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We use numbers to solve problems. A simple problem like "If Ellen has 3 walls on which to watch the paint dry, and she watches 1 of them, how many walls are left to watch?" has a simple answer. As problems get more interesting, as surely they must, so do the numbers we need in order to solve them.
We'll go over each and every one of the main types of numbers you'll need in an algebra class. We'll cover natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. We're going to skip over wrong numbers, because you can just hang up on them anyway.
We'll also talk about different ways to express the numbers we already do know, usually in the form of fractions, decimals, and percents. Because it can get boring to always write "1" as "1," and sometimes we just need to write it as 432,588 ÷ 432,588 to keep life interesting. Don't put that as the number of dependents on your 1040, though. The IRS tends to find such shenanigans unamusing. Lighten up, IRS.
If someone lived in ancient Egypt and wanted to count how many cats she had (cat ladies were in large supply back then), all she'd need would be the set of natural numbers, also known as counting numbers. These are, as the second name implies, the numbers 1, 2, 3, 4, etc. These numbers are very useful for counting things, as you might imagine. Who would've guessed, right? We'll try to make things a bit more interesting from here on out.
Natural Numbers on a Number Line
Let's try to answer the following questions using just the natural numbers:
John has 3 sheep. Mary gives him 4 more sheep. How many sheep does John have now? Do a little addition, and 3 + 4 = 7. Don't worry about Mary though...she held onto a little lamb.
John has 5 goats. He gives Mary 2 goats. How many goats does John have now? Answer: 5 - 2 = 3 goats. And he's got those listed on eBay, so hopefully he'll soon be 100% goat-free.
John has 5 cows. He gives Mary 5 cows. How many cows does John have?
Whoa, there, Bessie! Before you get a little too cocky and start shouting "zero" from the rooftops (which is dangerous, by the way...why not just open a window?), notice that 0 is not a counting number. We can't really answer this question as written. Unfortunately, in order to list 0 as an answer, we would need to be able to include whole numbers, which aren't all that different from natural numbers. Read on...
Zero is the only whole number that's not also a natural number. If we start there and take a look at our new collection of numbers, we have 0, 1, 2, 3, 4, and so on. You can easily remember this because it looks as if we literally added a hole (whole) to the beginning of the set. Not to mention that 0 and a hole have something in common in addition to their shape: they're a whole lot of nothing.
Whole Numbers on a Number Line
Now let's go back and try again to answer that question about the cows:
John has 5 cows. He gives Mary 5 cows. How many cows does John have? 5 - 5 = 0.
Well...that's a shame. Where's he going to get his whole milk? See how it always comes back to "whole"?
It probably seems obvious to you that 0 is a number, but the symbol 0 actually took quite a while to work its way into mathematics. Think about the Romans. They had their fancy Roman numerals, but they didn't have a symbol for 0. It took years for that concept to occur to them. Zero wasn't built in a day.
If you're fascinated by this topic, you're in luck, because the number 0 is the star of several books available for your reading pleasure:
The Nothing That Is: A Natural History of Zero, by Robert Kaplan
Zero: The Biography of a Dangerous Idea, by Charles Seife
Now that this makes sense to you and you've got it locked away inside your brain, we're going to throw you a curve ball. Isn't that nice of us?
Weirdly enough, mathematicians haven't been able to agree unanimously on whether or not 0 is, in fact, a natural number. The very fact that it took so long for anyone to think it up indicates to us that it's pretty unnatural, but who are we to say? Our point is this: if you're chatting someone up and the topic turns to natural and/or whole numbers, make sure you and your partner in discourse are on the same page when it comes to 0. Many a bestie has been lost in the midst of a heated natural number/whole number squabble.
Now that John has unloaded practically all of his animals on poor Mary, he feels like he should probably compensate her a little for taking them off his hands.
John has $50, but he promises Mary he'll pay her $70.
The question "how much money does John have?" doesn't have a clear answer in this case. Technically, John has $50; if he were to give all his money to Mary, he won't have any left, but he'll still be $20 in debt. Jeez, John. Ever heard of a credit card?
To deal with this sort of idea, we need a new piece of notation: the sign of a number. We're not talking about figuring out if John's money is an Aries or Sagittarius. This is math, not astrology. Lucky for you, because your outlook for this week is kind of bleak.
Anyway, back to the problem. John has $50, but he owes Mary $70. We represent this by the following equation:
50 – 70 = -20
We would read this as "fifty minus seventy equals negative twenty." The symbol "-" is called a negative sign. Notice that it looks suspiciously similar to the minus sign. Although the two symbols are related, they are different. They're like identical twins who have totally different tastes in clothes.
Be careful: John doesn't technically have "negative twenty dollars." Instead, the negative sign means that John is lacking 20 bucks that he needs to settle his debts. Hey, don't laugh at the idea; the U.S. is technically lacking 15 trillion dollars. Just think about how many sheep they had to give away to hit that mark. So if you find a briefcase containing that amount, please forward to the White House. There's a $200 reward.
Try this: take a nice, clean sheet of paper and draw a positive number line with 0 landing at the very edge of the page. Now hold your number line up to a mirror. You're not really doing this, are you? Spoilsport. Once you've gotten over the beauty of your own reflection, you'll see the natural numbers reflected in the mirror. All those numbers heading off into Mirrorland? Those are negative numbers. You'll just have to imagine the negative signs in front of them. Your mirror can only do so much.
So the numbers to the left of 0 are negative, and the numbers to the right of 0 are positive. But 0 is neither negative nor positive; 0 is just 0. It's non-committal. Moderate. If it were allowed to vote, it would probably do so for a third-party candidate.
If we take 0, the natural numbers, and their negative reflections, we get the integers. That's a pretty big group. If they got seated in a restaurant, gratuity would almost certainly be included.
Food for Thought
What are the non-positive integers?
Answer: 0 and the negative integers
What are the non-negative integers?
Answer: 0 and the positive integers
Think of the distance from one integer to another as being one step. If you start at zero and take two steps to the right, you get to 2. If you start at 0 and take two steps to the left, you get to -2. If you take two steps forward, we take two steps back. We come together, 'cuz opposites attract. In any case, we go the same distance.
Each integer (except 0) has two pieces of information: its distance from 0, and its direction from 0.
The distance of an integer from 0 is called its magnitude or its absolute value. We indicate "the absolute value of x" by putting vertical bars around x, like this: |x|. Because absolute value is basically a measurement rather than a number, it's never negative. Can you imagine how difficult it would be, for example, to track down drapes that would fit a window measuring -3 feet long? Not even the helpful people at Target would be able to help you out with that one.
The sign of the number is there to tell us which direction from 0 we're stepping. By the way, don't freak out; there's actually very little physical activity involved in solving these problems. If there's no sign in front of the number, it means the number is positive. If there's one negative sign in front of the number, it means we reflected the number in our mirror, so it's negative. Also, it looks like it's getting ready to brush its teeth.
Of course, things can get more complicated than that. We can go crazy with the negative signs and write things like -(-(-3)). Ever slipped up and said something like, "I don't want no socks for Christmas," and then your grammar-stickler uncle gets you socks just to prove a point? What you did was use a double-negative—because you said that you do not want no socks, that must mean that you do want socks.
Same thing with a number like -(-3), which is the same as positive 3. However, with three negative signs as in the earlier example, the number would once again become negative. Now it's like you're saying, "I don't want none of no socks for Christmas." Wow. You really need to spend some more quality time with your uncle.
Remember, one negative sign meant we reflected a number into the mirror once.
Two negative signs means we reflect the negative number back again, so now we're back to the right:
And we could reflect again to get the following:
Be Careful: Taking the negative of a number doesn't always give us a negative number, as the previous examples demonstrate. So before assuming a number is negative just because you see a negative sign, make sure that there's only one of them. If there are multiple negative signs, the number may be negative or positive, depending on how many negative signs there are. We know that you may not don't never can't not want to deal with too many negative signs, but it's just a part of life.
If we want to be extra clear that a number is positive, we can write an extra "+" in front of it. However, if there's no + or - sign, the number is understood to be positive. If a number is preceded by the symbol ©, then it's copyrighted. You may not reproduce, retransmit or rebroadcast this number without the express written consent of Major League Baseball.
What's the value of -|-5|?
We've got two negative signs and an absolute value sign. Let's work our way from the inside out. Remember, anything inside those absolute value bars is positive.
-|-5| = -(5) = -5
You didn't pull anything doing that exercise, did you? We warned you to stretch first...
Integers are fine and dandy, but not everything in this world works out to a perfectly even whole number. That pair of boots you've had your eye on may not be exactly $64. That recipe may not call for exactly 1 cup of sugar. And good luck finding a book in the library using the Dewey Whole Number System.
Take a look at the problems below, and see if you can solve them by using only integers.
1. Divide 25 cookies between 5 people. How many cookies does each person get? Answer: 5, assuming Santa isn't in the building.
2. Divide 100 pennies into piles of 10. How many piles do we have? That's 10. Make cents?
3. Divide 3 cheesecakes between 4 people. How many cheesecakes does each person get?
First of all, are you not aware of the obesity problem in this country? Are there no carrots in this hypothetical refrigerator?
Okay, so we can't answer this question with a whole number, because each person gets of a cheesecake. The number is an example of a rational number: a number that can be written as the quotient of two integers. Mathematicians say a rational number can be written as a quotient p/q where q is nonzero. We call p the numerator and q the denominator. When we write a rational number in the form p/q where q is not equal to 0 or 1, we call it a fraction. The reason we can't have q equal 0 is because we'd be dividing by 0, which gives us an undefined number. Go ahead—just try looking up 0/0 in the dictionary. You won't find it. And if we divide it by one, we've just got ourselves a whole number, and not a fraction. In this case, we don't even have to worry about minding our p's and q's.
Fractions are examples of division problems where we simply don't carry out the long division. We just let the two numbers kinda hang out there, one in the top bunk and one in the bottom bunk. They could use the rest. You could, too, if you spent so much of your time getting divided up into smaller pieces like that.
Even though we don't write integers as fractions, they're still considered rational numbers. This is because we can think of all integers as a quotient with a denominator of 1. For example, we can write 5 as 5/1, 11 as 11/1, and -25,693 as -25,693/1. Note that this doesn't necessarily work in reverse, as not all rational numbers are integers—for example: 5/2. This is the only example we could think of, but since there's literally an infinite amount of rational numbers where this is the case, we probably weren't trying very hard.
Not only can we think of 5 as 5/1, but we can also write it as 25/5, or 50/10. Similarly, we can write 10 as 100/10, or as 10/1. As you can see, there are many different fractions that can be used to express any particular rational number. We'll talk later about how to tell when two different fractions are representing the same number, but don't worry your little head about that for now. Just close your eyes, go to sleep, and dream of puppies and rainbows.
All right, so switching gears...why can't the denominator q be zero?
Think about it this way. If you have 3 cheeseburgers and divide them between zero people, how many cheeseburgers does each person get? First of all, who made you the Grand High Cheeseburger Distributor? And second, how come you don't get any? That seems sort of cruel.
Third, and more to the point, if there aren't any people at all, what does it even mean to "divide the cheeseburgers among people?" This question doesn't make sense. We can put zero into 3 any number of times that we want and still have plenty of zero leftover! This number just can't be defined by any single value.
If that example didn't drive the idea home, here's another. If you have p pennies and put them in piles of zero pennies each, how many piles do you have? No matter how many piles of zero pennies you have, you'll still have zero pennies. Which is depressing, because you're never going to be able to afford the latest, greatest, and dreamiest smartphone at this rate.
Even infinitely many piles of zero pennies will not get you to the right number. So this doesn't make sense either. Hope that makes sense. That it doesn't make sense, we mean. Oy.
Have your parents ever accused you of being irrational? Probably when you were doing something like throwing a hissy-fit in Toys R Us just because they wouldn't buy you a new Thomas the Tank Engine train, right? That was just last week.
Well, irrational numbers can be just as big a pain to deal with. They're a little weird, in that they can't be written as fractions. How in the world did anyone ever find a need for a number that can't be written as a fraction, you ask? You ask a lot of good questions.
Let's take a look at how and why mathematicians stumbled upon irrational numbers.
What number times itself equals 1? That's easy: 1 × 1 = 1. What number times itself equals 9? Boom: 3 × 3 = 9. What number times itself equals 2? Ahhh...now you see where we're going with this. There is no rational number that, when multiplied by itself, gives us an answer of 2. For this reason, mathematicians made up just such a number, and called it √2. Mathematicians don't generally have a ton of friends, so they have to resort to naming everything around them so they won't feel so alone. Sometimes they'll even take √2 out on walks.
Why do we need √2 so badly? Here's an example of when it may come in handy.
The Pythagorean Theorem states that a2 + b2 = c2. In other words, one leg of a right triangle multiplied by itself plus the other leg multiplied by itself equals the hypotenuse multiplied by itself. In a very basic right triangle where each of the legs has a measure of 1, the hypotenuse would be exactly √2, since 12 + 12 = (√2)2.
So say you work at Subway and you're in charge of cutting those perfect right triangle cheese slices. How in the world would you ever be able to do your job if you couldn't figure out the length of the cheese's hypotenuse?
The sequence 1/2, 2/3, 3/4, 4/5, 5/6,... gets closer and closer to 1, but as long as you keep following this pattern, you're never going to hit it precisely.
The sequence 1, 3/2, 7/5, 17/12, 41/29,... approaches √2, but √2 is not rational. This means that our set of rational numbers has a "hole" in it! If we try to fill in the hole, we're filling it with something that's not a rational number. And if we try to patch up the hole, we're probably going to run out of fabric. And we really need the rest of that in order to make those doilies.
Here are some of the holes in our number line:
By taking a look at different sequences we can find lots of other "holes" in the rational numbers, and each time we fill them in with something, we create an irrational number. Whenever possible, we should try filling it in with Crack-B-Gone liquid cement, because that stuff is amazing.
This is going to sound really crazy (like, crazier than filling in a sequence of numbers with liquid cement), but it turns out that there are a lot more irrational numbers than rational numbers. The reason this seems a little crazy is that we know there are infinitely many rational numbers, so how can there be more of anything else?
The thing is, you can systematically count rational numbers, but there are so many irrational numbers in between each pair of rational ones that there are just way too many to even systematically count them. For this reason, we say that even though both the rationals and the irrationals are infinite, the irrationals are a much bigger infinity than the rationals. How's that for brain stretching? Just when you thought you had this whole "infinity" thing figured out, right?
A couple other popular irrational numbers are π (approximately 3.14159...), which is the ratio of the perimeter of a circle to its diameter, and √5. What we're getting at is that irrational numbers are pretty abstract.
Real numbers are what we get when we combine all the irrational and rational numbers. These numbers are "real" because they're useful for measuring things in the real world such as money, distance, temperature, and Weight Watcher points.
Absolute values and negative signs work the same way for real numbers as they do for integers. We can still use the number line to think about these concepts, but now we can take partial steps. Quick, short little shuffles of the feet, if you will. Fractions, decimals, weird square roots, and even π are all happily having a picnic under the "real number" umbrella.
But here's a question: what number times itself equals -1?
Unfortunately, there's no real number that, multiplied by itself, gives -1 or any other negative integer, for that matter. Any positive real number multiplied by itself is positive, and any negative real number multiplied by itself is also positive: (-1)(-1) = 1, and so on. So if we want a number that creates a negative when multiplied by itself, we actually have to make up a new number—just pull one out of thin air. Whoever said there was no place for imagination in mathematics?
That's where the famous imaginary number i comes in. When you multiply i by itself you get -1.
i2 = -1
In other words, the square root of -1 is i. Now let's hop onto the back of our imaginary pet Hippogriff and fly on over to some exercises.