Absolute Value

Numbers such as [latex]5[/latex] and [latex]-5[/latex] are opposites because they are the same distance from [latex]0[/latex] on the real number line. They are both five units from [latex]0[/latex]. The distance between [latex]0[/latex] and any number on the number line is called the absolute value of that number.

Because distance is never negative, the absolute value of any number is never negative.

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[/latex] is written as [latex]\big |5\big |[/latex], and the absolute value of [latex]-5[/latex] is written as [latex]\big |-5\big |[/latex] as shown below.

 

In the video below we show another example of how to find the absolute value of an integer.

Addition of Integers

One way to think of positive and negative integers that may be helpful is to think about the numbers in terms of money.

Think of positive numbers as money you have and negative numbers as money you spend.  This will help you determine if your answer is positive or negative.  (-4) + 7 would be spending $4 and having $7, once you settle up, you still have $3.  So the answer would be positive 3.

Another example is (-3) + (-5).  This means you spend $3 and you spend an additional $5, so you have spent $8, which would be represented by -8.

If you have 6 + (-10) and we think in terms of money, you have $6 but you spend $10.  Once you settle up, you are in debt $4.  This gives you an answer of -4.

Another way to think about integers is yards gained or lost in a football game. Positive numbers represent yards gained, while negative numbers represent yards lost. (-4) + 7 means we lose 4 yards then gain 7 yards, for a net gain of 3 yards. So, (-4) + 7 = 3.

Likewise, (-3) + (-5) means we lose 3 yards then lose another 5 yards for a net loss of 8 yards. So, (-3) + (-5) = -8. And, 6 + (-10) means we gain 6 yards the lose 10 yards for a net loss of 4 yards. So, 6 + (-10) = -4.

We know that [latex]2+3=3+2[/latex]. But is [latex]-2+3=3+\left(-2\right)[/latex]? Well [latex]-2+3=1[/latex] and  [latex]3+\left(-2\right)=1[/latex]. So, [latex]-2+3=3+\left(-2\right)[/latex]. In fact, this is true for all integer values and is called the commutative property of addition. The order that we add integers doesn’t matter.

It is also true that adding zero to any integer has no effect on the integer. For example, [latex]-5+0=-5[/latex]. Because [latex]0[/latex] does not change the identity of any integer it is added to, [latex]0[/latex] is called the additive identity.

Adding more than two integers

When we add more than two integers, we add them two at a time.

Another way to solve the problem [latex]6+\left(-10\right)+\left(-7\right)+5[/latex] is to add up all the positive and negatives separately then add the sums. This requires reorganizing the problem using the commutative property to: [latex]6+5+\left(-10\right)+\left(-7\right)[/latex]. Then [latex]6+5=11[/latex] and [latex]\left(-10\right)+\left(-7\right)=-17[/latex]. Adding these sums gives: [latex]11+\left(-17\right)=-6[/latex]. The same answer as before. Regrouping the numbers is an example of the associative property of addition.

 

Subtraction of Integers

If we continue to think about integers as money, we can start to understand the subtraction process. If you have $5 and spend $3, you will have $2 left over. That can be written as [latex]5-3=2[/latex].

But if you have $5 and you spend $12, you end up in debt by $7. The debt can be expressed as a negative number. This can be written as [latex]5-12=-7[/latex].

What if you are already in debt and you continue to spend money. Suppose you are in debt by $10 and you spend $7 more. Now you are further in debt, and your debt totals $17. This can be written as [latex]-10-7=-17[/latex].

We can also think about the subtraction of money as the addition of debt. This leads us to start to think of subtraction as the addition of opposites. You will often see this idea, the Subtraction Property, written as follows:

Consider these examples.

If we have $6 and spend $4, we have $2 left.

If we have $6 and we add a debt of $4, we have $2 left.

This illustrates that [latex]6 - 4[/latex] is equivalent to [latex]6+\left(-4\right)[/latex].

Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already know how to subtract [latex]6 - 4[/latex]. But knowing that [latex]6 - 4[/latex] gives the same answer as [latex]6+\left(-4\right)[/latex] helps when we are subtracting negative numbers.

Now consider what happens when we subtract a negative.

Suppose we have have $8 and we subtract a $5 debt (which means we gain money). Subtracting a debt ultimately means that we're adding to the amount of money that we have.

[latex]8-\left(-5\right)[/latex] gives the same result as [latex]8+5[/latex]. Subtracting a negative number is like adding a positive. In other words, subtracting a negative is equivalent to adding the opposite, a positive.

In the following video we show another example of subtracting two digit integers.

Properties

When we looked at the addition of integers, we discovered certain properties. The additive identity, [latex]0[/latex]; the commutative property of addition; the associative property of addition.

[latex]-3+5=2[/latex] and [latex]5+(-3)=2[/latex], the commutative property of addition.

[latex]-4+(3+(-5))=-4+(-2)=-6[/latex] and [latex](-4+(3))+(-5)=-1+(-5)=-6[/latex], the associative property of addition.

But, beware, for if we leave the problems as subtraction, the properties do not apply!

[latex]3-7=-4[/latex] but [latex]7-3=4[/latex]. The commutative property does not work for subtraction!

[latex](8-3)-5=5-5=0[/latex] but [latex]8-(3-5)=8-(-2)=10[/latex]. The associative property does not work for subtraction!

We must write subtraction as addition of the opposite for the properties of addition to apply.