In mathematics, the associative property means that when three or more numbers are added or multiplied, the grouping of numbers (without changing their order) does not change the result.

Formally, for any numbers a, b, and c, the associative property is defined as follows:

- Addition: (a + b) + c = a + (b + c)
- Multiplication: (a * b) * c = a * (b * c)

The associative property applies broadly to many types of numbers including natural numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers. The term “associative property” likely was coined around 1844 by Irish mathematician and scientist William Rowan Hamilton in a discussion regarding octonions, which are non-associative.

### The Associative Property in Addition and Multiplication

Consider the following examples of the associative property:

- For addition: If we take three numbers, say, 2, 3, and 5, the associative property says that (2 + 3) + 5 = 2 + (3 + 5). In the first instance, we get 5 + 5 = 10 and in the second, we get 2 + 8 = 10. Hence, the grouping does not affect the final outcome.
- For multiplication: Take 2, 3, and 4 as our numbers. So, (2 * 3) * 4 = 2 * (3 * 4). In the first instance, we have 6 * 4 = 24 and in the second, we have 2 * 12 = 24. Again, the result does not depend on the grouping.

### Limitations of the Associative Property

While the associative property works for addition and multiplication, it doesn’t apply to subtraction and division.

To illustrate this, let’s use the numbers 10, 5, and 2:

- For subtraction: If we group the first two numbers, we get (10 – 5) – 2 = 5 – 2 = 3. But, if we group the last two numbers, we get 10 – (5 – 2) = 10 – 3 = 7. The results differ, showing that subtraction is not associative.
- For division: Similarly, if we group the first two numbers, we get (10 ÷ 5) ÷ 2 = 2 ÷ 2 = 1. Grouping the last two gives 10 ÷ (5 ÷ 2) = 10 ÷ 2.5 = 4. The results are different, proving division is also not associative.

### Comparison with the Commutative Property

Both the associative and commutative properties are essential to the understanding of basic arithmetic operations, but they serve different functions. While the associative property concerns the grouping of numbers in an operation, the commutative property pertains to the order of numbers.

The commutative property states that for addition and multiplication, the order of numbers does not change the result. Formally, for any numbers a and b, the commutative property is defined as follows:

- Addition: a + b = b + a
- Multiplication: a * b = b * a

Similar to the associative property, the commutative property does not apply to subtraction and division.

### Operations Closed under Both Properties

The operations that are closed under both the associative and commutative properties are addition and multiplication. This means for these operations, changing the grouping of the numbers (associative property) or the order of the numbers (commutative property) does not affect the result.

There is no standard binary operation that is closed under one of these properties but not the other, as these two properties tend to be fundamental to the same types of operations.

### Associative Property Examples

The associative property holds for various types of numbers, including natural (counting) numbers, whole numbers, integers, real numbers, complex numbers, and irrational numbers.

#### Whole Number Examples

For example, consider the whole numbers 2, 0, and 4:

- For addition: (2 + 0) + 4 = 2 + (0 + 4) = 6
- For multiplication: (2 x 0) x 4 = 2 x (0 x 4) = 0

#### Integer Examples

Let’s use the integers -2, 3, and 5:

- For addition: We have (-2 + 3) + 5 and -2 + (3 + 5). The first expression gives us 1 + 5 = 6, and the second expression gives us -2 + 8 = 6. The results are equal, verifying the associative property for addition with integers.
- For multiplication: We have (-2 * 3) * 5 and -2 * (3 * 5). The first expression gives us -6 * 5 = -30, and the second expression gives us -2 * 15 = -30. Again, the results are equal, showing that multiplication with integers is associative.

#### Real Number Examples

Let’s consider the real numbers -2.5, 3.4, and 5.1:

- For addition: We have (-2.5 + 3.4) + 5.1 and -2.5 + (3.4 + 5.1). The first expression gives us 0.9 + 5.1 = 6.0, and the second expression gives us -2.5 + 8.5 = 6.0. The results are equal, confirming the associative property for addition with real numbers.
- For multiplication: We have (-2.5 * 3.4) * 5.1 and -2.5 * (3.4 * 5.1). The first expression gives us -8.5 * 5.1 = -43.35, and the second expression gives us -2.5 * 17.34 = -43.35. Yet again, the results are equal, verifying that multiplication with real numbers is associative.

#### Complex Number Examples

Consider the complex numbers 2 + 3i, -1 + i, and 4 – 2i:

- For addition: We have [(2 + 3i) + (-1 + i)] + (4 – 2i) and (2 + 3i) + [(-1 + i) + (4 – 2i)]. Simplifying the first expression, we get (1 + 4i) + (4 – 2i) = 5 + 2i. For the second expression, we get (2 + 3i) + (3 – i) = 5 + 2i. Both results are the same, demonstrating the associative property for addition with complex numbers.
- For multiplication: We have [(2 + 3i) * (-1 + i)] * (4 – 2i) and (2 + 3i) * [(-1 + i) * (4 – 2i)]. Simplifying the first expression, we get (-5 – i) * (4 – 2i) = -18 – 12i. For the second expression, we get (2 + 3i) * (-4 – 6i) = -18 – 12i. Both results are the same, demonstrating the associative property for multiplication with complex numbers.

#### Irrational Number Examples

Irrational numbers are real numbers that cannot be expressed as a ratio of integers. Examples of irrational numbers include square roots of non-perfect squares, cube roots of non-perfect cubes, pi (π), Euler’s number (e), and the golden ratio (φ). The associative property also applies to irrational numbers.

Consider the irrational numbers √2, √3, and π:

- For addition: We have [(√2 + √3) + π] and [√2 + (√3 + π)]. In either case, the sum is √2 + √3 + π. The order of the addition doesn’t matter because of the associative property.
- For multiplication: Similarly, we have [(√2 * √3) * π] and [√2 * (√3 * π)]. In either case, the product is √2 * √3 * π. The order of the multiplication doesn’t matter because of the associative property.

### References

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