Properties of Integers

Integers are a fundamental part of mathematics, useful in counting, ordering, and basic arithmetic. Integers can be positive, negative, or zero. Understanding the properties of integers can help solve a variety of mathematical problems. In this lesson, we will discuss the essential properties of integers in depth, illustrating each with examples and visual aids.

Understanding integers

Before diving into the properties, let's briefly understand what integers are. Integers are a set of numbers that includes zero (0), positive numbers (1, 2, 3,...) and negative numbers (-1, -2, -3,...). They do not have fractional or decimal parts.

Properties of integers

1. Closing property

The closure property states that when you perform any operation (such as addition, subtraction, or multiplication) on any two integers, the result will always be an integer.

Addition: The sum of two integers will always be an integer.

For example, if you add 
3 + 5 = 8
-4 + (-6) = -10

Subtraction: The difference between two integers is also an integer.

Examples include:
5 - 3 = 2
-8 - (-3) = -5

Multiplication: The product of two integers is an integer.

For example:
4 * (-3) = -12
(-6) * (-2) = 12

Note: Division of integers sometimes does not give an integer (for example, 1 / 2 = 0.5, which is not an integer). Therefore, the closure property does not apply to division.

2. Exchangeable assets

The commutative property relates to addition and multiplication, showing that the order of the numbers does not change the result.

Addition: a + b = b + a

Example:
5 + 3 = 3 + 5
=> 8 = 8

Multiplication: a * b = b * a

Example: 
4 * (-2) = (-2) * 4
=> -8 = -8

The commutative property does not apply to subtraction and division:

5 - 3 ≠ 3 - 5
9 ÷ 3 ≠ 3 ÷ 9

3. Associative property

The associative property of integers also applies to addition and multiplication, that is, the way numbers are grouped does not change their sum or product.

Sum: (a + b) + c = a + (b + c)

Example:
(2 + 3) + 4 = 2 + (3 + 4)
=> 5 + 4 = 2 + 7
=> 9 = 9

Multiplication: (a * b) * c = a * (b * c)

Example: 
(5 * 2) * 3 = 5 * (2 * 3)
=> 10 * 3 = 5 * 6
=> 30 = 30

The associative property does not apply to subtraction and division as can be seen in the examples below:

(6 – 4) – 2 ≠ 6 – (4 – 2)
(12 ÷ 2) ÷ 2 ≠ 12 ÷ (2 ÷ 2)

4. Identity property

The identity property of integers describes numbers that do not change the value of another number when used in an operation.

Addition (Additive Identity): The number 0 is the additive identity because any integer remains unchanged when zero is added to it.

Example: 
7 + 0 = 7
-9 + 0 = -9

Multiplication (Multiplicative Identity): The number 1 is the multiplicative identity because any integer remains unchanged when multiplied by one.

Example:
8 * 1 = 8
-3 * 1 = -3

5. Distributive property

The distributive property connects addition and multiplication, and tells us how to multiply a sum by multiplying each sum separately and then adding the products.

a * (b + c) = a * b + a * c

Example:
2 * (3 + 4) = 2*3 + 2*4
=> 2 * 7 = 6 + 8
=> 14 = 14

Visual representation of properties

Let us illustrate the commutative property through a visual:

Here is an illustration showing distributed multiplication over addition:

Illustrative examples and exercises

Example problems

Let us further our understanding through example problems:

1. Example 1 - Using the Closure Property:

   If a = 7 and b = -3, what is a + b? Is the result an integer?

   a + b = 7 + (-3) = 4
   Since 4 is an integer, the closure property is true.
           

2. Example 2 – Use of Commutative Property:

   Verify: 5 + (-3) = -3 + 5

   5 + (-3) = 2
   -3 + 5 = 2
   Both expressions are equivalent, which confirms the commutative property.
           

3. Example 3 - Using Associative Property:

   Calculate and verify: (-1 + 4) + 2 = -1 + (4 + 2)

   (-1 + 4) + 2 = 3 + 2 = 5
   -1 + (4 + 2) = -1 + 6 = 5
   Both calculations give the result 5, which confirms the associative property.
           

4. Example 4 - Use of the Identity Property:

   Show that the result of 10 + 0 and -5 * 1 is the same number.

   10 + 0 = 10
   -5 * 1 = -5
   It shows both additive and multiplicative identity properties.
           

5. Example 5 - Using the Distributive Property:

   Verify: 3 * (2 + 4) = 3*2 + 3*4

   3 * (2 + 4) = 3 * 6 = 18
   3*2 + 3*4 = 6 + 12 = 18
   Both sides are equal, which confirms the distributive property.
           

Exercises for practice

• Use the Closure Property to verify that 8 - 5 is an integer.
• Demonstrate the commutative property with -4 + 10 and 10 + (-4).
• Use the associative property to solve: (-6 + 2) + 5 and -6 + (2 + 5).
• Apply the identity property to show that adding 0 to any number does not change it.
• Use the Distributive Property to Simplify 4 * (5 + 3)

Concluding remarks

The properties of integers, such as closure, commutative, associative, identity, and distributive, form the basis of arithmetic operations. These properties not only make calculations easier but also help in proving complex mathematical theorems. A thorough understanding of these properties gives students the necessary tools to handle numbers confidently and provides a solid foundation for more advanced mathematics.

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