Integers are whole numbers that are either positive, zero, or negative, and do not contain decimals.
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 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.
Addition
Adding two positive integers will always give you a positive integer.
Adding two negative integers will always give you a negative integer.
Adding one positive and one negative integer will give you:
Multiplication
The product of a positive integer and a negative integer will always give you a negative integer.
The product of two positive integers will always be a positive-valued integer.
The product of two negative integers will always be a positive integer.
Division
Dividing two positive integers will always give you a positive value.
Dividing two negative integers will give a positive value.
Dividing a negative integer by a positive integer will give you a negative value, and the opposite applies too.
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 additions
Using 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.
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