Integers and Negative Numbers
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