You use them every day. They’re all around us: in your house, on street signs, in the newspaper, at the store, on your parent’s bank statements, everywhere! What are these useful gadgets that help us to tell time, understand our finances, and keep track of so many things in our lives? What are integers?
Back to Basics: Counting
From an early age, most likely in your first year of life, your parents exposed you to the integers. Well to be fair, they probably started with the natural numbers first.
Natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …
The natural or counting numbers are what we use to enumerate sets of distinct objects. In other words, they are labels that answer the question, “How many?”. All natural numbers are integers, but not all integers are natural numbers (as we shall see).
There is no largest natural number (or integer, for that matter). For example, if you tell me that a trillion (1,000,000,000,000) is the largest number, then all I have to do is add 1 to get a bigger one: 1,000,000,000,001. In other words, there are an infinite amount of natural numbers.
(Some might say that infinity (∞) is the largest number, but that’s not quite right. Infinity is not a number at all! Instead, it’s a more complicated mathematical concept that we use to talk about numbers.
Adding Nothing to the Discussion
How many apples are in an empty bucket? “What a silly question!” you might be thinking to yourself. An empty bucket contains no apples of course! But there’s no natural number that describes the situation.
For a long time, humankind got along very well without a true concept of zero as a number. But ancient peoples in Mesopotamia and elsewhere began finding a use for “nothing,” especially when recording business transactions and tallying resources.
Some say that the earliest use of zero happened sometime between 400 BC and 300 BC (or even earlier, depending on whom you ask). But it took until the 5th century AD before mathematicians in India gave zero (0) the status of a full-fledged number.
The number system including both zero and the natural numbers is called the whole number system. All of the whole numbers are integers; in particular, 0 is an integer.
Whole numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …
As it turns out, the number 0 makes a great “starting point.” For example, after you were born but before your 1st birthday came around, you were 0 years old!
You Win Some, You Lose Some
As any gambler will tell you, sometimes you walk away richer, but more often you walk away poorer. So what’s the best way to keep track of wins and losses?
Suppose I walk into a casino and win $100 on Monday and then lose $20 on Tuesday. Assuming I didn’t spend any of the winnings yet, I should have $80 in my pocket by the end of Tuesday. This is simple arithmetic: $100 − $20 = $80.
But what if the situation was different? Suppose instead that I won only $20 on Monday and then lost $100 on Tuesday. How much money do I now have? In other words, what is $20 − $100? This time, I would owe the casino $80. A debt of $80 can be represented using negative numbers: −$80.
Every counting number has a negative (or opposite). Together with the whole numbers, the negatives comprise the remaining half of the set of integers.
Integers: … −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …
Notice that we place the negatives on the left of the positives. Also note that negative integers are just as infinite as positive integers.
Arithmetic on the Integers
Now that you know what an integer is, let’s talk about how to work with these numbers.
First of all, it’s important to know how to compare two integers. Just think of the number line. Left means less.
For example, −3 < −1 because −3 shows up to the left of −1 on the number line.
Be careful! It’s tempting to say that −3 is “bigger” than −1 because we all know that 3 is larger than 1, but negative numbers work in the opposite way. Which is better from your viewpoint, owing someone $1 (−1 dollars), or owing them $3 (−3 dollars)?
Adding and Subtracting Integers
There are a few rules to keep in mind when adding and subtracting integers. Basically, we need to be careful with our negatives.
- Positive + Positive = Positive
- Negative + Negative = Negative
- Positive + Negative or Negative + Positive turns into subtraction. The sign of the result will be the same as the sign of the number with the larger absolute value.
Remember, the absolute value of a number is the distance of that number from 0. For all practical purposes, that means just to ignore a negative sign. So |5| = 5, |0| = 0, and |−2| = 2, for example.
- 5 + 18. Easy peasy — two positives add in the usual way.
- −5 + (−18) = −23. Just add 5 + 18, and keep the negative sign!
- −5 + 18 = 13. Here, the different signs turn this into subtraction: 18 − 5 = 13. Keep the answer positive because |18| > |−5|.
- 5 + (−18) = −13. Again, different signs imply subtraction: 18 − 5 = 13. But now the answer is negative because |−18| > |5|.
Next, to do subtraction, simply rewrite the problem as addition.
A – B = A + (−B)
Don’t forget that in math, two negatives make a positive! In other words, −(−A) = +A.
Here are a few examples:
- 21 − 35 = 21 + (−35) = −14
- −7 − (−9) = −7 + 9 = 2
It helps to think in terms of money. Positive numbers are income (or winnings), while negative numbers represent debt (or losses). Two debts added together will result in a bigger debt (larger in the sense of absolute value, of course). Income and debt together will partially cancel one another out (by subtracting one from the other).
Even subtracting a negative amount makes better sense in terms of money. Why is subtracting a negative number the same as adding a positive? Well, if you owe me $10, and I tell you that I’m willing to forgive $6 of the debt (take away a $6 debt), how much do you owe me now?
−$10 − (−$6) = −$10 + $6 = −$4
Multiplying and Dividing Integers
The sign rules for multiplying and dividing are fairly straightforward.
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
In fact, it works the exact same way for division.
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Negative ÷ Positive = Negative
I like to think of a light switch. Multiplying by a negative number switches the light from on to off (+ to −) or from off to on (− to +).
In particular, multiplying by the number −1 only switches the sign of the number. And if you multiply by −1 twice, then that’s like flipping the light switch twice—the number goes back to its original sign.
5 × (−1) = -5
−5 × (−1) = 5
Here are a few more examples to help your understanding.
- 6 × (−3) = (−6) × 3 = −18
- 6 × 3 = (−6) × (−3) = 18
- −15 ÷ 5 = 15 ÷ (−5) = −3
- 15 ÷ 5 = (−15) ÷ (−5) = 3
The Special Case of the Integer Zero
Zero has been an enigma ever since it was introduced. The trouble is that sometimes it doesn’t seem to do anything, but in other cases it can have profound effects.
Zero in Addition and Subtraction
Zero (0) is the additive identity element for the set of Integers. Adding 0 to any other integer does not change its value.
Additive Identity Property: A + 0 = 0 + A = A
Related to this, every integer A has an opposite or (additive inverse), –A, that when added together with the original number results in 0.
Additive Inverse Property: A + (−A) = (−A) + A = 0.
These two properties become very important later on in mathematics. But the main takeaway here is that 0 acts like a neutral “starting point” for the set of integers.
By the way, if you subtract 0, you get the same effect as adding it.
A − 0 = A
However, reversing the order of the subtraction makes a big difference (no pun intended!). You obtain the opposite number.
0 − A = −A
Zero in Multiplication and Division
If you think that 0 isn’t very powerful because it can’t change the value of a number when you add it, well think again! Multiplying by 0 causes any number to become 0. Yes, our friend zero can produce an army of clones just by multiplying!
A × 0 = 0 × A = 0
It seems to work the same way when you divide 0 by any number, but there’s a catch.
0 ÷ A = 0, if A ≠ 0
In fraction notation, 0/A = 0, if A ≠ 0
However, division by 0 is undefined.
So, 0/1 = 0, and 0/(−3,450,120) = 0. But 1/0 and (−3,450,120)/0 are simply undefined. In fact, 0/0 is also undefined!By NASA images
- The integers consist of all natural numbers, their negatives, and zero
- To compare two integers, imagine them on the number line and remember that “left” means “less”
- When adding, different signs will result in a subtraction. The sign of the answer is the sign of the number having the larger absolute value.
- To subtract, just add the opposite
- Sign rules for multiplying and dividing are the same: (+)(+) = (−)(−) = +, and (+)(−) = (−)(+) = −
- Zero is the additive identity
- Zero times anything is zero, and zero divided by any non-zero number is zero
- Never divide by zero!