Facts about integers
Integers can be generated from the set of counting numbers and the subtraction operation. For example, when you subtract a larger natural number from a smaller one, you have a negative number. When a natural number is also subtracted from itself, you have zero.
The result of adding, subtracting, or multiplying integers is always an integer. This cannot be true with dividing integers. Dividing 5 by 2 will give you 2.5, which isn't an integer.
Positive integers are known as natural numbers. An important characteristic of natural numbers can be seen in the equation a + x = b. This only has a solution if b> a, as a and x can only be positive and their addition will produce a larger number. In the realm of integers, the equation a + x = b will always have an answer.
A good way to represent integers on a number line is shown in the figure below.
An image of an integer number line
A set of integers is denoted by Z, which is written as Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}. Z here has a property that shows that it has an infinite number of elements (...- 4, -3, -2, -1, 0,) in figure 1 in the set.
Real-life example of integers
Integers help to capture values in every field.
Consecutive integers
Consecutive integers are integer numbers that follow each other in a sequence without gaps. They represent an unbroken sequence of numbers where one follows the other by the addition of one. If we had x as an integer, then x + 1 and x + 2 will be the two consecutive integers. These numbers are in ascending order, and some examples are:
-5, -4, -3, -2, -1, 0, 1, 2
200, 201, 202, 203, 204, 205
-1, 0, 1, 2, 3, 4, 5
-13, -12, -11, -10, -9, -8, -7
Assuming you had to solve a mathematical equation and you know the sum of two consecutive integers is 97. What are the two integers?
Answer:
Let's assume that the first integer is x. We know from the description of a consecutive integer that the second must be x + 1. We can write an equation for this.
\(x + (x + 1) = 97 \rightarrow 2x + 1 = 97 \rightarrow 2x = 97 - 1 \rightarrow x = 48\)
This means the first integer is 48. And the second will be 48 + 1, which is 49.
Odd consecutive integers
These are odd integers that follow each other yet differ by two. When x is an odd integer, then consecutive odd integers are x + 2, x + 4, x + 6. Examples are:
{5, 7, 9, 11, 13...}
{-7, -5, -3, -1, 1..}
Even consecutive integers
These are also even integers that follow each other yet differ by two. When x is an even integer, then consecutive even integers are x + 2, x + 4, x + 6. Examples are:
{2, 4, 6, 8, 10, 12..}
{-10, -8, 6, 4..}
Integer rules for mathematical operations
It's useful to learn the rules for integers in mathematical operations.
Adding and subtracting integers
Let's take a few examples to get familiar with these operations.
Sam owes his friend Frank $5. He goes to borrow an additional $3, how much will he owe in all?.
Answer:
This is quite simple. We add both and know he owes $8.
However, this can be expressed mathematically as - $5 + (- $3) = - $8. This can in turn be written as: $5 - $3 = - $8
This can be confusing – using a number line makes it much easier.
An image of a number line expressing integer additions
Number line expressing integer additionsUsing your first figure as a reference point, move three steps back on the integer number line. Whilst positive values move right (forward), negative ones move left (backward). And with our example, we have -8 as our answer again.
Let's say Sam eventually pays back $4 out of the $8 he owes. How much is left to pay?
Answer:
This is another simple calculation. Intuitively, we know that the answer is $4.
However, we can write this mathematically as - $ 8 + $ 4 = - $ 4, as well as draw a number line again.

Using your first figure as a reference, move four steps forward on the integer number line. This shows that -4 is our answer.
You might be presented with an equation like \(-3 - (-6) = x\).
Answer:
When two negative signs meet as they do in this equation, they both become positive.
So we can have \(-3 + 6 = x \rightarrow x = 3\)
Multiplying and dividing integers
Let's look at examples that prove the rule of multiplication.
What is the product of -3 and 7?
\(-3 \cdot 7 = -21\)
Remember – the product of a positive and a negative integer will be a negative one.
What is the product of 5 and 4?
\(5 \cdot 4 = 20\)
As we mentioned the product of two positive integers, will be a positive one, in this case 20.
What is the product of -6 and -8?
\(-6 \cdot -8 = 48\)
Divide \(\frac{16}{8} = 2\)
Remember, dividing two positive integers will give you a positive integer.
Divide \(\frac{-28}{-4} = 7\)
Integers - Key takeaways
- Integers are whole numbers that are either positive, zero, or negative.
- The result of adding, subtracting, or multiplying integers is always an integer.
- Consecutive integers are integer numbers that follow each other in a sequence or in order without gaps.
- A set of integers is denoted by Z.
- You cannot always have an integer when two integers are divided.