1.2 Working with Integers

Introduction

When working with positive integers, all of the basic operations of addition, subtraction, multiplication, and division work exactly as you've learned in elementary school and usually pose no problems.

It's when we incorporate negative integers that things sometimes get confusing and even seasoned mathematicians occasionally make mistakes.

In this section, we will go over the basic concepts so that you can generally avoid these issues.

Adding & Subtracting Integers

It's always helpful to link a concept in math with something you can visualize that has similar behavior.

Here's an image that might be useful. It is a box that has balloons lifting it up into the air and bags of sand pulling it down.

Suppose you wanted the box to go down, what could you do? There are actually two options. You can add sandbags or remove balloons. Either of those actions would make the box go down.

But what if you wanted the box to go higher? In that case, removing sandbags or adding balloons would do the trick.

If we imagine the balloons represent positive integers and the sandbags represent negative integers, then the rules for addition and subtraction will follow the same pattern.

Adding a negative integer (sandbag) is the same as subtracting a positive integer (balloon). In symbols, \(a+(-b) = a - b\).

Subtracting a negative integer (sandbag) is the same as adding a positive integer (balloon). In symbols, \(a-(-b) = a + b\).

If the balloon and sandbag analogy doesn't make sense to you, you can use similar scenarios like gaining or losing money, riding up or down an elevator, or walking forward or backward to help illustrate and remember the rules.

It can sometimes be helpful to see a few concrete examples, so here are five calculations involving positive and negative \(6\) and \(8\).

As you encounter exercises with negative integers be very careful to write down the signs of the numbers correctly. Keeping track of the signs of the integers is crucial to getting the correct answers.

Multiplying & Dividing Integers

The key to multiplying integers is to remember that multiplication is simply a shorthand notation for repeated addition. For example, \(2 \times 4\) is shorthand for two \(4\)'s added together, while \(3 \times 4\) is short for three \(4\)'s added together.

Using this concept, we can find the product \(2 \times (-4)\) by adding \(-4\) two times.

\begin{align} 2 \times (-4) &= (-4) + (-4) && \small \color{#5fa2ce} {\text{using the definition of multiplication}}\\ &= -8 && \small \color{#5fa2ce} {\text{using the rule for adding two negative integers}} \end{align}

Notice that the result is a negative number. Anytime we multiply a positive and a negative together the result will always be negative. This is because it is a repeated addition of negative numbers.

But what if we multiply two negative integers? Consider the product \((-2)(-4)\). If we rewrite negative \(2\) as the opposite of \(2\), taking advantage of the fact that \(-2=-(2)\), then the answer is easier to see.

\begin{align} (-2)(-4) &= -(2)(-4) && \small \color{#5fa2ce} {\text{since }(-2)=-(2)} \\ &= -(2(-4)) && \small \color{#5fa2ce} {\text{focusing on multiplying }(2(-4))} \\ &= -(-8) && \small \color{#5fa2ce} {\text{since }[2(-4)]=-8} \\ &= 8 \end{align}

As this example demonstrates, the product of two negative numbers is always positive.

Here are a few more examples:

We've been focusing on multiplication, but the process for division is the same. The reason is that division of one number by another is equivalent to multiplying by the reciprocal of the second number. For example, \(8 \div 2 = 4\) is equivalent to \(8 \cdot \frac{1}{2} = 4\).

Since any division \(a \div b\) can be written as the multiplication \(a \cdot \frac{1}{b}\), the sign rules for multiplication also apply to division.

Here are a few examples:

Division Involving \(0\)

Since \(0\) is an integer, we will take a moment to discuss two situations where division involves \(0\). The two cases are easily confused, but you can always check your answer by remembering that any division can be rewritten as multiplication. For example, \(\frac{\color{red}{15}}{\color{blue}{3}}=5\) because \(5(\color{blue}{3})=\color{red}{15}\).

To see how this helps, consider \(\frac{\color{red}{0}}{\color{blue}{2}}\), what would it equal? Since \(0 \cdot {\color{blue}{2}}=\color{red}{0}\) it must be that \(\frac{\color{red}{0}}{\color{blue}{2}}=0\). As a rule, \(0\) divided by any nonzero number is always \(0\).

But how about \(\frac{\color{red}{2}}{\color{blue}{0}}\), what does it equal? To find an answer we would need to identify the exact number \(n\) so that \(n \cdot {\color{blue}{0}}=\color{red}{2}\). But no such number exists since \(0\) times anything is still \(0\). We must conclude, therefore, that division by \(0\) is undefined.

Powers of Integers

We've seen that multiplication is a shorthand notation for repeated addition. There is also a shorthand for repeated multiplication: exponents. For instance, if we wanted to multiply the number \(5\) by itself \(6\) times, we can write either write it longhand as \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\) or use the much shorter notation \(5^6\).

In the example \(5^6\), the number \(5\) is the base and \(6\) is the exponent. The exponent tells us how many times the base is multiplied by itself. We might also say that \(5^6\) is \(6 \text{th}\) power of \(5\).

Notice that \(b^1\) means there is just one \(b\), so we do not usually write the exponent. Two powers have specific names that you will hear frequently. We often refer to \(b^2\)as "\(b\) squared" and \(b^3\) as "\(b\) cubed". These names come from geometry formulas for the area of a square and the volume of a cube.

When a positive integer is raised to a positive exponent the result is always a positive number, so there's nothing special to be aware of. However, when a negative integer is raised to an exponent the result could be positive or it could be negative—it depends on whether the exponent is an even number or an odd number.

Most calculators have an exponent button (which often looks like or depending on the model) to help you evaluate powers quickly. When using a calculator, you will want to take note of the parentheses. If there are parentheses around the negative base then the power applies to the entire integer.

If there are no parentheses then the power does not apply to the negative sign. It's very easy to confuse expressions like\(-3^2\) and \((-3)^2\), but the two are different:

\[-3^2=-(3^2)=-9\]

while

\[(-3)^2=(-3)(-3)=9\]

Use the examples below to practice evaluating powers of integers with your calculator.

Roots of Integers

We know that powers are used to evaluate expressions like \(5^2=25\), but what if we wanted to go in the reverse direction, start with \(25\) and getting back \(5\)? That is what roots do. Just as subtraction is the opposite of addition, roots are defined as the inverses of powers.

Using this definition, we can see that \(\sqrt{9} = 3\) because \(3 \times 3 = 9\), and \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\).

Here are a some other examples of roots.

Special care should be taken when evaluating roots of negative integers. For instance, if we were to try to find \(\sqrt{-9}\), then we're essentially asking, "What number, when multiplied by itself, equals \(-9\)?"

Since the product of any two real numbers is always, positive, we'll never find one that, when squared, equals \(-9\). Square roots of negative numbers lead to the realm of imaginary and complex numbers, which are not addressed in this course.

While square roots of negative numbers lead to imaginary numbers, cube roots do not pose the same challenge.

Consider, as an example, \(\sqrt[3]{-8}\). Here we are asking "What number, when multiplied by itself three times, equals -8?" In this case, there is a real number solution because \((-2) \cdot (-2) \cdot (-2) = -8\). Unlike square roots, cube roots of negative numbers do not require the introduction of imaginary numbers or complex numbers.

Conclusion

In this section we've covered basic operations with integers. In the coming chapters, we'll use those operations to form and simplify expressions and solve equations. But first, we'll turn our attention to operations with fractions.