Integers

An integer is simply a positive or negative whole number.

These are all examples of integers:

These are not integers:

When you add, subtract or multiply two integers you get another integer.

When you divide one integer by another you do not necessarily get an integer.

• $32/8=4$ (integer)
• $8/5=1.6$ (not)

Most people are pretty good at adding, subtracting, and multiplying positive integers. Division gets a little trickier, but what is even more difficult for many is dealing with negative integers \( (-8+2=?) \). Let’s go into this with more detail.

Addition and subtraction with negative numbers

It can be very helpful to imagine a number line when familiarizing yourself with negative integer operations. You likely know that adding a positive integer to a number leads to a rightward movement on the number line, and subtracting a positive integer will move your total to the left. When you add a negative number you move left, and when you subtract a negative number you move right. Adding a negative number is the same as subtraction:

Subtracting a negative number is like addition:

When you subtract from or add to a negative number you incrementally move along the number line in the same way. Here are some examples:

• \( -4+-3=-4-3=-7 \)
• \( -4- -3=-4+3=-1 \)
• \( -8+12=4 \)
• \( -3-2=-5 \)

Multiplication and division of negative integers

Here’s an easy trick for multiplying or dividing with negative integers. First, multiply or divide as you would if both were positive (ignore the signs). Now you must decide whether the result should be positive or negative. To do this, follow this simple rule: if the signs of your original two numbers matched then the result will be positive, if the signs did not match then your result will be negative.

• \( +, + = + \)
• \( +, - = - \)
• \( -, + = - \)
• \( -, - = + \)

And now here are some examples with actual numbers:

• \( 24/-8=-3 \)
• \( -24/-8=3 \)
• \( -24/8=-3 \)
• \( 3\cdot-10=-30 \)
• \( -5\cdot-7=35 \)

Understanding all these integer concepts is crucial to developing a solid math foundation. You should understand each how to solve each of the practice questions in this section before proceeding to the next lesson.

Practice questions

1. Which of the following are integers? \( 10, -5, 0.5, \frac{3}{4}, -1.8, 180 \)

Simplify

2. \( 12-5-6 \)
3. \( -2 \cdot 15 \)
4. \( -4 \cdot -3 \)
5. \( -100 - -30 \)
6. \( -12 - 5 \)
7. \( 100 / -4 \)
8. \( -30/-6 \)

Solutions

1. These are integers: \( 10, -5, 180 \)
2. \( 12-5-6=1 \)
3. \( -2 \cdot 15 = -30 \)
4. \( -4 \cdot -3 \)
5. \( -100 - -30 = -100 + 30 = -70 \)
6. \( -12 - 5 = -17 \)
7. \( 100 / -4 = -25\)
8. \( -30 / -6 = 5 \)