterward reflect your answer across 0 on the number line. You'll always put those negative signs back on as the finishing touch. The icing on the cake. The Hershey's syrup on the Cracklin' Oat Bran. (We haven't actually tried that but we're guessing it would be delicious.)

-4.2 ÷ 2.1 = -2

= 4.2 ÷ (-2.1) = -2

If both the divisor and dividend have negative signs, perform the division and ignore the signs. The answer needs to be reflected across 0 twice, which gets us the same answer as if both the divisor and dividend were positive. In this case, we can simply ignore the negative signs entirely. Which is rude, we guess, but whatever.

-4.2 ÷ (-2.1) = 2

Remember that division is an abbreviation for multiplying by the multiplicative inverse of a number. This is most useful when fractions are involved. Check it: Here's the thing, though: division is not commutative.

For example, is different from . Both are correct, but they're saying different things. Like your parents when they argue. See, that's the problem.

Division is also not associative. Let's see why.

### Sample Problem

1 ÷ 2 ÷ 3

If we evaluate 1 ÷ 2 first, we get one thing: But if we evaluate 2 ÷ 3 first, we get something else: So always remember to go left to right when dividing multiple terms.

Be Careful: Division by zero is undefined, which we talked about in the section on rational numbers. If you ever feel tempted to divide by zero even though you know it's wrong, we're pretty sure there's a hotline you can call.

• ### Long Division Remainder

Here's a refresher on long division, by way of example. We're hoping monkey see, monkey do. Not that you in any way resemble a monkey. Well, maybe a little around the eyes.

### Sample Problem

What is 250 divided by 4?

We need some words to help us refer to different parts of the division problem. If you've read the sections leading up to this one, they're pretty well drilled into your brain by now, but if not, here you go:

• The dividend is the thing being divided up. Think of it this way: the word "dividend" ends with "end," and your end is divided up into two parts.

• The divisor is the thing that performs the dividing. Definitely the alpha male in this relationship.

• The quotient is the answer. All hail the quotient. Another useful word is multiple. A "multiple of 4" is any number that's evenly divisible by 4. For example, 4, 8, 12, and 16 are multiples of 4. We have to mention here that 396 is also a multiple of 4. He asked us to, and we owed him a favor.

To do the division problem, we're going to work our way across the dividend from left to right, and see how many times the divisor "fits" into each part of it.

First, consider just the first digit of the dividend. Above that digit, we'll write the number of times the divisor fits into that digit: Hmm...4 doesn't fit into 2 at all (not without a healthy dose of elbow grease), since 4 is bigger than 2. So we write a 0 in that space: We then subtract the multiple of 4 that fits into 2. In this case, that's 0: We address the next digit of the dividend by "carrying it down." Lift with your legs, not with your back. Now, we write how many times 4 fits into 25 in the space above the 5 in the dividend. Since 24 (4 × 6) is the largest multiple of 4 that fits into 25, we write 6 in that space: Then we subtract the multiple of 4 that fits into 25: Now we deal with the next digit of the dividend by "carrying it down": Since 4 fits into 10 twice, we write 2 in the next space of the quotient and then subtract 8 (4 × 2): While it appears we're out of digits in the dividend, we're not. Looks like we had an extra emergency supply in the basement. We can stick a decimal point onto the end and as many zeros as we need to the right of that. Remember to write the decimal point in the quotient, too. Then we can "carry down" any extra zeros: We see 4 fits into 20 an even 5 times, so we can write the appropriate number in the appropriate place: We stop when the subtraction results in a zero. The quotient is our "final answer," Meredith.

250 ÷ 4 = 62.5

We can check our answer by making sure that the quotient multiplied by the divisor equals the dividend: 62.5 × 4 = 250. Since this equation is correct, we know we got the right answer! And we're having a good hair day! Will wonders never cease?

• ### Exponents and Powers - Whole Numbers

This sequence shows up a lot in math and computer science, so take note. Especially if you like computer science—you know, taking various chemicals in eye droppers and dripping them onto your PC and whatnot.

2
2 × 2 = 4
2 × 2 × 2 = 8
2 × 2 × 2 × 2 = 16
2 × 2 × 2 × 2 × 2 = 32

Writing out all these 2s gets boring quickly. Who wants to write out twenty 2s, all multiplied together? (If this is you, please put your hand down. No one can see you right now anyway.)

Thankfully, there's a shortcut. We write 2n, pronounced "2 to the n," "2 raised to the n," or "2 to the nth power," which all mean n copies of 2 multiplied together. And to help you remember that we're "raising it," we even literally raise it up a little bit next to the number we're multiplying. Aren't mathematicians thoughtful? They even sent you flowers on your birthday. Remember that?

If we've got 2n, that little n is called an exponent or power, 2 is called the base, and the process of raising a number to a power is called exponentiation. The numbers 2, 22, 23, and so on are called powers of 2. If you see something like 2love, that's the power of love.

Be Careful: When raising a negative number to a power, keep careful track of your negative signs. Clip and tag them if you have to. If it's the negative number that's being raised to the power, we get one thing:

(-2)4 = (-2)(-2)(-2)(-2) = 16

If not, we group it differently and get something else:

-24 = -(24) = -16

### Sample Problem

Jen wrote -32 = 9. What did Jen do wrong?

The negative sign isn't being squared, so the answer should be -9. It would only be positive 9 if we had (-3)2. We're really, really sorry if your name is Jen. It's a total coincidence, we swear.

### A Little Bit About Zero

If we raise 0 to any positive exponent, we still get 0. This makes sense, because if you multiply one or more copies of 0 together, you'll just get 0. Turns out it's hard for 0 to become anything other than 0. Even if he really applies himself.

Any nonzero number raised to the 0 power is 1. Think about it this way:

24 = 16
23 = 8
22 = 4
21 = 2

As the exponent drops by 1, the answer is divided in half. If we drop the exponent by 1 once more and divide the answer in half again, we get 20 = 1. We can't believe how many times you just dropped that exponent. Can't you be more careful?

So here's the deal:

20 = 1
30 = 1
150 = 1
(-36.25)0 = 1

It's 1s all the way down: raise any number to the power of 0, and the answer is 1.

Well, except for one weird exception. What's 00? Zero is a troublesome number. We want 0 raised to any power to be 0, but we also want any number raised to the 0 power to be 1. There's no way to win! This means that 00 is undefined. If it's not too late, don't think about this too hard. It'll make your head hurt.

### Multiplication

What is 25 × 27?

This means that you need to multiply 5 copies of 2 together, and then multiply that result by 7 copies of 2. That's a total of 12 copies of 2. So 25 × 27 = 212. Why so many copies of 2? What are you, passing them out at a meeting?

If we have the same base with two different exponents and we're multiplying these numbers, as in the above example, the exponents get added together. In symbols, if a, b, and c are real numbers, then:

ab × ac = a(b + c)

### Negative Exponents

So far, we've only looked at exponents that are positive integers. Let's try to figure out what a number would be when raised to a negative exponent.

Suppose we want to understand what 3-1 means. Let's use what we know about multiplying exponents. Since we add exponents during multiplication, 31 × 3-1 would be 31 + (-1) = 30 = 1. This tells us that 3-1 is the multiplicative inverse, or reciprocal, of 3. So . Did you follow that? If not, double back and read this paragraph again until it sinks in. It won't kill you.

Now what happens if we take bigger powers? Like 5-7, for example. In this case, we'll look at 57 × 5-7 = 57 + (-7) = 50 = 1. So 5-7 is the same as (1/5)7. Are you loving this stuff as much as we think you are?

### Division

What's 25 ÷ 22?

This means , so we're just canceling out two of our 2s. Buh-bye, guys. You shall be missed.

After reducing, our fraction equals 23.

In general, ab ÷ ac = a(b – c), because we start out with b copies of a, divide out c copies, and are left with b c copies.

Heads up, though: a can't be 0.

Notice that if b > c, you're left with a positive exponent. But if b < c, you have a negative exponent. Which shouldn't stress you out any, as you now know what to do with them.

### Sample Problem

What's 42 ÷ 44?

This translates to: See what we did there on the end? Always look for ways that an expression can be further simplified.

### Sample Problem

What's 63 ÷ 67?

What this really means is "3 copies of 6 divided by 7 copies of 6": Cancel out 3 copies of 6 from the top and bottom of the fraction to get .

Be careful: In order to use the properties above, the base of the exponents has to be the same. For example, we can't combine 43 and 52. That's unfortunately as nice as it gets with exponent notation. Which isn't very nice. Hear that, Santa?

### Exponentiation

What is (25)3?

This really means (2 × 2 × 2 × 2 × 2)3. You can't just add the 5 and the 3 together in this instance, because what we're actually being asked to do is take 3 copies of (2 × 2 × 2 × 2 × 2), or 15 copies of 2 multiplied together. Looks a little like we're going to be multiplying exponents here. In fact, it looks a lot like that.

(25)3 = 25 × 3 = 215

So, in general, (ab)c = ab × c.

### Raising Products to a Power

What's (6 × 7)3?

Obviously we could just multiply 6 by 7 to get (42)3, but let's see what happens if we leave 'em separated.

(6 × 7)3 = (6 × 7)(6 × 7)(6 × 7) = 63 × 73.

In general, if a and b are real numbers and c is a whole number, (a × b)c = ac × bc.

### Raising Quotients to a Power If a and b are real numbers and c is a whole number, . Just slap that exponent on the numerator and the denominator separately.

• ### Prime Factorization

A prime number is a number greater than 1 that's only divisible by itself and 1. It's like someone fed it into a factor compactor.

Here are some examples:

1 is not prime.
2 is prime.
3 is prime.
4 is not prime because it's divisible by 2.
5 is prime.
6 is not prime because it's divisible by 2 and 3.

As it turns out, 2 is the only even number that's prime. Whoop-dee-doo for number 2. For any other even number n, 2 divides into n, so n is not prime.

Click here to see a list of the first 1000 primes. It's good to be able to recognize the prime numbers at least up to 31 or so. However, if you want to memorize all 1000 of them, we won't stop you. Having the ability to rattle them off will be a great party trick, if nothing else. By the way, how do you get invited to such cool parties?

Every single whole number can be written uniquely (in only one way) as a product of primes.

For example, 12 breaks down like this:

12 = 2 × 2 × 3

We can reorder the product and write 12 = 2 × 3 × 2, or 12 = 3 × 2 × 2, but we can't write 12 as a product using any other prime numbers. We have to use two copies of 2 and one copy of 3. It's the law.

To find the prime factorization of a number, you can "pull out" one prime at a time. Put your back into it.

We'll illustrate what this means by an example, mostly because we're terrible at drawing stuff.

### Sample Problem

Find the prime factorization of 120.

Okay, 120 is divisible by 2, so first we "pull out" a 2:

120 = 2 × 60

Always look first to see if you can pull out a 2. That's always our "prime suspect." Oh, sure, groan away.

Now we move on to the 60. Since 60 is also divisible by 2, we "pull out" 2 from 60:

120 = 2 × 2 × 30

Ah, but 30 is also divisible by 2:

120 = 2 × 2 × 2 × 15

So far, so good. We can't divide that 15 by 2, but we can divide it by 3 because 15 = 3 × 5, and 3 and 5 are both prime numbers.

120 = 2 × 2 × 2 × 3 × 5

We've reached the end at last. All those factors are now prime numbers, so we can't split 120 up any further.

Another way to find the prime factorization of a number is to simply recognize the number as a product of two smaller numbers, and factor each of the smaller numbers. Better recognize.

### Sample Problem

What's the prime factorization of 200?

200 = 20 × 10
= (4 × 5) × (2 × 5)
= (2 × 2 × 5) × (2 × 5)
= 2 × 2 × 2 × 5 × 5

Since there's only one way to write any particular number as a product of primes, it doesn't matter what method you use to find those primes. There are certain methods that are slower, such as counting out that number of pennies and then dividing them into neat, even piles, but you'll still arrive at the correct answer eventually. Now get yourself to a Coinstar, you penny hoarder.

• ### Order of Operations

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, Exponents, 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.

### Sample Problem First we evaluate things in parentheses. Uh, we can't simplify (4) any more than it already is, so let's move on to the exponents. Then multiplication and division:

6 – 1

5

When adding and subtracting, 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 Problem

What's 4 – 6 – 2?

We don't have any parentheses, exponents, multiplication, division, or addition, so we jump right away to subtraction. As usual, though, we've gotta move left to right.

4 – 6 – 2 =
(4 – 6) – 2 =
-2 – 2 = -4

Notice that, if we'd subtracted 6 – 2 first, we would get a totally different (and wrong) answer:

4 – (6 – 2) =
4 – 4 = 0

We also work from left to right when evaluating multiplication and division.

### Sample Problem

What is 3 × 4 ÷ 2 ÷ 6?

We only have multiplication and division here, so let's roll along from left to right.

3 × 4 ÷ 2 ÷ 6 =
12 ÷ 2 ÷ 6 =
6 ÷ 6 = 1

If we worked from right to left, we would get a different answer: 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.

### Sample Problem Now work out each of the pieces: =
6 + 2 – 0 – 4

Then combine the answers to the pieces:

6 + 2 – 0 – 4 = 4

### Sample Problem Yeesh, what a beast. Let's break it down into smaller chunks, each separated by a plus or minus sign (since addition and subtraction come last). =
3 + 6 × 16 – 6 × 1

Now we handle that multiplication.

3 + 6 × 16 – 6 × 1 =
3 + 96 – 6

And finally, we rock the addition and subtraction, left to right.

3 + 96 – 6 = 93

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 subtraction = add a negative. Yeah, either way, you're going to have to remember some stuff. C'est l'algebra.

Let's work out one last example.

### Sample Problem Ok, so let's start by dealing with the stuff inside the parentheses: . We wanna change it into something more manageable before we square it. 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've 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's 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!

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