Integers
5.1 Addition and Subtraction of Integers
What Integers Are
Historically speaking, natural and whole numbers were good enough for a very long time. But when commerce became more commonplace and more sophisticated, it became necessary to keep track of debts and credits. The need to represent a negative quantity led to the development of integers, numbers that can have direction. Negative numbers are most easily understood as representing the opposite of the number. A negative number is denoted by a $-$ sign in front of it and is in the opposite direction of a positive number. For example, $-2$ is the opposite of $2$. Whole numbers together with their opposites form the integers.
Definition: Integers and Additive Inverse
The set of integers is
$$\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}.$$
For any integer $a$, the opposite of $a$ is $-a$. The opposite is formally called the additive inverse because adding a number and its opposite result in 0.
Fact: Opposites
For any integer $a$:
- $a+(-a)=0$
- $-(-a)=a$
- 0 is its own opposite
Example: Finding opposites
Find the opposite of each number: $3$, $-5$, $0$, $23$.
Show Solution
The opposite flips sign and keeps size.
- Opposite of $3$ is $-3$.
- Opposite of $-5$ is $5$.
- Opposite of $0$ is $0$.
- Opposite of $23$ is $-23$.
A quick check is to add a number and its opposite and confirm the result is 0.
MyOpenMath: MOM: Integers and opposites
Absolute Value
When two numbers are opposites, they are the same distance from zero on the number line. They are the same size, but in opposite directions. This notion of size, or distance from zero, is called the absolute value of a number.
Definition: Absolute Value
For an integer $x$,
So $|x|$ is the distance from 0 to $x$ on the number line.
Fact: Absolute Value Facts
For integers $a,b$:
- $|a|\ge 0$
- $|a|=0$ iff $a=0$
- $|-a|=|a|$
Example: Evaluating absolute values
Evaluate: $|20|$, $|-5|$, $|0|$, solve $|x|=3$, solve $|x-5|=1$.
Show Solution
- $|20|=20$
- $|-5|=5$
- $|0|=0$
- $|x|=3$ gives $x=3$ or $x=-3$
- $|x-5|=1$ gives two cases:
- $x-5=1\Rightarrow x=6$
- $x-5=-1\Rightarrow x=4$
So the final solutions are $x\in{4,6}$ for the last one.
MyOpenMath: MOM: Absolute value
Negative numbers are often used to represent debt, or a deficit of some kind.
Example: Debt interpretation
If Omar has balance $-50$USD, what does that mean?
Show Solution
A negative balance means he owes money, not that he has money available.
- Balance: $-50$USD
- Debt magnitude: $|{-50}|=50$
So Omar owes $50$USD.
Integer Addition
Chip Model
Representing and teaching integers can be difficult, as students struggle with the idea of negative numbers. One way to help students is with a physical model know as the chip model. In the chip model, chips of different colors are used to represent positive and negative values. The net total of the chips in a given pile is the value of the number. For the ease of reading, I will represent positive chips with $\oplus$ and negative chips with $\ominus$. Lets begin by just representing a single integer a few different ways.
The number $3$ can be respresented in the simplest way as $3$ positive chips, or
$$\oplus \oplus \oplus$$
However, added an additional $\oplus$ and $\ominus$ chip would not change the value of the number.
$$\oplus \oplus \oplus \oplus \ominus$$
still represents $3$, because the single $\oplus$ and $\ominus$ negate each other. A number ands its opposite cancel out. This can come as a downside, as numbers represented using a chip model are not unique. But, its actually an upside as we'll soon find out. Here's one more way to write $3$.
Example: Represent Integer with Chip model
Represent each of the following integers using a chip model.
- $5$
- $-3$
Show Solution
- $\oplus \oplus \oplus \oplus \oplus$
- $\ominus \ominus \ominus$
Representing integers this way is useful because we can give a physical representation to adding negative numbers.
Example: Chip model for integer addition
Use a chip interpretation to evaluate $4+(-7)$.
Show Solution
Start with $4$ positive chips.
$$\oplus \oplus \oplus \oplus$$
Now we add in $7$ negative chips.
$$\oplus \oplus \oplus \oplus$$ $$\ominus \ominus \ominus \ominus \ominus \ominus \ominus $$
Now we can remove pairs of $\oplus$ and $\ominus$ to cancel them out.
$$\cancel{\oplus \oplus \oplus \oplus}$$ $$\cancel{\ominus \ominus \ominus \ominus} \ominus \ominus \ominus $$
Which leaves us with
$$\ominus \ominus \ominus$$ which is a representation of $-3$. So it must be the case that $4+(-7)=-3$.
MyOpenMath: Chip Model Addition
Number Line Model
The number line model is another way to represent integers. We simply add arrows going in the negative direction, left of zero, to represent negative numbers. It can also be used as a physical model where students imagine that they are standing on the number line and to represent a number they walk forwards or backwards a number of steps depending on the sign of the integer. The rules to this method are as follows:
- Always start at zero, facing the positive direction.
- If the number is positive, walk forward that many steps to represent it.
- IF the number is negative, walk backwards that many steps to represent it.
Example: Number line for addition
Compute $(-6)+(-2)$ using movement language.
Show Solution
Start at 0.
- Add $-6$: move 6 units left to $-6$.
- Add $-2$: move 2 more units left to $-8$.
You end at $-8$, so the sum is $-8$.
MyOpenMath: Chip Model Addition
Properties of Integer Addition
Just like whole numbers, integers also have properties for addition. Many of them are the same, although we have a new addition in that every integer has an additive inverse.
Fact: Properties of Integer Addition
For integers $a,b,c$:
- Closure: $a+b\in\mathbb{Z}$
- Commutative: $a+b=b+a$
- Associative: $(a+b)+c=a+(b+c)$
- Identity: $a+0=a$
- Additive inverse: for each $a$, there exists $-a$ with $a+(-a)=0$
Example: Applying addition properties
Give an example of each property of addition given above using your preferred integer addition model.
Show Solution
Lets stick with the chip model for this solution. Lets also pick some small numbers, say $a,b,c=1,-2,3$
For closure, this one is vacuous. Meaning, theres nothing to check. If I take some chips and add more chips, I'll always be able to count the chips up after canceling out pairs. Theres no way for me to get a representation that is not another integer. Adding itegers will always yield another integer.
To show $a+b=b+a$ we can use $1+(-2)=(-2)+1$. Observe
$$\oplus + \ominus \ominus$$ and $$\ominus \ominus + \oplus$$
both represent the same integer, $-1$.
To show $(a+b)+c=a+(b+c)$ we can show $(1+(-2))+3=1+((-2)+3)$
Parenthesis says we should compute $(1+(-2))$ first,
$$\oplus + \ominus \ominus = \ominus$$
Which gives us $-1$. now we add $3$. $$\ominus \oplus \oplus \oplus = \oplus \oplus$$
to get out final answer of $2$. To compute the other grouping $1+((-2)+3)$ we start with $((-2)+3)$.
$$\ominus \ominus + \oplus \oplus \oplus = \oplus$$
Which gets us $1$. Now we add $1$ to it,
$$\oplus + \oplus$$
Leaves us with $2$ again! So the grouping didn't matter.
$1+0$ is easy enough. Adding nothing to $\oplus$ leaves it the same.
And finally, the additive inverse.
$\oplus + \ominus$ represents $1+(-1)=0$ which indeed cancels out.
MyOpenMath: Integer Addition
Integer Subtraction
Subtraction has a nice intepretation in integers, because we can define it as adding the opposite of a number. When we define it as addition, we get the keep the commutative and associative properties that whole numbers lose with subtraction.
Definition: Integer Subtraction
For integers $a,b$, the difference $a-b$ is the unique integer $n$ such that
$$b+n=a.$$
Fact: Subtraction as Addition
For all integers $a,b$,
$$a-b=a+(-b).$$
Example: Rewrite as addition, then evaluate
Compute each:
- $3-10$
- $-3-8$
- $20-(-5)$
Show Solution
Rewrite first:
- $3-10=3+(-10)=-7$
- $-3-8=-3+(-8)=-11$
- $20-(-5)=20+5=25$
This is why subtraction with integers is mostly sign management after the rewrite.
MyOpenMath: Integer Subtraction
Chip Model for Subtraction
For our chip model, you can take two approaches to modeling subtraction. We'll skip subtracting positive numbers, because thats just using natrual numbers and start with a positive number minus a negative number.
Example: Chip Model Subtraction
Use the chip model to subtract $4-(-2)$
Show Solution
We start like normal by representing the $4$ with chips. $$\oplus \oplus \oplus \oplus$$
What we would do next is remove the next number for subtraction, but what we are removing is two negative chips, and there are no negative chips! But thats OK because we can add some negative chips in to remove as long as we add in positive chips to balance it out.
$$\oplus \oplus \oplus \oplus \oplus \oplus \ominus \ominus$$
There, now we have a representation of $4$ from which we can take away $-2$.
$$\oplus \oplus \oplus \oplus \oplus \oplus$$
Alternatively, we can think about $4-(-2)$ as $4$ plus the opposite of $-2$ which would be $4+2$, which is clearly $6$.
MyOpenMath: Chip Model for Integer Subtraction
Number Line for Subtraction
As for our numberline, we can think of the subtraction operation as turning around before continuing. Recall addition works by continuing to walk in the same direction:
$$2+(-3)$$
Would mean:
- Start at $0$
- Walk forwards $2$ to represent $2$
- Walk backwards $3$ to represent $-3$
Which leaves us at $2+(-3)=-1$. Where subtraction would look something like this:
Example: Integer Subtraction with Numberline
Evaluate $4-(-2)$ using a numberline.
Show Solution
- Start at $0$
- Walk forwards $4$ to represent $4$.
- We see a minus sign so we turn around.
- Walk backwards $2$ to represent $-2$
And we end up on $6$ to show $4-(-2)=6$.
MyOpenMath: Numberline Model for Integer Subtraction
Example: Simplify expressions
Simplify:
- $2-(5-x)$
- $5-(x-3)$
- $-(x-y)-y$
Show Solution
Change the subtractions to adding the opposite:
- The opposite of $5-x$ is $-5+x$ so, $$2-(5-x)=2-5+x=x-3$$
- The opposite of $x-3$ is $-x+3$ so, $$5-(x-3)=5-x+3=8-x$$
- Once more, $$-(x-y)-y=-x+y-y=-x$$
Always slow down at parentheses with a leading minus sign.